wpmath0000002_13

Maze_generation_algorithm.html

  1. O ( α ( V ) ) O(\alpha(V))
  2. α ( x ) < 5 \alpha(x)<5
  3. x x

McCarthy_91_function.html

  1. M ( n ) = { n - 10 , if n > 100 M ( M ( n + 11 ) ) , if n 100 M(n)=\left\{\begin{matrix}n-10,&\mbox{if }~{}n>100\mbox{ }\\ M(M(n+11)),&\mbox{if }~{}n\leq 100\mbox{ }\end{matrix}\right.
  2. f ( n ) f(n)
  3. f ( n - 1 ) f(n-1)

Meagre_set.html

  1. Y Y
  2. W W
  3. Y Y
  4. W W
  5. X X
  6. Y Y
  7. X , Y , W X,Y,W
  8. P 1 P_{1}
  9. P 2 P_{2}
  10. W W
  11. W 1 W 2 W 3 W_{1}\supset W_{2}\supset W_{3}\supset\cdots
  12. X X
  13. P 1 P_{1}
  14. P 2 P_{2}
  15. W W
  16. P 2 P_{2}
  17. X X

Mean_free_path.html

  1. = μ - 1 = ( ( μ / ρ ) ρ ) - 1 , \ell=\mu^{-1}=((\mu/\rho)\rho)^{-1},
  2. l = 2 d 3 Φ Q s l=\frac{2d}{3\Phi Q_{s}}
  3. l = 4 V S l=\frac{4V}{S}
  4. = ( σ n ) - 1 \ell=(\sigma n)^{-1}
  5. n n
  6. σ σ
  7. n n
  8. P ( stopping within d x ) = Area atoms Area slab = σ n L 2 d x L 2 = n σ d x P(\mathrm{stopping\ within\ }dx)=\frac{\mathrm{Area_{atoms}}}{\mathrm{Area_{% slab}}}=\frac{\sigma nL^{2}\,dx}{L^{2}}=n\sigma\,dx
  9. σ σ
  10. d I = - I n σ d x dI=-In\sigma dx
  11. d I d x = - I n σ = def - I \frac{dI}{dx}=-In\sigma\ \stackrel{\mathrm{def}}{=}\ -\frac{I}{\ell}
  12. I = I 0 e - x / I=I_{0}e^{-x/\ell}
  13. x x
  14. x x
  15. x + d x x+dx
  16. d P ( x ) = I ( x ) - I ( x + d x ) I 0 = 1 e - x / d x . dP(x)=\frac{I(x)-I(x+dx)}{I_{0}}=\frac{1}{\ell}e^{-x/\ell}dx.
  17. x x
  18. x = def 0 x d P ( x ) = 0 x e - x / d x = \langle x\rangle\ \stackrel{\mathrm{def}}{=}\ \int_{0}^{\infty}xdP(x)=\int_{0}% ^{\infty}\frac{x}{\ell}e^{-x/\ell}\,dx=\ell
  19. T = I I 0 = e - x / T=\frac{I}{I_{0}}=e^{-x/\ell}
  20. x x
  21. x = d x x=dx
  22. = ( n σ ) - 1 , \ell=(n\sigma)^{-1},
  23. = ( 2 n σ ) - 1 . \ell=(\sqrt{2}\,n\sigma)^{-1}.\,
  24. n = N / V = p / ( k B T ) n=N/V=p/(k_{\rm B}T)
  25. σ = π ( 2 r ) 2 = π d 2 \sigma=\pi(2r)^{2}=\pi d^{2}
  26. r r
  27. = k B T 2 π d 2 p \ell=\frac{k_{\rm B}T}{\sqrt{2}\pi d^{2}p}
  28. B {}_{B}
  29. = μ p π k B T 2 m , \ell=\frac{\mu}{p}\sqrt{\frac{\pi k_{\mathrm{B}}T}{2m}},

Mean_squared_error.html

  1. Y ^ \hat{Y}
  2. n n
  3. Y Y
  4. MSE = 1 n i = 1 n ( Y i ^ - Y i ) 2 \operatorname{MSE}=\frac{1}{n}\sum_{i=1}^{n}(\hat{Y_{i}}-Y_{i})^{2}
  5. 1 n i = 1 n \frac{1}{n}\sum_{i=1}^{n}
  6. ( Y i ^ - Y i ) 2 (\hat{Y_{i}}-Y_{i})^{2}
  7. θ ^ \hat{\theta}
  8. θ \theta
  9. MSE ( θ ^ ) = E [ ( θ ^ - θ ) 2 ] . \operatorname{MSE}(\hat{\theta})=\operatorname{E}\big[(\hat{\theta}-\theta)^{2% }\big].
  10. θ \theta
  11. MSE ( θ ^ ) = Var ( θ ^ ) + ( Bias ( θ ^ , θ ) ) 2 . \operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+\left(% \operatorname{Bias}(\hat{\theta},\theta)\right)^{2}.
  12. MSE ( θ ^ ) 𝔼 ( ( θ ^ - θ ) 2 ) = 𝔼 [ ( θ ^ - 𝔼 ( θ ^ ) + 𝔼 ( θ ^ ) - θ ) 2 ] = 𝔼 [ ( θ ^ - 𝔼 ( θ ^ ) ) 2 + 2 ( ( θ ^ - 𝔼 ( θ ^ ) ) ( 𝔼 ( θ ^ ) - θ ) ) + ( 𝔼 ( θ ^ ) - θ ) 2 ] = 𝔼 [ ( θ ^ - 𝔼 ( θ ^ ) ) 2 ] + 2 𝔼 [ ( θ ^ - 𝔼 ( θ ^ ) ) ( 𝔼 ( θ ^ ) - θ This is a constant, so it can be pulled out. ) ] + 𝔼 [ ( 𝔼 ( θ ^ ) - θ ) 2 This is a constant, so its expected value is itself. ] = 𝔼 [ ( θ ^ - 𝔼 ( θ ^ ) ) 2 ] + 2 ( 𝔼 ( θ ^ ) - θ That first constant, now pulled out. ) 𝔼 ( θ ^ - 𝔼 ( θ ^ ) ) = 𝔼 ( θ ^ ) - 𝔼 ( θ ^ ) = 0 + ( 𝔼 ( θ ^ ) - θ ) 2 = 𝔼 [ ( θ ^ - 𝔼 ( θ ^ ) ) 2 ] + ( 𝔼 ( θ ^ ) - θ ) 2 = Var ( θ ^ ) + Bias ( θ ^ , θ ) 2 \begin{aligned}\displaystyle\operatorname{MSE}(\hat{\theta})\equiv\mathbb{E}((% \hat{\theta}-\theta)^{2})&\displaystyle=\mathbb{E}\left[\left(\hat{\theta}-% \mathbb{E}(\hat{\theta})+\mathbb{E}(\hat{\theta})-\theta\right)^{2}\right]\\ &\displaystyle=\mathbb{E}\left[\left(\hat{\theta}-\mathbb{E}(\hat{\theta})% \right)^{2}+2\left((\hat{\theta}-\mathbb{E}(\hat{\theta}))(\mathbb{E}(\hat{% \theta})-\theta)\right)+\left(\mathbb{E}(\hat{\theta})-\theta\right)^{2}\right% ]\\ &\displaystyle=\mathbb{E}\left[\left(\hat{\theta}-\mathbb{E}(\hat{\theta})% \right)^{2}\right]+2\mathbb{E}\Big[(\hat{\theta}-\mathbb{E}(\hat{\theta}))(% \overbrace{\mathbb{E}(\hat{\theta})-\theta}^{\begin{smallmatrix}\,\text{This % is}\\ \,\text{a constant,}\\ \,\text{so it can be}\\ \,\text{pulled out.}\end{smallmatrix}})\,\Big]+\mathbb{E}\Big[\,\overbrace{% \left(\mathbb{E}(\hat{\theta})-\theta\right)^{2}}^{\begin{smallmatrix}\,\text{% This is a}\\ \,\text{constant, so its}\\ \,\text{expected value}\\ \,\text{is itself.}\end{smallmatrix}}\,\Big]\\ &\displaystyle=\mathbb{E}\left[\left(\hat{\theta}-\mathbb{E}(\hat{\theta})% \right)^{2}\right]+2(\overbrace{\mathbb{E}(\hat{\theta})-\theta}^{\begin{% smallmatrix}\,\text{That first}\\ \,\text{constant, now}\\ \,\text{pulled out.}\end{smallmatrix}})\underbrace{\mathbb{E}(\hat{\theta}-% \mathbb{E}(\hat{\theta}))}_{=\mathbb{E}(\hat{\theta})-\mathbb{E}(\hat{\theta})% =0}+\left(\mathbb{E}(\hat{\theta})-\theta\right)^{2}\\ &\displaystyle=\mathbb{E}\left[\left(\hat{\theta}-\mathbb{E}(\hat{\theta})% \right)^{2}\right]+\left(\mathbb{E}(\hat{\theta})-\theta\right)^{2}\\ &\displaystyle=\operatorname{Var}(\hat{\theta})+\operatorname{Bias}(\hat{% \theta},\theta)^{2}\end{aligned}
  13. X 1 , , X n X_{1},\dots,X_{n}
  14. X ¯ = 1 n i = 1 n X i \overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}
  15. MSE ( X ¯ ) = E ( ( X ¯ - μ ) 2 ) = ( σ n ) 2 = σ 2 n \operatorname{MSE}(\overline{X})=\operatorname{E}((\overline{X}-\mu)^{2})=% \left(\frac{\sigma}{\sqrt{n}}\right)^{2}=\frac{\sigma^{2}}{n}
  16. σ 2 \sigma^{2}
  17. S n - 1 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 = 1 n - 1 ( i = 1 n X i 2 - n X ¯ 2 ) . S^{2}_{n-1}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\,\right)^{2}=% \frac{1}{n-1}\left(\sum_{i=1}^{n}X_{i}^{2}-n\overline{X}^{2}\right).
  18. σ 2 \sigma^{2}
  19. MSE ( S n - 1 2 ) = 1 n ( μ 4 - n - 3 n - 1 σ 4 ) = 1 n ( γ 2 + 2 n n - 1 ) σ 4 , \begin{aligned}\displaystyle\operatorname{MSE}(S^{2}_{n-1})&\displaystyle=% \frac{1}{n}\left(\mu_{4}-\frac{n-3}{n-1}\sigma^{4}\right)\\ &\displaystyle=\frac{1}{n}\left(\gamma_{2}+\frac{2n}{n-1}\right)\sigma^{4},% \end{aligned}
  20. μ 4 \mu_{4}
  21. γ 2 = μ 4 / σ 4 - 3 \gamma_{2}=\mu_{4}/\sigma^{4}-3
  22. σ 2 \sigma^{2}
  23. S n - 1 2 S^{2}_{n-1}
  24. S a 2 = n - 1 a S n - 1 2 = 1 a i = 1 n ( X i - X ¯ ) 2 \begin{aligned}\displaystyle S^{2}_{a}&\displaystyle=\frac{n-1}{a}S^{2}_{n-1}% \\ &\displaystyle=\frac{1}{a}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\,\right)^{2}% \end{aligned}
  25. MSE ( S a 2 ) = E ( ( n - 1 a S n - 1 2 - σ 2 ) 2 ) = n - 1 n a 2 [ ( n - 1 ) γ 2 + n 2 + n ] σ 4 - 2 ( n - 1 ) a σ 4 + σ 4 \begin{aligned}\displaystyle\operatorname{MSE}(S^{2}_{a})&\displaystyle=% \operatorname{E}\left(\left(\frac{n-1}{a}S^{2}_{n-1}-\sigma^{2}\right)^{2}% \right)\\ &\displaystyle=\frac{n-1}{na^{2}}[(n-1)\gamma_{2}+n^{2}+n]\sigma^{4}-\frac{2(n% -1)}{a}\sigma^{4}+\sigma^{4}\end{aligned}
  26. a = ( n - 1 ) γ 2 + n 2 + n n = n + 1 + n - 1 n γ 2 . a=\frac{(n-1)\gamma_{2}+n^{2}+n}{n}=n+1+\frac{n-1}{n}\gamma_{2}.
  27. γ 2 = 0 \gamma_{2}=0
  28. a = n + 1 a=n+1
  29. γ 2 = - 2 \gamma_{2}=-2
  30. a = n - 1 + 2 / n a=n-1+2/n
  31. S n - 1 2 . S^{2}_{n-1}.
  32. θ ^ \hat{\theta}
  33. X ¯ = 1 n i = 1 n ( X i ) \overline{X}=\frac{1}{n}\sum_{i=1}^{n}(X_{i})
  34. MSE ( X ¯ ) = E ( ( X ¯ - μ ) 2 ) = ( σ n ) 2 \operatorname{MSE}(\overline{X})=\operatorname{E}((\overline{X}-\mu)^{2})=% \left(\frac{\sigma}{\sqrt{n}}\right)^{2}
  35. θ ^ \hat{\theta}
  36. S n - 1 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 S^{2}_{n-1}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\,\right)^{2}
  37. MSE ( S n - 1 2 ) = E ( ( S n - 1 2 - σ 2 ) 2 ) = 2 n - 1 σ 4 \operatorname{MSE}(S^{2}_{n-1})=\operatorname{E}((S^{2}_{n-1}-\sigma^{2})^{2})% =\frac{2}{n-1}\sigma^{4}
  38. θ ^ \hat{\theta}
  39. S n 2 = 1 n i = 1 n ( X i - X ¯ ) 2 S^{2}_{n}=\frac{1}{n}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\,\right)^{2}
  40. MSE ( S n 2 ) = E ( ( S n 2 - σ 2 ) 2 ) = 2 ( n - 1 ) n 2 σ 4 \operatorname{MSE}(S^{2}_{n})=\operatorname{E}((S^{2}_{n}-\sigma^{2})^{2})=% \frac{2(n-1)}{n^{2}}\sigma^{4}
  41. θ ^ \hat{\theta}
  42. S n + 1 2 = 1 n + 1 i = 1 n ( X i - X ¯ ) 2 S^{2}_{n+1}=\frac{1}{n+1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\,\right)^{2}
  43. MSE ( S n + 1 2 ) = E ( ( S n + 1 2 - σ 2 ) 2 ) = 2 n + 1 σ 4 \operatorname{MSE}(S^{2}_{n+1})=\operatorname{E}((S^{2}_{n+1}-\sigma^{2})^{2})% =\frac{2}{n+1}\sigma^{4}
  44. X i N ( μ , σ 2 ) X_{i}\sim\operatorname{N}(\mu,\sigma^{2})
  45. ( n - 1 ) S n - 1 2 σ 2 χ n - 1 2 \frac{(n-1)S^{2}_{n-1}}{\sigma^{2}}\sim\chi^{2}_{n-1}
  46. S n - 1 2 S^{2}_{n-1}
  47. χ n - 1 2 \chi^{2}_{n-1}
  48. 2 n - 2 2n-2
  49. S n - 1 2 S^{2}_{n-1}
  50. S n + 1 2 S^{2}_{n+1}
  51. S n 2 S^{2}_{n}
  52. S n + 1 2 S^{2}_{n+1}
  53. 2 n σ 2 \frac{2}{n}\sigma^{2}
  54. θ ^ \hat{\theta}
  55. θ \theta

Measurable_cardinal.html

  1. 𝔠 {\mathfrak{c}}
  2. 𝔠 {\mathfrak{c}}
  3. 𝔠 {\mathfrak{c}}

Mechanical_puzzle.html

  1. ( A + B + C ) / 4 (A+B+C)/4

Mechanical_ventilation.html

  1. P T A = ( P A O ) - ( P A L V ) P_{TA}=(P_{AO})-(P_{ALV})
  2. V D m e c h = V T - V D p h y s - P a C O 2 ( V T - V D - V D m e c h ) P A C O 2 V_{Dmech}=V_{T}-V_{Dphys}-\frac{PaCO2(V_{T}-V_{D}-V_{Dmech})}{P_{ACO_{2}}}
  3. V D V T = P a C O 2 - P E ¯ C O 2 P a C O 2 \frac{V_{D}}{V_{T}}=\frac{PaCO_{2}-P\bar{E}CO_{2}}{PaCO_{2}}
  4. V ˙ A = ( V T - V D S p h y s ) × f \dot{V}_{A}=\ (V_{T}-V_{DSphys})\times f
  5. P a C O 2 = 0.863 × V ˙ C O 2 V ˙ A PaCO_{2}=\frac{0.863\times\dot{V}_{CO_{2}}}{\dot{V}_{A}}
  6. V A = V T - V f V_{A}=V_{T}-V_{f}
  7. Q S P Q T = C c O 2 - C a O 2 5 + ( C c O 2 - C a O 2 ) \frac{Q_{SP}}{Q_{T}}=\frac{CcO_{2}-CaO_{2}}{5+(CcO_{2}-CaO_{2})}

Medical_ultrasound.html

  1. 0.5 dB cm depth MHz \textstyle 0.5\frac{\mbox{dB}~{}}{\mbox{cm depth}~{}\cdot\mbox{MHz}~{}}

Menger_sponge.html

  1. l o g 20 l o g 3 \frac{log 20}{log 3}
  2. M := n M n M:=\bigcap_{n\in\mathbb{N}}M_{n}
  3. M n + 1 := { ( x , y , z ) 3 : i , j , k { 0 , 1 , 2 } : ( 3 x - i , 3 y - j , 3 z - k ) M n and at most one of i , j , k is equal to 1 } . M_{n+1}:=\left\{\begin{matrix}(x,y,z)\in\mathbb{R}^{3}:&\begin{matrix}\exists i% ,j,k\in\{0,1,2\}:(3x-i,3y-j,3z-k)\in M_{n}\\ \mbox{and at most one of }~{}i,j,k\mbox{ is equal to 1}\end{matrix}\end{matrix% }\right\}.

Mercer's_theorem.html

  1. K : [ a , b ] × [ a , b ] K:[a,b]\times[a,b]\rightarrow\mathbb{R}
  2. i = 1 n j = 1 n K ( x i , x j ) c i c j 0 \sum_{i=1}^{n}\sum_{j=1}^{n}K(x_{i},x_{j})c_{i}c_{j}\geq 0
  3. [ T K φ ] ( x ) = a b K ( x , s ) φ ( s ) d s . [T_{K}\varphi](x)=\int_{a}^{b}K(x,s)\varphi(s)\,ds.
  4. K ( s , t ) = j = 1 λ j e j ( s ) e j ( t ) K(s,t)=\sum_{j=1}^{\infty}\lambda_{j}\,e_{j}(s)\,e_{j}(t)
  5. λ i e i ( t ) = [ T K e i ] ( t ) = a b K ( t , s ) e i ( s ) d s . \lambda_{i}e_{i}(t)=[T_{K}e_{i}](t)=\int_{a}^{b}K(t,s)e_{i}(s)\,ds.
  6. i = 1 λ i | e i ( t ) e i ( s ) | sup x [ a , b ] | K ( x , x ) | 2 , \sum_{i=1}^{\infty}\lambda_{i}|e_{i}(t)e_{i}(s)|\leq\sup_{x\in[a,b]}|K(x,x)|^{% 2},
  7. i = 1 λ i e i ( t ) e i ( s ) \sum_{i=1}^{\infty}\lambda_{i}e_{i}(t)e_{i}(s)
  8. a b K ( t , t ) d t = i λ i . \int_{a}^{b}K(t,t)\,dt=\sum_{i}\lambda_{i}.
  9. trace ( T K ) = a b K ( t , t ) d t . \operatorname{trace}(T_{K})=\int_{a}^{b}K(t,t)\,dt.
  10. K ( s , t ) = j = 1 λ j e j ( s ) e j ( t ) K(s,t)=\sum_{j=1}^{\infty}\lambda_{j}\,e_{j}(s)\,e_{j}(t)
  11. K L μ μ 2 ( X × X ) . K\in L^{2}_{\mu\otimes\mu}(X\times X).
  12. T K φ , ψ = X × X K ( y , x ) φ ( y ) ψ ( x ) d [ μ μ ] ( y , x ) . \langle T_{K}\varphi,\psi\rangle=\int_{X\times X}K(y,x)\varphi(y)\psi(x)\,d[% \mu\otimes\mu](y,x).
  13. K ( y , x ) = i λ i e i ( y ) e i ( x ) K(y,x)=\sum_{i\in\mathbb{N}}\lambda_{i}e_{i}(y)e_{i}(x)

Meromorphic_function.html

  1. f ( z ) = 1 sin z . f(z)=\frac{1}{\sin z}.
  2. f / g f/g
  3. g ( z ) = 0 g(z)=0
  4. f ( z 1 , z 2 ) = z 1 / z 2 f(z_{1},z_{2})=z_{1}/z_{2}
  5. ( 0 , 0 ) (0,0)
  6. f ( z ) = z 3 - 2 z + 10 z 5 + 3 z - 1 , f(z)=\frac{z^{3}-2z+10}{z^{5}+3z-1},
  7. f ( z ) = e z z and f ( z ) = sin z ( z - 1 ) 2 f(z)=\frac{e^{z}}{z}\,\text{ and }f(z)=\frac{\sin{z}}{(z-1)^{2}}
  8. f ( z ) = e 1 z f(z)=e^{\frac{1}{z}}
  9. { 0 } \mathbb{C}\setminus\{0\}
  10. f ( z ) = ln ( z ) f(z)=\ln(z)
  11. f ( z ) = 1 sin ( 1 z ) f(z)=\frac{1}{\sin\left(\frac{1}{z}\right)}
  12. z = 0 z=0
  13. f ( z ) = sin 1 z f(z)=\sin\frac{1}{z}

Method_of_Fluxions.html

  1. x ˙ \dot{x}

Metric_tensor.html

  1. r ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) \vec{r}(u,v)=(x(u,v),y(u,v),z(u,v))
  2. r ( u ( t ) , v ( t ) ) \scriptstyle{\vec{r}(u(t),v(t))}
  3. s \displaystyle s
  4. \left\|\cdot\right\|
  5. r u = r u \scriptstyle\vec{r}_{u}=\tfrac{\partial\vec{r}}{\partial u}
  6. r v = r v \scriptstyle\vec{r}_{v}=\tfrac{\partial\vec{r}}{\partial v}
  7. r ( u , v ) \scriptstyle{\vec{r}(u,v)}
  8. d s 2 = [ d u d v ] [ E F F G ] [ d u d v ] \begin{aligned}\displaystyle ds^{2}&\displaystyle=\begin{bmatrix}du&dv\end{% bmatrix}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}du\\ dv\end{bmatrix}\\ \end{aligned}
  9. E = r u r u , F = r u r v , G = r v r v . E^{\prime}=\vec{r}_{u^{\prime}}\cdot\vec{r}_{u^{\prime}},\quad F^{\prime}=\vec% {r}_{u^{\prime}}\cdot\vec{r}_{v^{\prime}},\quad G^{\prime}=\vec{r}_{v^{\prime}% }\cdot\vec{r}_{v^{\prime}}.
  10. [ E F F G ] = [ u u u v v u v v ] T [ E F F G ] [ u u u v v u v v ] \begin{bmatrix}E^{\prime}&F^{\prime}\\ F^{\prime}&G^{\prime}\end{bmatrix}=\begin{bmatrix}\frac{\partial u}{\partial u% ^{\prime}}&\frac{\partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}^{\mathrm{T}}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{% \partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}
  11. J = [ u u u v v u v v ] . J=\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{% \partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}.
  12. [ E F F G ] \begin{bmatrix}E&F\\ F&G\end{bmatrix}
  13. [ d u d v ] = [ u u u v v u v v ] [ d u d v ] \begin{bmatrix}du\\ dv\end{bmatrix}=\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{% \partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}\begin{bmatrix}du^{\prime}\\ dv^{\prime}\end{bmatrix}
  14. d s 2 = [ d u d v ] [ E F F G ] [ d u d v ] = [ d u d v ] [ u u u v v u v v ] T [ E F F G ] [ u u u v v u v v ] [ d u d v ] = [ d u d v ] [ E F F G ] [ d u d v ] = ( d s ) 2 . \begin{aligned}\displaystyle ds^{2}&\displaystyle=\begin{bmatrix}du&dv\end{% bmatrix}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}du\\ dv\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}du^{\prime}&dv^{\prime}\end{bmatrix}\begin{% bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{\partial v^{% \prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}^{\mathrm{T}}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{% \partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}\begin{bmatrix}du^{\prime}\\ dv^{\prime}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}du^{\prime}&dv^{\prime}\end{bmatrix}\begin{% bmatrix}E^{\prime}&F^{\prime}\\ F^{\prime}&G^{\prime}\end{bmatrix}\begin{bmatrix}du^{\prime}\\ dv^{\prime}\end{bmatrix}\\ &\displaystyle=(ds^{\prime})^{2}.\end{aligned}
  15. 𝐩 = p 1 r u + p 2 r v \mathbf{p}=p_{1}\vec{r}_{u}+p_{2}\vec{r}_{v}
  16. 𝐚 = a 1 r u + a 2 r v \mathbf{a}=a_{1}\vec{r}_{u}+a_{2}\vec{r}_{v}
  17. 𝐛 = b 1 r u + b 2 r v \mathbf{b}=b_{1}\vec{r}_{u}+b_{2}\vec{r}_{v}
  18. 𝐚 𝐛 = a 1 b 1 r u r u + a 1 b 2 r u r v + b 1 a 2 r v r u + a 2 b 2 r v r v = a 1 b 1 E + a 1 b 2 F + b 1 a 2 F + a 2 b 2 G = [ a 1 a 2 ] [ E F F G ] [ b 1 b 2 ] . \begin{aligned}\displaystyle\mathbf{a}\cdot\mathbf{b}&\displaystyle=a_{1}b_{1}% \vec{r}_{u}\cdot\vec{r}_{u}+a_{1}b_{2}\vec{r}_{u}\cdot\vec{r}_{v}+b_{1}a_{2}% \vec{r}_{v}\cdot\vec{r}_{u}+a_{2}b_{2}\vec{r}_{v}\cdot\vec{r}_{v}\\ &\displaystyle=a_{1}b_{1}E+a_{1}b_{2}F+b_{1}a_{2}F+a_{2}b_{2}G\\ &\displaystyle=\begin{bmatrix}a_{1}&a_{2}\end{bmatrix}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}\end{aligned}.
  19. g ( 𝐚 , 𝐛 ) = a 1 b 1 E + a 1 b 2 F + b 1 a 2 F + a 2 b 2 G . g(\mathbf{a},\mathbf{b})=a_{1}b_{1}E+a_{1}b_{2}F+b_{1}a_{2}F+a_{2}b_{2}G.
  20. g ( 𝐚 , 𝐛 ) = g ( 𝐛 , 𝐚 ) . g(\mathbf{a},\mathbf{b})=g(\mathbf{b},\mathbf{a}).
  21. g ( λ 𝐚 + μ 𝐚 , 𝐛 ) = λ g ( 𝐚 , 𝐛 ) + μ g ( 𝐚 , 𝐛 ) , and g(\lambda\mathbf{a}+\mu\mathbf{a^{\prime}},\mathbf{b})=\lambda g(\mathbf{a},% \mathbf{b})+\mu g(\mathbf{a^{\prime}},\mathbf{b}),\quad\,\text{and}
  22. g ( 𝐚 , λ 𝐛 + μ 𝐛 ) = λ g ( 𝐚 , 𝐛 ) + μ g ( 𝐚 , 𝐛 ) g(\mathbf{a},\lambda\mathbf{b}+\mu\mathbf{b^{\prime}})=\lambda g(\mathbf{a},% \mathbf{b})+\mu g(\mathbf{a},\mathbf{b^{\prime}})
  23. 𝐚 = g ( 𝐚 , 𝐚 ) \|\mathbf{a}\|=\sqrt{g(\mathbf{a},\mathbf{a})}
  24. cos θ = g ( 𝐚 , 𝐛 ) 𝐚 𝐛 . \cos\theta=\frac{g(\mathbf{a},\mathbf{b})}{\|\mathbf{a}\|\,\|\mathbf{b}\|}.
  25. r ( u , v ) \vec{r}(u,v)
  26. D | r u × r v | d u d v \iint_{D}\left|\vec{r}_{u}\times\vec{r}_{v}\right|\,du\,dv
  27. D ( r u r u ) ( r v r v ) - ( r u r v ) 2 d u d v = D E G - F 2 d u d v = D det [ E F F G ] d u d v \begin{aligned}\displaystyle\iint_{D}&\displaystyle\sqrt{(\vec{r}_{u}\cdot\vec% {r}_{u})(\vec{r}_{v}\cdot\vec{r}_{v})-(\vec{r}_{u}\cdot\vec{r}_{v})^{2}}\,du\,% dv\\ &\displaystyle\quad=\iint_{D}\sqrt{EG-F^{2}}\,du\,dv\\ &\displaystyle\quad=\iint_{D}\sqrt{\operatorname{det}\begin{bmatrix}E&F\\ F&G\end{bmatrix}}\,du\,dv\end{aligned}
  28. g p ( a U p + b V p , Y p ) = a g p ( U p , Y p ) + b g p ( V p , Y p ) , and g_{p}(aU_{p}+bV_{p},Y_{p})=ag_{p}(U_{p},Y_{p})+bg_{p}(V_{p},Y_{p}),\ \ \,\text% {and}
  29. g p ( Y p , a U p + b V p ) = a g p ( Y p , U p ) + b g p ( Y p , V p ) . g_{p}(Y_{p},aU_{p}+bV_{p})=ag_{p}(Y_{p},U_{p})+bg_{p}(Y_{p},V_{p}).\,
  30. g p ( X p , Y p ) = g p ( Y p , X p ) . g_{p}(X_{p},Y_{p})=g_{p}(Y_{p},X_{p}).\,
  31. Y p g p ( X p , Y p ) Y_{p}\mapsto g_{p}(X_{p},Y_{p})
  32. g ( X , Y ) ( p ) = g p ( X p , Y p ) g(X,Y)(p)=g_{p}(X_{p},Y_{p})\,
  33. g i j [ 𝐟 ] = g ( X i , X j ) . g_{ij}[\mathbf{f}]=g\left(X_{i},X_{j}\right).
  34. v = i = 1 n v i X i , w = i = 1 n w i X i v=\sum_{i=1}^{n}v^{i}X_{i},\quad w=\sum_{i=1}^{n}w^{i}X_{i}
  35. g ( v , w ) = i , j = 1 n v i w j g ( X i , X j ) = i , j = 1 n v i w j g i j [ 𝐟 ] g(v,w)=\sum_{i,j=1}^{n}v^{i}w^{j}g\left(X_{i},X_{j}\right)=\sum_{i,j=1}^{n}v^{% i}w^{j}g_{ij}[\mathbf{f}]
  36. g ( v , w ) = 𝐯 [ 𝐟 ] T G [ 𝐟 ] 𝐰 [ 𝐟 ] = 𝐰 [ 𝐟 ] T G [ 𝐟 ] 𝐯 [ 𝐟 ] g(v,w)=\mathbf{v}[\mathbf{f}]^{\mathrm{T}}G[\mathbf{f}]\mathbf{w}[\mathbf{f}]=% \mathbf{w}[\mathbf{f}]^{\mathrm{T}}G[\mathbf{f}]\mathbf{v}[\mathbf{f}]
  37. 𝐟 𝐟 = ( k X k a k 1 , , k X k a k n ) = 𝐟 A \mathbf{f}\mapsto\mathbf{f}^{\prime}=\left(\sum_{k}X_{k}a_{k1},\dots,\sum_{k}X% _{k}a_{kn}\right)=\mathbf{f}A
  38. G [ 𝐟 A ] = A T G [ 𝐟 ] A G[\mathbf{f}A]=A^{\mathrm{T}}G[\mathbf{f}]A
  39. g i j [ 𝐟 A ] = k , = 1 n a k i g k [ 𝐟 ] a j . g_{ij}[\mathbf{f}A]=\sum_{k,\ell=1}^{n}a_{ki}g_{k\ell}[\mathbf{f}]a_{\ell j}.
  40. 𝐟 = ( X 1 = x 1 , , X n = x n ) . \mathbf{f}=\left(X_{1}=\frac{\partial}{\partial x^{1}},\dots,X_{n}=\frac{% \partial}{\partial x^{n}}\right).
  41. g i j [ 𝐟 ] = g ( x i , x j ) . g_{ij}[\mathbf{f}]=g\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{% \partial x^{j}}\right).
  42. y i = y i ( x 1 , x 2 , , x n ) , i = 1 , 2 , , n y^{i}=y^{i}(x^{1},x^{2},\dots,x^{n}),\quad i=1,2,\dots,n
  43. g i j [ 𝐟 ] = g ( y i , y j ) . g_{ij}[\mathbf{f}^{\prime}]=g\left(\frac{\partial}{\partial y^{i}},\frac{% \partial}{\partial y^{j}}\right).
  44. y i = k = 1 n x k y i x k \frac{\partial}{\partial y^{i}}=\sum_{k=1}^{n}\frac{\partial x^{k}}{\partial y% ^{i}}\frac{\partial}{\partial x^{k}}
  45. g i j [ 𝐟 ] = k , = 1 n x k y i g k [ 𝐟 ] x y j . g_{ij}[\mathbf{f^{\prime}}]=\sum_{k,\ell=1}^{n}\frac{\partial x^{k}}{\partial y% ^{i}}g_{k\ell}[\mathbf{f}]\frac{\partial x^{\ell}}{\partial y^{j}}.
  46. G [ 𝐟 ] = ( ( D y ) - 1 ) T G [ 𝐟 ] ( D y ) - 1 G[\mathbf{f}^{\prime}]=\left((Dy)^{-1}\right)^{\mathrm{T}}G[\mathbf{f}](Dy)^{-% 1}\,
  47. q m ( X m ) = g m ( X m , X m ) , X m T m M . q_{m}(X_{m})=g_{m}(X_{m},X_{m}),\quad X_{m}\in T_{m}M.
  48. q m ( i ξ i X i ) = ( ξ 1 ) 2 + ( ξ 2 ) 2 + + ( ξ p ) 2 - ( ξ p + 1 ) 2 - - ( ξ n ) 2 q_{m}\left(\sum_{i}\xi^{i}X_{i}\right)=(\xi^{1})^{2}+(\xi^{2})^{2}+\cdots+(\xi% ^{p})^{2}-(\xi^{p+1})^{2}-\cdots-(\xi^{n})^{2}
  49. g i j [ 𝐟 ] = g ( X i , X j ) . g_{ij}[\mathbf{f}]=g(X_{i},X_{j}).\,
  50. G [ 𝐟 A ] - 1 = A - 1 G [ 𝐟 ] - 1 ( A - 1 ) T . G[\mathbf{f}A]^{-1}=A^{-1}G[\mathbf{f}]^{-1}(A^{-1})^{\mathrm{T}}.
  51. α p ( a X p + b Y p ) = a α p ( X p ) + b α p ( Y p ) . \alpha_{p}(aX_{p}+bY_{p})=a\alpha_{p}(X_{p})+b\alpha_{p}(Y_{p}).\,
  52. p α p ( X p ) p\mapsto\alpha_{p}(X_{p})
  53. α i = α ( X i ) , i = 1 , 2 , , n . \alpha_{i}=\alpha(X_{i}),\quad i=1,2,\dots,n.
  54. α [ 𝐟 ] = [ α 1 α 2 α n ] . \alpha[\mathbf{f}]=\left[\alpha_{1}\ \ \alpha_{2}\ \ \dots\ \ \alpha_{n}\right].
  55. α [ 𝐟 A ] = α [ 𝐟 ] A . \alpha[\mathbf{f}A]=\alpha[\mathbf{f}]A.
  56. g ~ ( α , β ) = α [ 𝐟 ] G [ 𝐟 ] - 1 β [ 𝐟 ] T . \tilde{g}(\alpha,\beta)=\alpha[\mathbf{f}]G[\mathbf{f}]^{-1}\beta[\mathbf{f}]^% {\mathrm{T}}.
  57. α [ 𝐟 A ] G [ 𝐟 A ] - 1 β [ 𝐟 A ] T = ( α [ 𝐟 ] A ) ( A - 1 G [ 𝐟 ] - 1 ( A - 1 ) T ) A T β [ 𝐟 ] T = α [ 𝐟 ] G [ 𝐟 ] - 1 β [ 𝐟 ] T . \begin{aligned}\displaystyle\alpha[\mathbf{f}A]G[\mathbf{f}A]^{-1}\beta[% \mathbf{f}A]^{\mathrm{T}}&\displaystyle=(\alpha[\mathbf{f}]A)\left(A^{-1}G[% \mathbf{f}]^{-1}(A^{-1})^{\mathrm{T}}\right)A^{\mathrm{T}}\beta[\mathbf{f}]^{% \mathrm{T}}\\ &\displaystyle=\alpha[\mathbf{f}]G[\mathbf{f}]^{-1}\beta[\mathbf{f}]^{\mathrm{% T}}.\end{aligned}
  58. X = v 1 [ 𝐟 ] X 1 + v 2 [ 𝐟 ] X 2 + + v n [ 𝐟 ] X n = 𝐟 [ v 1 [ 𝐟 ] v 2 [ 𝐟 ] v n [ 𝐟 ] ] = 𝐟 v [ 𝐟 ] X=v^{1}[\mathbf{f}]X_{1}+v^{2}[\mathbf{f}]X_{2}+\dots+v^{n}[\mathbf{f}]X_{n}=% \mathbf{f}\begin{bmatrix}v^{1}[\mathbf{f}]\\ v^{2}[\mathbf{f}]\\ \vdots\\ v^{n}[\mathbf{f}]\end{bmatrix}=\mathbf{f}v[\mathbf{f}]\,
  59. X = 𝐟𝐀 v [ 𝐟𝐀 ] = 𝐟 v [ 𝐟 ] . X=\mathbf{fA}v[\mathbf{fA}]=\mathbf{f}v[\mathbf{f}].
  60. θ i [ 𝐟 ] ( X j ) = { 1 if i = j 0 if i j . \theta^{i}[\mathbf{f}](X_{j})=\begin{cases}1&\mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\not=j.\end{cases}
  61. θ [ 𝐟 ] = [ θ 1 [ 𝐟 ] θ 2 [ 𝐟 ] θ n [ 𝐟 ] ] . \theta[\mathbf{f}]=\begin{bmatrix}\theta^{1}[\mathbf{f}]\\ \theta^{2}[\mathbf{f}]\\ \vdots\\ \theta^{n}[\mathbf{f}]\end{bmatrix}.
  62. θ [ 𝐟 A ] = A - 1 θ [ 𝐟 ] . \theta[\mathbf{f}A]=A^{-1}\theta[\mathbf{f}].
  63. α = a 1 [ 𝐟 ] θ 1 [ 𝐟 ] + a 2 [ 𝐟 ] θ 2 [ 𝐟 ] + + a n [ 𝐟 ] θ n [ 𝐟 ] = [ a 1 [ 𝐟 ] a 2 [ 𝐟 ] a n [ 𝐟 ] ] θ [ 𝐟 ] = a [ 𝐟 ] θ [ 𝐟 ] \begin{aligned}\displaystyle\alpha&\displaystyle=a_{1}[\mathbf{f}]\theta^{1}[% \mathbf{f}]+a_{2}[\mathbf{f}]\theta^{2}[\mathbf{f}]+\cdots+a_{n}[\mathbf{f}]% \theta^{n}[\mathbf{f}]\\ &\displaystyle=\left[\frac{}{}a_{1}[\mathbf{f}]\ \ a_{2}[\mathbf{f}]\ \ \dots% \ \ a_{n}[\mathbf{f}]\right]\theta[\mathbf{f}]=a[\mathbf{f}]\theta[\mathbf{f}]% \end{aligned}
  64. α = a [ 𝐟 A ] θ [ 𝐟 A ] = a [ 𝐟 ] θ [ 𝐟 ] \alpha=a[\mathbf{f}A]\theta[\mathbf{f}A]=a[\mathbf{f}]\theta[\mathbf{f}]
  65. g p ( X p , - ) : Y p g p ( X p , Y p ) g_{p}(X_{p},-):Y_{p}\mapsto g_{p}(X_{p},Y_{p})
  66. a [ 𝐟 ] = v [ 𝐟 ] T G [ 𝐟 ] . a[\mathbf{f}]=v[\mathbf{f}]^{\mathrm{T}}G[\mathbf{f}].
  67. v [ 𝐟 A ] T G [ 𝐟 A ] = v [ 𝐟 ] T ( A - 1 ) T A T G [ 𝐟 ] A = v [ 𝐟 ] T G [ 𝐟 ] A v[\mathbf{f}A]^{\mathrm{T}}G[\mathbf{f}A]=v[\mathbf{f}]^{\mathrm{T}}(A^{-1})^{% \mathrm{T}}A^{\mathrm{T}}G[\mathbf{f}]A=v[\mathbf{f}]^{\mathrm{T}}G[\mathbf{f}]A
  68. a i [ 𝐟 ] = k = 1 n v k [ 𝐟 ] g k i [ 𝐟 ] a_{i}[\mathbf{f}]=\sum_{k=1}^{n}v^{k}[\mathbf{f}]g_{ki}[\mathbf{f}]
  69. v [ 𝐟 ] = G - 1 [ 𝐟 ] a [ 𝐟 ] T v[\mathbf{f}]=G^{-1}[\mathbf{f}]a[\mathbf{f}]^{\mathrm{T}}
  70. v [ 𝐟 A ] = A - 1 v [ 𝐟 ] . v[\mathbf{f}A]=A^{-1}v[\mathbf{f}].
  71. v i [ 𝐟 ] = k = 1 n g i k [ 𝐟 ] a k [ 𝐟 ] . v^{i}[\mathbf{f}]=\sum_{k=1}^{n}g^{ik}[\mathbf{f}]a_{k}[\mathbf{f}].
  72. v = v 1 𝐞 1 + + v n 𝐞 n v=v^{1}\mathbf{e}_{1}+\dots+v^{n}\mathbf{e}_{n}
  73. φ * ( v ) = i = 1 n a = 1 m v i φ a x i 𝐞 a . \varphi_{*}(v)=\sum_{i=1}^{n}\sum_{a=1}^{m}v^{i}\frac{\partial\varphi^{a}}{% \partial x^{i}}\mathbf{e}_{a}.
  74. g ( v , w ) = φ * ( v ) φ * ( w ) . g(v,w)=\varphi_{*}(v)\cdot\varphi_{*}(w).
  75. G ( 𝐞 ) = ( D φ ) T ( D φ ) G(\mathbf{e})=(D\varphi)^{\mathrm{T}}(D\varphi)
  76. D φ = [ φ 1 x 1 φ 1 x 2 φ 1 x n φ 2 x 1 φ 2 x 2 φ 2 x n φ m x 1 φ m x 2 φ m x n ] . D\varphi=\begin{bmatrix}\frac{\partial\varphi^{1}}{\partial x^{1}}&\frac{% \partial\varphi^{1}}{\partial x^{2}}&\dots&\frac{\partial\varphi^{1}}{\partial x% ^{n}}\\ \frac{\partial\varphi^{2}}{\partial x^{1}}&\frac{\partial\varphi^{2}}{\partial x% ^{2}}&\dots&\frac{\partial\varphi^{2}}{\partial x^{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\varphi^{m}}{\partial x^{1}}&\frac{\partial\varphi^{m}}{\partial x% ^{2}}&\dots&\frac{\partial\varphi^{m}}{\partial x^{n}}\\ \end{bmatrix}.
  77. g : T M × M T M 𝐑 g:TM\times_{M}TM\to\mathbf{R}
  78. g p : T p M × T p M 𝐑 . g_{p}:T_{p}M\times T_{p}M\to\mathbf{R}.
  79. g Γ ( ( T M T M ) * ) . g_{\otimes}\in\Gamma\left((TM\otimes TM)^{*}\right).
  80. g ( v w ) = g ( v , w ) g_{\otimes}(v\otimes w)=g(v,w)
  81. g τ = g g_{\otimes}\circ\tau=g_{\otimes}
  82. τ : T M T M T M T M \tau:TM\otimes TM\stackrel{\cong}{\to}TM\otimes TM
  83. ( T M T M ) * T * M T * M , (TM\otimes TM)^{*}\cong T^{*}M\otimes T^{*}M,
  84. g : E × M E 𝐑 g:E\times_{M}E\to\mathbf{R}
  85. g p : E p × E p 𝐑 . g_{p}:E_{p}\times E_{p}\to\mathbf{R}.
  86. E * E * \scriptstyle E^{*}\otimes E^{*}
  87. S g X p = d e f g ( X p , - ) , S_{g}X_{p}\,\stackrel{def}{=}\,g(X_{p},-),
  88. [ S g X p , Y p ] = g p ( X p , Y p ) [S_{g}X_{p},Y_{p}]=g_{p}(X_{p},Y_{p})\,
  89. [ S g X p , Y p ] = [ S g Y p , X p ] [S_{g}X_{p},Y_{p}]=[S_{g}Y_{p},X_{p}]\,
  90. g S ( X p , Y p ) = [ S X p , Y p ] . g_{S}(X_{p},Y_{p})=[SX_{p},Y_{p}].\,
  91. S g - 1 : T * M T M S_{g}^{-1}:T^{*}M\to TM
  92. [ S g - 1 α , β ] = [ S g - 1 β , α ] [S_{g}^{-1}\alpha,\beta]=[S_{g}^{-1}\beta,\alpha]
  93. T * M T * M 𝐑 T^{*}M\otimes T^{*}M\to\mathbf{R}
  94. T M T M . TM\otimes TM.
  95. γ ( t ) \gamma(t)
  96. L = a b i , j = 1 n g i j ( γ ( t ) ) ( d d t x i γ ( t ) ) ( d d t x j γ ( t ) ) d t . L=\int_{a}^{b}\sqrt{\sum_{i,j=1}^{n}g_{ij}(\gamma(t))\left({d\over dt}x^{i}% \circ\gamma(t)\right)\left({d\over dt}x^{j}\circ\gamma(t)\right)}\,dt.
  97. d s 2 = i , j = 1 n g i j ( p ) d x i d x j ds^{2}=\sum_{i,j=1}^{n}g_{ij}(p)dx^{i}dx^{j}
  98. L = a b | i , j = 1 n g i j ( γ ( t ) ) ( d d t x i γ ( t ) ) ( d d t x j γ ( t ) ) | d t . L=\int_{a}^{b}\sqrt{\left|\sum_{i,j=1}^{n}g_{ij}(\gamma(t))\left({d\over dt}x^% {i}\circ\gamma(t)\right)\left({d\over dt}x^{j}\circ\gamma(t)\right)\right|}\,% dt\ .
  99. E = 1 2 a b i , j = 1 n g i j ( γ ( t ) ) ( d d t x i γ ( t ) ) ( d d t x j γ ( t ) ) d t . E=\frac{1}{2}\int_{a}^{b}\sum_{i,j=1}^{n}g_{ij}(\gamma(t))\left({d\over dt}x^{% i}\circ\gamma(t)\right)\left({d\over dt}x^{j}\circ\gamma(t)\right)\,dt.
  100. Λ f = U f d μ g = φ ( U ) f φ - 1 ( x ) | det g | d x \Lambda f=\int_{U}f\,d\mu_{g}=\int_{\varphi(U)}f\circ\varphi^{-1}(x)\sqrt{|% \det g|}\,dx
  101. ω = | det g | d x 1 d x n \omega=\sqrt{|\det g|}\,dx^{1}\wedge\cdots\wedge dx^{n}
  102. x x
  103. y y
  104. g = [ 1 0 0 1 ] . g=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}.
  105. L = a b ( d x ) 2 + ( d y ) 2 . L=\int_{a}^{b}\sqrt{(dx)^{2}+(dy)^{2}}.
  106. ( r , θ ) (r,\theta)
  107. x = r cos θ x=r\cos\theta
  108. y = r sin θ y=r\sin\theta
  109. J = [ cos θ - r sin θ sin θ r cos θ ] . J=\begin{bmatrix}\cos\theta&-r\sin\theta\\ \sin\theta&r\cos\theta\end{bmatrix}.
  110. g = J T J = [ cos 2 θ + sin 2 θ - r sin θ cos θ + r sin θ cos θ - r cos θ sin θ + r cos θ sin θ r 2 sin 2 θ + r 2 cos 2 θ ] = [ 1 0 0 r 2 ] g=J^{\mathrm{T}}J=\begin{bmatrix}\cos^{2}\theta+\sin^{2}\theta&-r\sin\theta% \cos\theta+r\sin\theta\cos\theta\\ -r\cos\theta\sin\theta+r\cos\theta\sin\theta&r^{2}\sin^{2}\theta+r^{2}\cos^{2}% \theta\end{bmatrix}=\begin{bmatrix}1&0\\ 0&r^{2}\end{bmatrix}
  111. / x i \partial/\partial x^{i}
  112. q i q^{i}
  113. g i j = k l δ k l x k q i x l q j = k x k q i x k q j . g_{ij}=\sum_{kl}\delta_{kl}{\partial x^{k}\over\partial q^{i}}{\partial x^{l}% \over\partial q^{j}}=\sum_{k}\frac{\partial x^{k}}{\partial q^{i}}\frac{% \partial x^{k}}{\partial q^{j}}.
  114. ( θ , φ ) (\theta,\varphi)
  115. θ \theta
  116. φ \varphi
  117. g = [ 1 0 0 sin 2 θ ] . g=\left[\begin{array}[]{cc}1&0\\ 0&\sin^{2}\theta\end{array}\right].
  118. d s 2 = d θ 2 + sin 2 θ d φ 2 . ds^{2}=d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}.
  119. r μ ( x 0 , x 1 , x 2 , x 3 ) = ( c t , x , y , z ) , r^{\mu}\rightarrow(x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)\ ,
  120. g = [ 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ] . g=\begin{bmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{bmatrix}.
  121. d s 2 = c 2 d t 2 - d x 2 - d y 2 - d z 2 = d r μ d r μ = g μ ν d r μ d r ν ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=dr^{\mu}dr_{\mu}=g_{\mu\nu}dr^{\mu}dr^% {\nu}
  122. ( x 0 , x 1 , x 2 , x 3 ) = ( c t , r , θ , φ ) (x^{0},x^{1},x^{2},x^{3})=(ct,r,\theta,\varphi)
  123. G = ( g μ ν ) = [ ( 1 - 2 G M r c 2 ) 0 0 0 0 - ( 1 - 2 G M r c 2 ) - 1 0 0 0 0 - r 2 0 0 0 0 - r 2 sin 2 θ ] G=(g_{\mu\nu})=\begin{bmatrix}(1-\frac{2GM}{rc^{2}})&0&0&0\\ 0&-(1-\frac{2GM}{rc^{2}})^{-1}&0&0\\ 0&0&-r^{2}&0\\ 0&0&0&-r^{2}\sin^{2}\theta\end{bmatrix}\,

Michaelis–Menten_kinetics.html

  1. v v
  2. [ S ] [S]
  3. v = d [ P ] d t = V max [ S ] K M + [ S ] v=\frac{d[P]}{dt}=\frac{V_{\max}{[S]}}{K_{\mathrm{M}}+[S]}
  4. V max V_{\max}
  5. K M K_{\mathrm{M}}
  6. V max V_{\max}
  7. E + S k r k f E S k cat E + P E+S\,\overset{k_{f}}{\underset{k_{r}}{\rightleftharpoons}}\,ES\,\overset{k_{% \mathrm{cat}}}{\longrightarrow}\,E+P
  8. k f k_{f}
  9. k r k_{r}
  10. k cat k_{\mathrm{cat}}
  11. v = d [ P ] d t = V max [ S ] K M + [ S ] = k cat [ E ] 0 [ S ] K M + [ S ] v=\frac{d[P]}{dt}=V_{\max}\frac{[S]}{K_{\mathrm{M}}+[S]}=k_{\mathrm{cat}}[E]_{% 0}\frac{[S]}{K_{\mathrm{M}}+[S]}
  12. [ S ] [S]
  13. V max V_{\max}
  14. V max = k cat [ E ] 0 V_{\max}=k_{\mathrm{cat}}[E]_{0}
  15. [ E ] 0 [E]_{0}
  16. k cat k_{\mathrm{cat}}
  17. K M K_{\mathrm{M}}
  18. K M K_{\mathrm{M}}
  19. V max V_{\max}
  20. K M K_{\mathrm{M}}
  21. K M K_{\mathrm{M}}
  22. k cat k\text{cat}
  23. k cat / K M k\text{cat}/K_{\mathrm{M}}
  24. k cat / K M k\text{cat}/K_{\mathrm{M}}
  25. t t
  26. d [ E ] d t = - k f [ E ] [ S ] + k r [ E S ] + k c a t [ E S ] d [ S ] d t = - k f [ E ] [ S ] + k r [ E S ] d [ E S ] d t = k f [ E ] [ S ] - k r [ E S ] - k c a t [ E S ] d [ P ] d t = k c a t [ E S ] . \begin{aligned}\displaystyle\frac{d[E]}{dt}&\displaystyle=-k_{f}[E][S]+k_{r}[% ES]+k_{cat}[ES]\\ \displaystyle\frac{d[S]}{dt}&\displaystyle=-k_{f}[E][S]+k_{r}[ES]\\ \displaystyle\frac{d[ES]}{dt}&\displaystyle=k_{f}[E][S]-k_{r}[ES]-k_{cat}[ES]% \\ \displaystyle\frac{d[P]}{dt}&\displaystyle=k_{cat}[ES].\end{aligned}
  27. [ E ] + [ E S ] = [ E ] 0 [E]+[ES]=[E]_{0}
  28. k f [ E ] [ S ] = k r [ E S ] k_{f}[E][S]=k_{r}[ES]
  29. [ E ] = [ E ] 0 - [ E S ] [E]=[E]_{0}-[ES]
  30. k f ( [ E ] 0 - [ E S ] ) [ S ] = k r [ E S ] k_{f}([E]_{0}-[ES])[S]=k_{r}[ES]
  31. [ E S ] = [ E ] 0 [ S ] K d + [ S ] [ES]=\frac{[E]_{0}[S]}{K_{d}+[S]}
  32. K d = k r / k f K_{d}=k_{r}/k_{f}
  33. v v
  34. v = d [ P ] d t = V max [ S ] K d + [ S ] v=\frac{d[P]}{dt}=\frac{V_{\max}{[S]}}{K_{d}+[S]}
  35. V max = k cat [ E ] 0 V_{\max}=k_{\mathrm{cat}}[E]_{0}
  36. k f [ E ] [ S ] = k r [ E S ] + k cat [ E S ] k_{f}[E][S]=k_{r}[ES]+k_{\mathrm{cat}}[ES]
  37. [ E S ] = [ E ] 0 [ S ] K M + [ S ] [ES]=\frac{[E]_{0}[S]}{K_{\mathrm{M}}+[S]}
  38. K M = k r + k cat k f K_{\mathrm{M}}=\frac{k_{r}+k_{\mathrm{cat}}}{k_{f}}
  39. k r k_{r}
  40. k cat k_{\mathrm{cat}}
  41. k f {k_{f}}
  42. v v
  43. v = d [ P ] d t = V max [ S ] K M + [ S ] v=\frac{d[P]}{dt}=\frac{V_{\max}{[S]}}{K_{\mathrm{M}}+[S]}
  44. K d K_{d}
  45. K M K_{\mathrm{M}}
  46. ϵ d = k cat k r 1 \epsilon_{d}=\frac{k_{\mathrm{cat}}}{k_{r}}\ll 1
  47. ϵ m = [ E ] 0 [ S ] 0 + K M 1 \epsilon_{m}=\frac{[E]_{0}}{[S]_{0}+K_{\mathrm{M}}}\ll 1
  48. K M K_{\mathrm{M}}
  49. ϵ \epsilon\,\!
  50. E + S k r 1 k f 1 E S k r 2 k f 2 E + P E+S\,\overset{k_{f_{1}}}{\underset{k_{r_{1}}}{\rightleftharpoons}}\,ES\,% \overset{k_{f_{2}}}{\underset{k_{r_{2}}}{\rightleftharpoons}}\,E+P
  51. [ S ] [ P ] [S]\gg[P]
  52. Δ G 0 \Delta{G}\ll 0
  53. V max V_{\max}
  54. K M K_{\mathrm{M}}
  55. [ S ] [S]
  56. v 0 v_{0}
  57. [ S ] K M = W [ F ( t ) ] \frac{[S]}{K_{\mathrm{M}}}=W[F(t)]\,
  58. F ( t ) = [ S ] 0 K M exp ( [ S ] 0 K M - V max K M t ) F(t)=\frac{[S]_{0}}{K_{\mathrm{M}}}\exp\!\left(\frac{[S]_{0}}{K_{\mathrm{M}}}-% \frac{V_{\max}}{K_{\mathrm{M}}}\,t\right)\,
  59. V max V_{\max}
  60. K M K_{\mathrm{M}}

Michelson–Morley_experiment.html

  1. c c
  2. T = 0 T=0
  3. L L
  4. v v
  5. T 1 T_{1}
  6. c T 1 cT_{1}
  7. v T 1 vT_{1}
  8. c T 1 = L + v T 1 cT_{1}=L+vT_{1}
  9. T 1 = L / ( c - v ) T_{1}=L/(c-v)
  10. v v
  11. c T 2 = L - v T 2 cT_{2}=L-vT_{2}
  12. T 2 = L / ( c + v ) T_{2}=L/(c+v)
  13. T l = T 1 + T 2 T_{l}=T_{1}+T_{2}
  14. T l = L c - v + L c + v T_{l}=\frac{L}{c-v}+\frac{L}{c+v}
  15. = 2 L c 1 1 - v 2 c 2 =\frac{2L}{c}\frac{1}{1-\frac{v^{2}}{c^{2}}}
  16. 2 L c ( 1 + v 2 c 2 ) \approx\frac{2L}{c}\left(1+\frac{v^{2}}{c^{2}}\right)
  17. T t = 2 L c T_{t}=\frac{2L}{c}
  18. c c
  19. T 3 T_{3}
  20. c T 3 cT_{3}
  21. v T 3 vT_{3}
  22. L L
  23. v T 3 vT_{3}
  24. L 2 + ( v T 3 ) 2 \scriptstyle\sqrt{L^{2}+\left(vT_{3}\right)^{2}}
  25. c T 3 = L 2 + ( v T 3 ) 2 \scriptstyle cT_{3}=\sqrt{L^{2}+\left(vT_{3}\right)^{2}}
  26. T 3 = L / c 2 - v 2 \scriptstyle T_{3}=L/\sqrt{c^{2}-v^{2}}
  27. T t = 2 T 3 T_{t}=2T_{3}
  28. T t = 2 L c 2 - v 2 = 2 L c 1 1 - v 2 c 2 2 L c ( 1 + v 2 2 c 2 ) T_{t}=\frac{2L}{\sqrt{c^{2}-v^{2}}}=\frac{2L}{c}\frac{1}{\sqrt{1-\frac{v^{2}}{% c^{2}}}}\approx\frac{2L}{c}\left(1+\frac{v^{2}}{2c^{2}}\right)
  29. T l - T t = 2 c ( L 1 - v 2 c 2 - L 1 - v 2 c 2 ) T_{l}-T_{t}=\frac{2}{c}\left(\frac{L}{1-\frac{v^{2}}{c^{2}}}-\frac{L}{\sqrt{1-% \frac{v^{2}}{c^{2}}}}\right)
  30. Δ 1 = 2 ( L 1 - v 2 c 2 - L 1 - v 2 c 2 ) \Delta_{1}=2\left(\frac{L}{1-\frac{v^{2}}{c^{2}}}-\frac{L}{\sqrt{1-\frac{v^{2}% }{c^{2}}}}\right)
  31. Δ 2 = 2 ( L 1 - v 2 c 2 - L 1 - v 2 c 2 ) \Delta_{2}=2\left(\frac{L}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-\frac{L}{1-\frac{v^{2% }}{c^{2}}}\right)
  32. Δ 1 - Δ 2 \Delta_{1}-\Delta_{2}
  33. n = Δ 1 - Δ 2 λ 2 L v 2 λ c 2 n=\frac{\Delta_{1}-\Delta_{2}}{\lambda}\approx\frac{2Lv^{2}}{\lambda c^{2}}
  34. c c
  35. v v
  36. c - v c-v
  37. c + v c+v
  38. T 1 T_{1}
  39. T 2 T_{2}
  40. c 2 - v 2 \sqrt{c^{2}-v^{2}}
  41. T 3 T_{3}
  42. L / γ L/\gamma
  43. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  44. L L
  45. T l T_{l}
  46. T l = 2 L 1 - v 2 c 2 c 1 1 - v 2 c 2 = 2 L c 1 1 - v 2 c 2 = T t T_{l}=\frac{2L\sqrt{1-\frac{v^{2}}{c^{2}}}}{c}\frac{1}{1-\frac{v^{2}}{c^{2}}}=% \frac{2L}{c}\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=T_{t}
  47. γ \gamma
  48. L 1 L_{1}
  49. L 2 L_{2}
  50. L 1 = L 2 L^{\prime}_{1}=L^{\prime}_{2}
  51. L 2 L 1 = L 2 ϕ / L 1 γ ϕ = γ \frac{L_{2}}{L_{1}}=\frac{L^{\prime}_{2}}{\phi}\left/\frac{L^{\prime}_{1}}{% \gamma\phi}\right.=\gamma
  52. ϕ \phi
  53. ϕ = 1 \phi=1
  54. L 1 L_{1}
  55. ϕ = 1 / γ \phi=1/\gamma
  56. L 2 L_{2}
  57. x = γ ϕ ( x - v t ) , y = ϕ y , z = ϕ z , t = γ ϕ ( t - v x c 2 ) x^{\prime}=\gamma\phi(x-vt),\ y^{\prime}=\phi y,\ z^{\prime}=\phi z,\ t^{% \prime}=\gamma\phi\left(t-\frac{vx}{c^{2}}\right)
  58. ϕ \phi
  59. ϕ = 1 \phi=1
  60. ϕ \phi
  61. ϕ = 1 \phi=1
  62. \approx
  63. \approx
  64. 10 - 15 \lesssim 10^{-15}
  65. 10 - 16 \lesssim 10^{-16}
  66. 10 - 17 \lesssim 10^{-17}

Mie_scattering.html

  1. I = I 0 ( 1 + cos 2 θ 2 R 2 ) ( 2 π λ ) 4 ( n 2 - 1 n 2 + 2 ) 2 ( d 2 ) 6 , I=I_{0}\left(\frac{1+\cos^{2}\theta}{2R^{2}}\right)\left(\frac{2\pi}{\lambda}% \right)^{4}\left(\frac{n^{2}-1}{n^{2}+2}\right)^{2}\left(\frac{d}{2}\right)^{6},
  2. Q = 2 - 4 p sin p + 4 p 2 ( 1 - cos p ) , Q=2-\frac{4}{p}\sin{p}+\frac{4}{p^{2}}(1-\cos{p}),
  3. ϵ r > 78 ( 38 ) \scriptstyle\epsilon_{\mathrm{r}}>78(38)

Mikhail_Ostrogradsky.html

  1. R ( x ) P ( x ) d x = T ( x ) S ( x ) + X ( x ) Y ( x ) d x , \int{R(x)\over P(x)}\,dx={T(x)\over S(x)}+\int{X(x)\over Y(x)}\,dx,
  2. V ( P x + Q y + R z ) d x d y d z = Σ ( P + Q + R ) d Σ \iiint_{V}\left({\partial P\over\partial x}+{\partial Q\over\partial y}+{% \partial R\over\partial z}\right)dx\,dy\,dz=\iint_{\Sigma}\left(P+Q+R\right)\,d\Sigma

Miller–Rabin_primality_test.html

  1. x 2 1 ( mod p ) x^{2}\equiv 1\;\;(\mathop{{\rm mod}}p)
  2. ( x - 1 ) ( x + 1 ) 0 ( mod p ) . (x-1)(x+1)\equiv 0\;\;(\mathop{{\rm mod}}p).
  3. a d 1 ( mod n ) a^{d}\equiv 1\;\;(\mathop{{\rm mod}}n)
  4. a 2 r d - 1 ( mod n ) a^{2^{r}\cdot d}\equiv-1\;\;(\mathop{{\rm mod}}n)
  5. a n - 1 1 ( mod n ) . a^{n-1}\equiv 1\;\;(\mathop{{\rm mod}}n).
  6. a d 1 ( mod n ) a^{d}\not\equiv 1\;\;(\mathop{{\rm mod}}n)
  7. a 2 r d - 1 ( mod n ) a^{2^{r}d}\not\equiv-1\;\;(\mathop{{\rm mod}}n)
  8. a d 2 r - 1 = ( a d 2 r - 1 - 1 ) ( a d 2 r - 1 + 1 ) a^{d2^{r}}-1=(a^{d2^{r-1}}-1)(a^{d2^{r-1}}+1)
  9. a n - 1 1 ( mod n ) a^{n-1}\not\equiv 1\;\;(\mathop{{\rm mod}}n)
  10. X ¯ \overline{X}
  11. P ( Y k | X ) P(Y_{k}|X)
  12. P ( X | Y k ) P(X|Y_{k})
  13. P ( X | Y k ) = P ( Y k | X ) P ( X ) P ( Y k | X ) P ( X ) + P ( Y k | X ¯ ) P ( X ¯ ) P(X|Y_{k})=\frac{P(Y_{k}|X)P(X)}{P(Y_{k}|X)P(X)+P(Y_{k}|\overline{X})P(% \overline{X})}
  14. a [ 2 , min ( n - 1 , 2 ( ln n ) 2 ) ] a\in[2,\min(n-1,\lfloor 2(\ln n)^{2}\rfloor)]
  15. a d 1 ( mod n ) and a 2 r d - 1 ( mod n ) for all r [ 0 , s - 1 ] a^{d}\not\equiv 1\;\;(\mathop{{\rm mod}}n)\,\text{ and }a^{2^{r}\cdot d}\not% \equiv-1\;\;(\mathop{{\rm mod}}n)\,\text{ for all }r\in[0,s-1]

Millimeter_cloud_radar.html

  1. r r
  2. ( r ; θ ; ϕ ) (r;\theta;\phi)
  3. E i ( r , θ , ϕ , t ) = A i ( θ , ϕ ) e i ϕ t r e i ( k c r - ω c t ) U ( t - 2 r / c ) E_{i}(r,\theta,\phi,t)=\frac{A_{i}(\theta,\phi)e^{i\phi t}}{r}e^{i(k_{c}r-% \omega_{c}t)}U(t-2r/c)
  4. ω c \omega_{c}
  5. t t
  6. c c
  7. r r
  8. k c k_{c}
  9. λ c \lambda_{c}
  10. A i A_{i}
  11. U ( t - 2 r / c ) U(t-2r/c)
  12. τ \tau
  13. T p w T_{pw}
  14. E r ( t ) = m = 0 N s A r , m e i ( 2 k c r m - ( ω c - ω d , m ) t + ϕ s , m + ϕ t ) U ( t - r / c ) E_{r}(t)=\sum_{m=0}^{N_{s}}A_{r,m}e^{i({2k_{c}r_{m}-(\omega_{c}-\omega_{d,m})t% +\phi_{s,m}+\phi_{t}})}U(t-r/c)
  15. A r , m A_{r,m}
  16. r m r_{m}
  17. ω c \omega_{c}
  18. ω d , m t \omega_{d,m}t
  19. ϕ s , m \phi_{s,m}
  20. ϕ t \phi_{t}
  21. ϕ s , m \phi_{s,m}
  22. τ s \tau_{s}
  23. τ s \tau_{s}
  24. N g N_{g}
  25. V ( τ s ) = m = 0 N g A r , m e i ( 2 k c r m - ( ω d , m ) t + ϕ s , m ) V(\tau_{s})=\sum_{m=0}^{N_{g}}A_{r,m}e^{i(2k_{c}r_{m}-(\omega_{d,m})t+\phi_{s,% m})}
  26. τ s \tau_{s}
  27. T s T_{s}
  28. I ( τ s , T s ) = R e [ i = 0 N g A r , m e i ( 2 k c r m - ( ω d , m ) t + ϕ s , m ) ] I(\tau_{s},T_{s})=Re[\sum_{i=0}^{N_{g}}A_{r,m}e^{i(2k_{c}r_{m}-(\omega_{d,m})t% +\phi_{s,m})}]
  29. Q ( τ s , T s ) = I m [ i = 0 N g A r , m e i ( 2 k c r m - ( ω d , m ) t + ϕ s , m ) ] Q(\tau_{s},T_{s})=Im[\sum_{i=0}^{N_{g}}A_{r,m}e^{i(2k_{c}r_{m}-(\omega_{d,m})t% +\phi_{s,m})}]
  30. N f I n + i Q n N_{f}I_{n}+iQ_{n}
  31. N f N_{f}
  32. N f N_{f}
  33. S c o m p l ( k ) S_{compl}(k)
  34. k k
  35. S ( k ) = S c o m p l * S c o m p l * ( k ) S(k)=S_{compl}*S_{compl}^{*}(k)
  36. S ( v d ) S(v_{d})
  37. v d v_{d}
  38. v m i n v_{min}
  39. v m a x v_{max}
  40. S ( v d ) > 0 S(v_{d})>0
  41. d v dv
  42. Z Z
  43. Z = v m i n v m a x S ( v d ) d v d Z=\int\limits_{v_{min}}^{v_{max}}S(v_{d})dv_{d}
  44. S ( v d ) S(v_{d})
  45. Z Z
  46. V = 1 Z v m i n v m a x v d S ( v d ) d v d V=\frac{1}{Z}\int\limits_{v_{min}}^{v_{max}}v_{d}S(v_{d})dv_{d}
  47. W = 1 Z v m i n v m a x ( v d - V ) 2 S ( v d ) d v d W=\sqrt{\frac{1}{Z}\int\limits_{v_{min}}^{v_{max}}(v_{d}-V)^{2}S(v_{d})dv_{d}}
  48. W W
  49. V V
  50. W W
  51. S k = 1 Z v m i n v m a x ( v d - V ) 3 | S ( v d ) | 2 d v d W 3 Sk=\frac{\frac{1}{Z}\int_{v_{min}}^{v_{max}}(v_{d}-V)^{3}|S(v_{d})|^{2}~{}% \mathrm{d}v_{d}}{W^{3}}
  52. K = 1 Z v m i n v m a x ( v d - V ) 4 | S ( v d ) | 2 d v d W 4 K=\frac{\frac{1}{Z}\int_{v_{min}}^{v_{max}}(v_{d}-V)^{4}|S(v_{d})|^{2}~{}% \mathrm{d}v_{d}}{W^{4}}

Minimal_surface.html

  1. ( 1 + u x 2 ) u y y - 2 u x u y u x y + ( 1 + u y 2 ) u x x = 0 (1+u_{x}^{2})u_{yy}-2u_{x}u_{y}u_{xy}+(1+u_{y}^{2})u_{xx}=0
  2. d d x ( z x 1 + z x 2 + z y 2 ) + d d y ( z y 1 + z x 2 + z y 2 ) = 0 \frac{d}{dx}\left(\frac{z_{x}}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}}\right)+\frac{d}{% dy}\left(\frac{z_{y}}{\sqrt{1+z_{x}^{2}+z_{y}^{2}}}\right)=0
  3. ( 1 + z x 2 ) z y y - 2 z x z y z x y + ( 1 + z y 2 ) z x x = 0 (1+z_{x}^{2})z_{yy}-2z_{x}z_{y}z_{xy}+(1+z_{y}^{2})z_{xx}=0

Minimum_message_length.html

  1. E E
  2. length ( E ) \operatorname{length}(E)
  3. E E
  4. P ( E ) P(E)
  5. length ( E ) = - log 2 ( P ( E ) ) \operatorname{length}(E)=-\log_{2}(P(E))
  6. H H
  7. E E
  8. P ( E | H ) P ( H ) P(E|H)P(H)
  9. P ( H and E ) P(H\and E)
  10. length ( H and E ) = - log 2 ( P ( H and E ) ) \operatorname{length}(H\and E)=-\log_{2}(P(H\and E))
  11. - log 2 ( P ( H and E ) ) = - log 2 ( P ( H ) ) + - log 2 ( P ( E | H ) ) -\log_{2}(P(H\and E))=-\log_{2}(P(H))+-\log_{2}(P(E|H))

Minkowski_inequality.html

  1. f + g p f p + g p \|f+g\|_{p}\leq\|f\|_{p}+\|g\|_{p}
  2. f p = sup g q = 1 | f g | d μ , 1 p + 1 q = 1 \|f\|_{p}=\sup_{\|g\|_{q}=1}\int|fg|d\mu,\qquad\tfrac{1}{p}+\tfrac{1}{q}=1
  3. ( k = 1 n | x k + y k | p ) 1 p ( k = 1 n | x k | p ) 1 p + ( k = 1 n | y k | p ) 1 p \left(\sum_{k=1}^{n}|x_{k}+y_{k}|^{p}\right)^{\frac{1}{p}}\leq\left(\sum_{k=1}% ^{n}|x_{k}|^{p}\right)^{\frac{1}{p}}+\left(\sum_{k=1}^{n}|y_{k}|^{p}\right)^{% \frac{1}{p}}
  4. | f + g | p 2 p - 1 ( | f | p + | g | p ) . |f+g|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p}).
  5. h ( x ) = x p h(x)=x^{p}
  6. p > 1 p>1
  7. | 1 2 f + 1 2 g | p | 1 2 | f | + 1 2 | g | | p 1 2 | f | p + 1 2 | g | p . \left|\tfrac{1}{2}f+\tfrac{1}{2}g\right|^{p}\leq\left|\tfrac{1}{2}|f|+\tfrac{1% }{2}|g|\right|^{p}\leq\tfrac{1}{2}|f|^{p}+\tfrac{1}{2}|g|^{p}.
  8. | f + g | p 1 2 | 2 f | p + 1 2 | 2 g | p = 2 p - 1 | f | p + 2 p - 1 | g | p . |f+g|^{p}\leq\tfrac{1}{2}|2f|^{p}+\tfrac{1}{2}|2g|^{p}=2^{p-1}|f|^{p}+2^{p-1}|% g|^{p}.
  9. ( f + g p ) (\|f+g\|_{p})
  10. ( f + g p ) (\|f+g\|_{p})
  11. f + g p p = | f + g | p d μ = | f + g | | f + g | p - 1 d μ ( | f | + | g | ) | f + g | p - 1 d μ = | f | | f + g | p - 1 d μ + | g | | f + g | p - 1 d μ ( ( | f | p d μ ) 1 p + ( | g | p d μ ) 1 p ) ( | f + g | ( p - 1 ) ( p p - 1 ) d μ ) 1 - 1 p Hölder’s inequality = ( f p + g p ) f + g p p f + g p \begin{aligned}\displaystyle\|f+g\|_{p}^{p}&\displaystyle=\int|f+g|^{p}\,% \mathrm{d}\mu\\ &\displaystyle=\int|f+g|\cdot|f+g|^{p-1}\,\mathrm{d}\mu\\ &\displaystyle\leq\int(|f|+|g|)|f+g|^{p-1}\,\mathrm{d}\mu\\ &\displaystyle=\int|f||f+g|^{p-1}\,\mathrm{d}\mu+\int|g||f+g|^{p-1}\,\mathrm{d% }\mu\\ &\displaystyle\leq\left(\left(\int|f|^{p}\,\mathrm{d}\mu\right)^{\frac{1}{p}}+% \left(\int|g|^{p}\,\mathrm{d}\mu\right)^{\frac{1}{p}}\right)\left(\int|f+g|^{(% p-1)\left(\frac{p}{p-1}\right)}\,\mathrm{d}\mu\right)^{1-\frac{1}{p}}&&% \displaystyle\,\text{ Hölder's inequality}\\ &\displaystyle=\left(\|f\|_{p}+\|g\|_{p}\right)\frac{\|f+g\|_{p}^{p}}{\|f+g\|_% {p}}\end{aligned}
  12. f + g p f + g p p . \frac{\|f+g\|_{p}}{\|f+g\|_{p}^{p}}.
  13. [ S 2 | S 1 F ( x , y ) d μ 1 ( x ) | p d μ 2 ( y ) ] 1 p S 1 ( S 2 | F ( x , y ) | p d μ 2 ( y ) ) 1 p d μ 1 ( x ) , \left[\int_{S_{2}}\left|\int_{S_{1}}F(x,y)\,d\mu_{1}(x)\right|^{p}d\mu_{2}(y)% \right]^{\frac{1}{p}}\leq\int_{S_{1}}\left(\int_{S_{2}}|F(x,y)|^{p}\,d\mu_{2}(% y)\right)^{\frac{1}{p}}d\mu_{1}(x),
  14. f 1 + f 2 p = ( S 2 | S 1 F ( x , y ) d μ 1 ( x ) | p d μ 2 ( y ) ) 1 p S 1 ( S 2 | F ( x , y ) | p d μ 2 ( y ) ) 1 p d μ 1 ( x ) = f 1 p + f 2 p . \|f_{1}+f_{2}\|_{p}=\left(\int_{S_{2}}\left|\int_{S_{1}}F(x,y)\,d\mu_{1}(x)% \right|^{p}d\mu_{2}(y)\right)^{\frac{1}{p}}\leq\int_{S_{1}}\left(\int_{S_{2}}|% F(x,y)|^{p}\,d\mu_{2}(y)\right)^{\frac{1}{p}}d\mu_{1}(x)=\|f_{1}\|_{p}+\|f_{2}% \|_{p}.

Minkowski_space.html

  1. 1 ¯ \overline{−1}
  2. 4 4
  3. ( , + , + , + ) (−,+,+,+)
  4. ( + , , , ) (+,−,−,−)
  5. p p
  6. p p
  7. c c→∞
  8. ( 0 , 2 ) (0,2)
  9. 4 × 4 4×4
  10. L L
  11. g g
  12. p p
  13. L L
  14. 4 × 4 4×4
  15. M M
  16. n = 4 n=4
  17. ( 3 , 1 ) (3,1)
  18. ( 1 , 3 ) (1,3)
  19. M M
  20. η η
  21. ( 0 , 2 ) (0,2)
  22. p p
  23. M M
  24. M M
  25. u , v u,v
  26. u u
  27. v v
  28. M M
  29. v v
  30. M M
  31. 𝐯 \mathbf{v}
  32. 3 3
  33. 4 4
  34. u v = η ( u , v ) u\cdot v=\eta(u,v)
  35. M M
  36. η ( a u + v , w ) = a η ( u , w ) + η ( v , w ) , u , v M , a (linearity in first slot) \eta(au+v,w)=a\eta(u,w)+\eta(v,w),\quad\forall u,v\in M,\forall a\in\mathbb{R}% \qquad\,\text{(linearity in first slot)}
  37. η ( u , v ) = η ( v , u ) (symmetry) \eta(u,v)=\eta(v,u)\qquad\,\text{(symmetry)}
  38. η ( u , v ) = 0 v M u = 0 (non-degeneracy) \eta(u,v)=0\quad\forall v\in M\Rightarrow u=0\qquad\,\text{(non-degeneracy)}
  39. v v
  40. w w
  41. η ( v , w ) = 0 η(v,w)=0
  42. e e
  43. η ( e , e ) = ± 1 η(e,e)=±1
  44. M M
  45. 4 4
  46. 1 , 2 1,2
  47. ± [ c 2 ( t 1 - t 2 ) 2 - ( x 1 - x 2 ) 2 - ( y 1 - y 2 ) 2 - ( z 1 - z 2 ) 2 ] , \pm\left[c^{2}(t_{1}-t_{2})^{2}-(x_{1}-x_{2})^{2}-(y_{1}-y_{2})^{2}-(z_{1}-z_{% 2})^{2}\right],
  48. ± ±
  49. η η
  50. ± [ c 2 t 2 - x 2 - y 2 - z 2 ] \pm\left[c^{2}t^{2}-x^{2}-y^{2}-z^{2}\right]
  51. ± ±
  52. x y = ± [ c 2 t 1 t 1 - x 1 x 2 - y 1 y 2 - z 1 z 2 ] . x\cdot y=\pm\left[c^{2}t_{1}t_{1}-x_{1}x_{2}-y_{1}y_{2}-z_{1}z_{2}\right].
  53. x y = x T [ η ] y , x\cdot y=x^{\mathrm{T}}[\eta]y,
  54. η η ηη
  55. 4 × 4 4×4
  56. η η
  57. η η ηη
  58. η η
  59. η = ± ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) , \eta=\pm\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix},
  60. u v = η ( u , v ) , u\cdot v=\eta(u,v),
  61. ( , + , + , + ) (−,+,+,+)
  62. - η ( e 0 , e 0 ) = η ( e 1 , e 1 ) = η ( e 2 , e 2 ) = η ( e 3 , e 3 ) = 1. -\eta(e_{0},e_{0})=\eta(e_{1},e_{1})=\eta(e_{2},e_{2})=\eta(e_{3},e_{3})=1.
  63. η ( e μ , e ν ) = η μ ν . \eta(e_{\mu},e_{\nu})=\eta_{\mu\nu}.
  64. v v
  65. v v
  66. 4 4
  67. v v
  68. 3 3
  69. v v
  70. w w
  71. η ( v , w ) = η μ ν v μ w ν = v 0 w 0 + v 1 w 1 + v 2 w 2 + v 3 w 3 = v μ w μ = v μ w μ , \eta(v,w)=\eta_{\mu\nu}v^{\mu}w^{\nu}=v^{0}w_{0}+v^{1}w_{1}+v^{2}w_{2}+v^{3}w_% {3}=v^{\mu}w_{\mu}=v_{\mu}w^{\mu},
  72. η ( v , v ) = η μ ν v μ v ν = v 0 v 0 + v 1 v 1 + v 2 v 2 + v 3 v 3 = v μ v μ . \eta(v,v)=\eta_{\mu\nu}v^{\mu}v^{\nu}=v^{0}v_{0}+v^{1}v_{1}+v^{2}v_{2}+v^{3}v_% {3}=v^{\mu}v_{\mu}.
  73. M M
  74. M M
  75. M M
  76. M M
  77. η η
  78. n 2 n≥2
  79. n n
  80. n n
  81. 4 4
  82. n > 4 n>4
  83. 1 + 1 1+1
  84. 4 4
  85. η η
  86. Φ Φ
  87. O ( 3 , 1 ) O(3,1)
  88. x x
  89. [ U 0 U 1 U 2 U 3 ] = [ γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ] [ U 0 U 1 U 2 U 3 ] , \begin{bmatrix}U^{\prime}_{0}\\ U^{\prime}_{1}\\ U^{\prime}_{2}\\ U^{\prime}_{3}\end{bmatrix}=\begin{bmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}\begin{bmatrix}U_{0}\\ U_{1}\\ U_{2}\\ U_{3}\end{bmatrix},
  90. γ = 1 1 - v 2 c 2 \gamma={1\over\sqrt{1-{v^{2}\over c^{2}}}}
  91. β = v c . \beta={v\over c}\,.
  92. O ( 3 , 1 ) O(3,1)
  93. P P
  94. T T
  95. ( P T ) (PT)
  96. v = ( c t , x , y , z ) = ( c t , 𝐫 ) v=(ct,x,y,z)=(ct,\mathbf{r})
  97. η ( v , v ) η(v,v)
  98. v v
  99. ( 0 , 0 , 0 , 0 ) (0,0,0,0)
  100. v + w v + w . \left\|v+w\right\|\geq\left\|v\right\|+\left\|w\right\|.
  101. T T

Minor_(linear_algebra).html

  1. ( - 1 ) i + j (-1)^{i+j}
  2. [ 1 4 7 3 0 5 - 1 9 11 ] \begin{bmatrix}\,\,\,1&4&7\\ \,\,\,3&0&5\\ -1&9&\!11\\ \end{bmatrix}
  3. M 23 = det [ 1 4 - 1 9 ] = det [ 1 4 - 1 9 ] = ( 9 - ( - 4 ) ) = 13 M_{23}=\det\begin{bmatrix}\,\,1&4&\Box\\ \,\Box&\Box&\Box\\ -1&9&\Box\\ \end{bmatrix}=\det\begin{bmatrix}\,\,\,1&4\\ -1&9\\ \end{bmatrix}=(9-(-4))=13
  4. C 23 = ( - 1 ) 2 + 3 ( M 23 ) = - 13. \ C_{23}=(-1)^{2+3}(M_{23})=-13.
  5. m = n m=n
  6. ( m k ) ( n k ) {m\choose k}\cdot{n\choose k}
  7. 1 i 1 < i 2 < < i k m 1\leq i_{1}<i_{2}<\ldots<i_{k}\leq m
  8. 1 j 1 < j 2 < < j k n 1\leq j_{1}<j_{2}<\ldots<j_{k}\leq n
  9. I I
  10. J J
  11. det ( ( A i p , j q ) p , q = 1 , , k ) \det\left((A_{i_{p},j_{q}})_{p,q=1,\ldots,k}\right)
  12. det I , J A \det_{I,J}A
  13. [ A ] I , J [A]_{I,J}
  14. M I , J M_{I,J}
  15. M i 1 , i 2 , , i k , j 1 , j 2 , , j k M_{i_{1},i_{2},\ldots,i_{k},j_{1},j_{2},\ldots,j_{k}}
  16. M ( i ) , ( j ) M_{(i),(j)}
  17. ( i ) (i)
  18. I I
  19. M i , j = det ( ( A p , q ) p i , q j ) M_{i,j}=\det\left(\left(A_{p,q}\right)_{p\neq i,q\neq j}\right)
  20. n × n n\times n
  21. ( a i j ) (a_{ij})
  22. det ( 𝐀 ) = a 1 j C 1 j + a 2 j C 2 j + a 3 j C 3 j + + a n j C n j = i = 1 n a i j C i j \ \det(\mathbf{A})=a_{1j}C_{1j}+a_{2j}C_{2j}+a_{3j}C_{3j}+...+a_{nj}C_{nj}=% \sum_{i=1}^{n}a_{ij}C_{ij}
  23. det ( 𝐀 ) = a i 1 C i 1 + a i 2 C i 2 + a i 3 C i 3 + + a i n C i n = j = 1 n a i j C i j \ \det(\mathbf{A})=a_{i1}C_{i1}+a_{i2}C_{i2}+a_{i3}C_{i3}+...+a_{in}C_{in}=% \sum_{j=1}^{n}a_{ij}C_{ij}
  24. 𝐂 = [ C 11 C 12 C 1 n C 21 C 22 C 2 n C n 1 C n 2 C n n ] \mathbf{C}=\begin{bmatrix}C_{11}&C_{12}&\cdots&C_{1n}\\ C_{21}&C_{22}&\cdots&C_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ C_{n1}&C_{n2}&\cdots&C_{nn}\end{bmatrix}
  25. 𝐀 - 1 = 1 det ( 𝐀 ) 𝐂 𝖳 . \mathbf{A}^{-1}=\frac{1}{\operatorname{det}(\mathbf{A})}\mathbf{C}^{\mathsf{T}}.
  26. 1 i 1 < i 2 < < i k n 1\leq i_{1}<i_{2}<\ldots<i_{k}\leq n
  27. 1 j 1 < j 2 < < j k n 1\leq j_{1}<j_{2}<\ldots<j_{k}\leq n
  28. n × n n\times n
  29. [ 𝐀 - 1 ] I , J = ± [ 𝐀 ] J , I det 𝐀 [\mathbf{A}^{-1}]_{I,J}=\pm\frac{[\mathbf{A}]_{J^{\prime},I^{\prime}}}{\det% \mathbf{A}}
  30. I , J I^{\prime},J^{\prime}
  31. I , J I,J
  32. 1 , , n 1,\ldots,n
  33. I I
  34. I I^{\prime}
  35. J J
  36. J J^{\prime}
  37. [ 𝐀 ] I , J [\mathbf{A}]_{I,J}
  38. I I
  39. J J
  40. [ 𝐀 ] I , J = det ( ( A i p , j q ) p , q = 1 , , k ) [\mathbf{A}]_{I,J}=\det\left((A_{i_{p},j_{q}})_{p,q=1,\ldots,k}\right)
  41. [ 𝐀 - 1 ] I , J ( e 1 e n ) = ± ( 𝐀 - 1 e j 1 ) ( 𝐀 - 1 e j k ) e i 1 e i n - k , [\mathbf{A}^{-1}]_{I,J}(e_{1}\wedge\ldots\wedge e_{n})=\pm(\mathbf{A}^{-1}e_{j% _{1}})\wedge\ldots\wedge(\mathbf{A}^{-1}e_{j_{k}})\wedge e_{i^{\prime}_{1}}% \wedge\ldots\wedge e_{i^{\prime}_{n-k}},
  42. e 1 , , e n e_{1},\ldots,e_{n}
  43. 𝐀 \mathbf{A}
  44. [ 𝐀 - 1 ] I , J det 𝐀 ( e 1 e n ) = ± ( e j 1 ) ( e j k ) ( 𝐀 e i 1 ) ( 𝐀 e i n - k ) = ± [ 𝐀 ] J , I ( e 1 e n ) . [\mathbf{A}^{-1}]_{I,J}\det\mathbf{A}(e_{1}\wedge\ldots\wedge e_{n})=\pm(e_{j_% {1}})\wedge\ldots\wedge(e_{j_{k}})\wedge(\mathbf{A}e_{i^{\prime}_{1}})\wedge% \ldots\wedge(\mathbf{A}e_{i^{\prime}_{n-k}})=\pm[\mathbf{A}]_{J^{\prime},I^{% \prime}}(e_{1}\wedge\ldots\wedge e_{n}).
  45. ( - 1 ) s = 1 k i s - s = 1 k j s (-1)^{\sum_{s=1}^{k}i_{s}-\sum_{s=1}^{k}j_{s}}
  46. I , J I,J
  47. [ 𝐀𝐁 ] I , J = K [ 𝐀 ] I , K [ 𝐁 ] K , J [\mathbf{AB}]_{I,J}=\sum_{K}[\mathbf{A}]_{I,K}[\mathbf{B}]_{K,J}\,
  48. ( 1 4 3 - 1 2 1 ) \begin{pmatrix}1&4\\ 3&\!\!-1\\ 2&1\\ \end{pmatrix}
  49. ( 𝐞 1 + 3 𝐞 2 + 2 𝐞 3 ) ( 4 𝐞 1 - 𝐞 2 + 𝐞 3 ) (\mathbf{e}_{1}+3\mathbf{e}_{2}+2\mathbf{e}_{3})\wedge(4\mathbf{e}_{1}-\mathbf% {e}_{2}+\mathbf{e}_{3})
  50. 𝐞 i 𝐞 i = 0 \mathbf{e}_{i}\wedge\mathbf{e}_{i}=0
  51. 𝐞 i 𝐞 j = - 𝐞 j 𝐞 i , \mathbf{e}_{i}\wedge\mathbf{e}_{j}=-\mathbf{e}_{j}\wedge\mathbf{e}_{i},
  52. - 13 𝐞 1 𝐞 2 - 7 𝐞 1 𝐞 3 + 5 𝐞 2 𝐞 3 -13\mathbf{e}_{1}\wedge\mathbf{e}_{2}-7\mathbf{e}_{1}\wedge\mathbf{e}_{3}+5% \mathbf{e}_{2}\wedge\mathbf{e}_{3}
  53. 𝐀 i j = ( - 1 ) i + j 𝐌 i j \mathbf{A}_{ij}=(-1)^{i+j}\mathbf{M}_{ij}
  54. 𝐀 - 1 = 1 det ( A ) [ A 11 A 21 A n 1 A 12 A 22 A n 2 A 1 n A 2 n A n n ] \mathbf{A}^{-1}=\frac{1}{\det(A)}\begin{bmatrix}A_{11}&A_{21}&\cdots&A_{n1}\\ A_{12}&A_{22}&\cdots&A_{n2}\\ \vdots&\vdots&\ddots&\vdots\\ A_{1n}&A_{2n}&\cdots&A_{nn}\end{bmatrix}

Mixed_radix.html

  1. i = 0 n ( ( [ i + 1 ] + 1 ) - 1 ) ( [ i ] + 1 ) ! = ( [ n + 1 ] + 1 ) ! - 1 \sum_{i=0}^{n}(([i+1]+1)-1)\cdot([i]+1)!=([n+1]+1)!-1
  2. i = 0 n ( m i + 1 - 1 ) M i = M n + 1 - 1 \sum_{i=0}^{n}(m_{i+1}-1)\cdot M_{i}=M_{n+1}-1
  3. M i = j = 1 i m j , m j > 1 , M 0 = 1 M_{i}=\prod_{j=1}^{i}m_{j},m_{j}>1,M_{0}=1
  4. i = 0 n ( p i + 1 - 1 ) p i # = p n + 1 # - 1 \sum_{i=0}^{n}(p_{i+1}-1)\cdot p_{i}\#=p_{n+1}\#-1
  5. p i # = j = 1 i p j p_{i}\#=\prod_{j=1}^{i}p_{j}

Mixed_tensor.html

  1. ( M N ) \scriptstyle{\left({{M}\atop{N}}\right)}
  2. T α β γ , T α β , γ T α , β γ T α , β γ T α , β γ T α , β γ T α β , γ T α β γ T_{\alpha\beta\gamma},\ T_{\alpha\beta}{}^{\gamma},\ T_{\alpha}{}^{\beta}{}_{% \gamma},\ T_{\alpha}{}^{\beta\gamma},\ T^{\alpha}{}_{\beta\gamma},\ T^{\alpha}% {}_{\beta}{}^{\gamma},\ T^{\alpha\beta}{}_{\gamma},\ T^{\alpha\beta\gamma}
  3. T α β = λ T α β γ g γ λ T_{\alpha\beta}{}^{\lambda}=T_{\alpha\beta\gamma}\,g^{\gamma\lambda}
  4. T α β λ T_{\alpha\beta}{}^{\lambda}
  5. T α β γ T_{\alpha\beta}{}^{\gamma}
  6. T α β δ λ λ = γ T α β γ T_{\alpha\beta}{}^{\lambda}\,\delta_{\lambda}{}^{\gamma}=T_{\alpha\beta}{}^{\gamma}
  7. T α = λ γ T α β γ g β λ , T_{\alpha}{}^{\lambda}{}_{\gamma}=T_{\alpha\beta\gamma}\,g^{\beta\lambda},
  8. T α = λ ϵ T α β γ g β λ g γ ϵ , T_{\alpha}{}^{\lambda\epsilon}=T_{\alpha\beta\gamma}\,g^{\beta\lambda}\,g^{% \gamma\epsilon},
  9. T α β = γ g γ λ T α β λ , T^{\alpha\beta}{}_{\gamma}=g_{\gamma\lambda}\,T^{\alpha\beta\lambda},
  10. T α = λ ϵ g λ β g ϵ γ T α β γ . T^{\alpha}{}_{\lambda\epsilon}=g_{\lambda\beta}\,g_{\epsilon\gamma}\,T^{\alpha% \beta\gamma}.
  11. g μ λ g λ ν = g μ = ν δ μ ν g^{\mu\lambda}\,g_{\lambda\nu}=g^{\mu}{}_{\nu}=\delta^{\mu}{}_{\nu}

Mixture.html

  1. h i = ( c i - c batch ) m i c batch m aver . h_{i}=\frac{(c_{i}-c\text{batch})m_{i}}{c\text{batch}m\text{aver}}.
  2. h i h_{i}
  3. c i c_{i}
  4. c batch c\text{batch}
  5. m i m_{i}
  6. m aver m\text{aver}
  7. i i
  8. i i
  9. i i
  10. V = 1 ( i = 1 N q i m i ) 2 i = 1 N q i ( 1 - q i ) m i 2 ( a i - j = 1 N q j a j m j j = 1 N q j m j ) 2 . V=\frac{1}{(\sum_{i=1}^{N}q_{i}m_{i})^{2}}\sum_{i=1}^{N}q_{i}(1-q_{i})m_{i}^{2% }\left(a_{i}-\frac{\sum_{j=1}^{N}q_{j}a_{j}m_{j}}{\sum_{j=1}^{N}q_{j}m_{j}}% \right)^{2}.
  11. V = 1 - q q M batch 2 i = 1 N m i 2 ( a i - a batch ) 2 . V=\frac{1-q}{qM\text{batch}^{2}}\sum_{i=1}^{N}m_{i}^{2}\left(a_{i}-a\text{% batch}\right)^{2}.

Mode-locking.html

  1. Δ ν = c 2 L \Delta\nu=\frac{c}{2L}
  2. Δ t = 0.441 N Δ ν . \Delta t=\frac{0.441}{N\Delta\nu}.

Modified_discrete_cosine_transform.html

  1. F : 𝐑 2 N 𝐑 N F\colon\mathbf{R}^{2N}\to\mathbf{R}^{N}
  2. X k = n = 0 2 N - 1 x n cos [ π N ( n + 1 2 + N 2 ) ( k + 1 2 ) ] X_{k}=\sum_{n=0}^{2N-1}x_{n}\cos\left[\frac{\pi}{N}\left(n+\frac{1}{2}+\frac{N% }{2}\right)\left(k+\frac{1}{2}\right)\right]
  3. y n = 1 N k = 0 N - 1 X k cos [ π N ( n + 1 2 + N 2 ) ( k + 1 2 ) ] y_{n}=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}\cos\left[\frac{\pi}{N}\left(n+\frac{1}{% 2}+\frac{N}{2}\right)\left(k+\frac{1}{2}\right)\right]
  4. w n 2 + w n + N 2 = 1 w_{n}^{2}+w_{n+N}^{2}=1
  5. w n = sin [ π 2 N ( n + 1 2 ) ] w_{n}=\sin\left[\frac{\pi}{2N}\left(n+\frac{1}{2}\right)\right]
  6. w n = sin ( π 2 sin 2 [ π 2 N ( n + 1 2 ) ] ) w_{n}=\sin\left(\frac{\pi}{2}\sin^{2}\left[\frac{\pi}{2N}\left(n+\frac{1}{2}% \right)\right]\right)
  7. cos [ π N ( - n - 1 + 1 2 ) ( k + 1 2 ) ] = cos [ π N ( n + 1 2 ) ( k + 1 2 ) ] \cos\left[\frac{\pi}{N}\left(-n-1+\frac{1}{2}\right)\left(k+\frac{1}{2}\right)% \right]=\cos\left[\frac{\pi}{N}\left(n+\frac{1}{2}\right)\left(k+\frac{1}{2}% \right)\right]
  8. cos [ π N ( 2 N - n - 1 + 1 2 ) ( k + 1 2 ) ] = - cos [ π N ( n + 1 2 ) ( k + 1 2 ) ] \cos\left[\frac{\pi}{N}\left(2N-n-1+\frac{1}{2}\right)\left(k+\frac{1}{2}% \right)\right]=-\cos\left[\frac{\pi}{N}\left(n+\frac{1}{2}\right)\left(k+\frac% {1}{2}\right)\right]
  9. ( A , B ) (A,B)
  10. ( B , C ) (B,C)
  11. ( B + B R ) / 2 + ( B - B R ) / 2 = B (B+B_{R})/2+(B-B_{R})/2=B
  12. ( W , W R ) (W,W_{R})
  13. W + W R 2 = ( 1 , 1 , ) W+W_{R}^{2}=(1,1,\ldots)
  14. ( A , B ) (A,B)
  15. ( W A , W R B ) (WA,W_{R}B)
  16. W R ( W R B + ( W R B ) R ) = W R ( W R B + W B R ) = W R 2 B + W W R B R W_{R}\cdot(W_{R}B+(W_{R}B)_{R})=W_{R}\cdot(W_{R}B+WB_{R})=W_{R}^{2}B+WW_{R}B_{R}
  17. ( B , C ) (B,C)
  18. W ( W B - W R B R ) = W 2 B - W W R B R W\cdot(WB-W_{R}B_{R})=W^{2}B-WW_{R}B_{R}
  19. ( W R 2 B + W W R B R ) + ( W 2 B - W W R B R ) = ( W R 2 + W 2 ) B = B , (W_{R}^{2}B+WW_{R}B_{R})+(W^{2}B-WW_{R}B_{R})=\left(W_{R}^{2}+W^{2}\right)B=B,

Modigliani–Miller_theorem.html

  1. V U = V L V_{U}=V_{L}\,
  2. V U V_{U}
  3. V L V_{L}
  4. r E ( L e v e r e d ) = r E ( U n l e v e r e d ) + D E ( r E ( U n l e v e r e d ) - r D ) r_{E}(Levered)=r_{E}(Unlevered)+\frac{D}{E}(r_{E}(Unlevered)-r_{D})
  5. r E r_{E}
  6. r D r_{D}
  7. D E \frac{D}{E}
  8. V L = V U + T C D V_{L}=V_{U}+T_{C}D\,
  9. V L V_{L}
  10. V U V_{U}
  11. T C D T_{C}D
  12. T C T_{C}
  13. T C D T_{C}D
  14. r E = r 0 + D E ( r 0 - r D ) ( 1 - T C ) r_{E}=r_{0}+\frac{D}{E}(r_{0}-r_{D})(1-T_{C})
  15. r E r_{E}
  16. r 0 r_{0}
  17. r D r_{D}
  18. D / E {D}/{E}
  19. T c T_{c}
  20. T C T_{C}

Modular_form.html

  1. S L ( 2 , 𝐙 ) = { ( a b c d ) | a , b , c , d 𝐙 , a d - b c = 1 } SL(2,\mathbf{Z})=\left\{\left.\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\right|a,b,c,d\in\mathbf{Z},\ ad-bc=1\right\}
  2. f ( a z + b c z + d ) = ( c z + d ) k f ( z ) f\left(\frac{az+b}{cz+d}\right)=(cz+d)^{k}f(z)
  3. z i z→i∞
  4. S = ( 0 - 1 1 0 ) S=\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right)
  5. T = ( 1 1 0 1 ) T=\left(\begin{array}[]{cc}1&1\\ 0&1\end{array}\right)
  6. f ( - 1 / z ) = z k f ( z ) f(-1/z)=z^{k}f(z)\,
  7. f ( z + 1 ) = f ( z ) f(z+1)=f(z)\,
  8. f ( z + 1 ) = f ( z ) f(z+1)=f(z)
  9. Λ = α , z \Lambda=\langle\alpha,z\rangle
  10. 1 , ω \langle 1,\omega\rangle
  11. E k ( Λ ) = λ Λ - 0 λ - k . E_{k}(\Lambda)=\sum_{\lambda\in\Lambda-0}\lambda^{-k}.
  12. ϑ L ( z ) = λ L e π i λ 2 z \vartheta_{L}(z)=\sum_{\lambda\in L}e^{\pi i\|\lambda\|^{2}z}
  13. ϑ L 8 × L 8 ( z ) = ϑ L 16 ( z ) , \vartheta_{L_{8}\times L_{8}}(z)=\vartheta_{L_{16}}(z),
  14. η ( z ) = q 1 / 24 n = 1 ( 1 - q n ) , q = e 2 π i z . \eta(z)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n}),\ q=e^{2\pi iz}.
  15. ( a b c d ) \left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)
  16. f ( a z + b c z + d ) = f ( z ) f\left(\frac{az+b}{cz+d}\right)=f(z)
  17. f ( z ) = n = - m a n e 2 i π n z . f(z)=\sum_{n=-m}^{\infty}a_{n}e^{2i\pi nz}.
  18. q = exp ( 2 π i z ) q=\exp(2\pi iz)
  19. f ( z ) = n = - m a n q n . f(z)=\sum_{n=-m}^{\infty}a_{n}q^{n}.
  20. a n a_{n}
  21. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  22. Γ 0 ( N ) = { ( a b c d ) S L 2 ( 𝐙 ) : c 0 ( mod N ) } \Gamma_{0}(N)=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_{2}(\mathbf{Z}):c\equiv 0\;\;(\mathop{{\rm mod}}N)\right\}
  23. Γ ( N ) = { ( a b c d ) S L 2 ( 𝐙 ) : c b 0 , a d 1 ( mod N ) } . \Gamma(N)=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_{2}(\mathbf{Z}):c\equiv b\equiv 0,a\equiv d\equiv 1\;\;% (\mathop{{\rm mod}}N)\right\}.
  24. dim 𝐂 M k ( S L ( 2 , 𝐙 ) ) = { k / 12 k 2 ( mod 12 ) k / 12 + 1 else \,\text{dim}_{\mathbf{C}}M_{k}(SL(2,\mathbf{Z}))=\left\{\begin{array}[]{ll}% \lfloor k/12\rfloor&k\equiv 2\;\;(\mathop{{\rm mod}}12)\\ \lfloor k/12\rfloor+1&\,\text{else}\end{array}\right.
  25. - \lfloor-\rfloor
  26. ε ( a , b , c , d ) \varepsilon(a,b,c,d)
  27. | ε ( a , b , c , d ) | = 1 \left|\varepsilon(a,b,c,d)\right|=1
  28. f ( a z + b c z + d ) = ε ( a , b , c , d ) ( c z + d ) k f ( z ) . f\left(\frac{az+b}{cz+d}\right)=\varepsilon(a,b,c,d)(cz+d)^{k}f(z).
  29. ε ( a , b , c , d ) ( c z + d ) k \varepsilon(a,b,c,d)(cz+d)^{k}
  30. Γ 0 ( N ) \Gamma_{0}(N)
  31. ( a b c d ) Γ 0 ( N ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma_{0}(N)
  32. f ( a z + b c z + d ) = χ ( d ) ( c z + d ) k f ( z ) f\left(\frac{az+b}{cz+d}\right)=\chi(d)(cz+d)^{k}f(z)
  33. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}

Modularity_theorem.html

  1. X 0 ( N ) X_{0}(N)
  2. L ( E , s ) = n = 1 a n n s . L(E,s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.
  3. a n a_{n}
  4. f ( E , q ) = n = 1 a n q n . f(E,q)=\sum_{n=1}^{\infty}a_{n}q^{n}.
  5. q = e 2 π i τ q=e^{2\pi i\tau}
  6. f ( E , τ ) f(E,\tau)
  7. f f

Module_(mathematics).html

  1. r ( x + y ) = r x + r y r\cdot(x+y)=r\cdot x+r\cdot y
  2. ( r + s ) x = r x + s x (r+s)\cdot x=r\cdot x+s\cdot x
  3. ( r s ) x = r ( s x ) (rs)\cdot x=r\cdot(s\cdot x)
  4. 1 R x = x . 1_{R}\cdot x=x.
  5. f ( r m + s n ) = r f ( m ) + s f ( n ) f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)

Molality.html

  1. n s o l u t e n_{solute}
  2. m s o l v e n t m_{solvent}
  3. b = n s o l u t e m s o l v e n t b=\frac{n_{solute}}{m_{solvent}}
  4. b M s o l u t e = m s o l u t e m s o l v e n t = w s o l u t e w s o l v e n t , b\,M_{solute}=\frac{m_{solute}}{m_{solvent}}=\frac{w_{solute}}{w_{solvent}},
  5. b M s o l v e n t = n s o l u t e n s o l v e n t = x s o l u t e x s o l v e n t . b\,M_{solvent}=\frac{n_{solute}}{n_{solvent}}=\frac{x_{solute}}{x_{solvent}}.
  6. b 0 = < m t p l > n 0 n 0 M 0 = M 0 - 1 . b_{0}=\frac{<}{m}tpl>{{n_{0}}}{{n_{0}M_{0}}}=M_{0}^{-1}.
  7. w w
  8. w = ( 1 + ( b M ) - 1 ) - 1 , b = w ( 1 - w ) M , w=(1+(b\,M)^{-1})^{-1},\ b=\frac{w}{(1-w)M},
  9. w i = w 0 b i M i , b i = w i w 0 M i , w_{i}=w_{0}\,b_{i}M_{i},\ b_{i}=\frac{w_{i}}{w_{0}M_{i}},
  10. w 0 = ( 1 + j = 1 n b j M j ) - 1 = 1 - j = 1 n w j . w_{0}=\left(1+\sum_{j=1}^{n}{b_{j}M_{j}}\right)^{-1}=1-\sum_{j=1}^{n}{w_{j}}.
  11. x = ( 1 + ( M 0 b ) - 1 ) - 1 , b = x M 0 ( 1 - x ) , x=(1+(M_{0}\,b)^{-1})^{-1},\ b=\frac{x}{M_{0}(1-x)},
  12. x i = x 0 M 0 b i , b i = b 0 x i x 0 , x_{i}=x_{0}M_{0}\,b_{i},\ b_{i}=\frac{b_{0}x_{i}}{x_{0}},
  13. x 0 = ( 1 + M 0 j = 1 n b j ) - 1 = 1 - j = 1 n x j . x_{0}=\left(1+M_{0}\sum_{j=1}^{n}{b_{j}}\right)^{-1}=1-\sum_{j=1}^{n}{x_{j}}.
  14. c = ρ b 1 + b M , b = c ρ - c M , c=\frac{\rho\,b}{1+bM},\ b=\frac{c}{\rho-cM},
  15. c i = c 0 M 0 b i , b i = b 0 c i c 0 , c_{i}=c_{0}M_{0}\,b_{i},\ b_{i}=\frac{b_{0}c_{i}}{c_{0}},
  16. c 0 = ρ b 0 1 + j = 1 n b j M j = ρ - j = 1 n c i M i M 0 . c_{0}=\frac{\rho\,b_{0}}{1+\displaystyle\sum_{j=1}^{n}{b_{j}M_{j}}}=\frac{\rho% -\displaystyle\sum_{j=1}^{n}{c_{i}M_{i}}}{M_{0}}.
  17. ρ s o l u t e = ρ b M 1 + b M , b = ρ s o l u t e M ( ρ - ρ s o l u t e ) , \rho_{solute}=\frac{\rho\,b\,M}{1+bM},\ b=\frac{\rho_{solute}}{M(\rho-\rho_{% solute})},
  18. ρ i = ρ 0 b i M i , b i = ρ i ρ 0 M i , \rho_{i}=\rho_{0}\,b_{i}M_{i},\ b_{i}=\frac{\rho_{i}}{\rho_{0}M_{i}},
  19. ρ 0 = ρ 1 + j = 1 n b j M j = ρ - j = 1 n ρ i . \rho_{0}=\frac{\rho}{1+\displaystyle\sum_{j=1}^{n}b_{j}M_{j}}=\rho-\sum_{j=1}^% {n}{\rho_{i}}.
  20. b i b j = x i x j = c i c j = ρ i M j ρ j M i = w i M j w j M i , \frac{b_{i}}{b_{j}}=\frac{x_{i}}{x_{j}}=\frac{c_{i}}{c_{j}}=\frac{\rho_{i}\,M_% {j}}{\rho_{j}\,M_{i}}=\frac{w_{i}\,M_{j}}{w_{j}\,M_{i}},
  21. w H N O 3 \displaystyle w_{HNO_{3}}
  22. M H N O 3 = 0.063 , M H F = 0.020 , M H 2 O = 0.018. M_{HNO_{3}}=0.063,\ M_{HF}=0.020,\ M_{H_{2}O}=0.018.
  23. b H 2 O = ( M H 2 O ) - 1 = 1 0.018 , b_{H_{2}O}=(M_{H_{2}O})^{-1}=\frac{1}{0.018},
  24. b H N O 3 b H 2 O = w H N O 3 M H 2 O w H 2 O M H N O 3 b H N O 3 = 18.83. \frac{b_{HNO_{3}}}{b_{H_{2}O}}=\frac{w_{HNO_{3}}M_{H_{2}O}}{w_{H_{2}O}M_{HNO_{% 3}}}\ \therefore b_{HNO_{3}}=18.83.
  25. b H F = w H F w H 2 O M H F = 2.19. b_{HF}=\frac{w_{HF}}{w_{H_{2}O}M_{HF}}=2.19.
  26. x H 2 O = ( 1 + M H 2 O ( b H N O 3 + b H F ) ) - 1 = 0.726 , \displaystyle x_{H_{2}O}=(1+M_{H_{2}O}(b_{HNO_{3}}+b_{HF}))^{-1}=0.726,
  27. V ~ 1 ϕ = 1 b ( 1 ρ - 1 ρ 0 0 ) + M 1 ρ {}^{\phi}\tilde{V}_{1}=\frac{1}{b}(\frac{1}{\rho}-\frac{1}{\rho_{0}^{0}})+% \frac{M_{1}}{\rho}
  28. V ~ i ϕ = 1 b j ( 1 ρ - 1 ρ 0 0 ) + b j M j b j ρ {}^{\phi}\tilde{V}_{i}=\frac{1}{\sum b_{j}}(\frac{1}{\rho}-\frac{1}{\rho_{0}^{% 0}})+\frac{\sum b_{j}M_{j}}{\sum b_{j}\rho}

Molar_concentration.html

  1. c = n V = N N A V = C N A . c=\frac{n}{V}=\frac{N}{N_{\rm A}\,V}=\frac{C}{N_{\rm A}}.
  2. × 10 2 3 \times 10^{2}3
  3. C i C_{i}
  4. C i = c i N A C_{i}=c_{i}\cdot N_{\rm A}
  5. N A N_{\rm A}
  6. × 10 2 3 \times 10^{2}3
  7. ρ i \rho_{i}
  8. ρ i = c i M i \rho_{i}=c_{i}\cdot M_{i}
  9. M i M_{i}
  10. i i
  11. x i x_{i}
  12. x i = c i M ρ = c i i x i M i ρ x_{i}=c_{i}\cdot\frac{M}{\rho}=c_{i}\cdot\frac{\sum_{i}x_{i}M_{i}}{\rho}
  13. x i = c i x j M j ρ - c i M i x_{i}=c_{i}\cdot\frac{\sum x_{j}M_{j}}{\rho-c_{i}M_{i}}
  14. M M
  15. ρ \rho
  16. x i = c i c = c i c i x_{i}=\frac{c_{i}}{c}=\frac{c_{i}}{\sum c_{i}}
  17. w i w_{i}
  18. w i = c i M i ρ w_{i}=c_{i}\cdot\frac{M_{i}}{\rho}
  19. b 2 = < m t p l > c 2 ρ - c 2 M 2 b_{2}=\frac{<}{m}tpl>{{c_{2}}}{{\rho-c_{2}\cdot M_{2}}}\,
  20. b i = < m t p l > c i ρ - c i M i b_{i}=\frac{<}{m}tpl>{{c_{i}}}{{\rho-\sum c_{i}\cdot M_{i}}}\,
  21. i c i V i ¯ = 1 \sum_{i}c_{i}\cdot\bar{V_{i}}=1
  22. c i = c i , T 0 < m t p l > ( 1 + α Δ T ) c_{i}=\frac{{c_{i,T_{0}}}}{<}mtpl>{{(1+\alpha\cdot\Delta T)}}
  23. c i , T 0 c_{i,T_{0}}
  24. α \alpha
  25. ρ \rho
  26. ρ \rho
  27. V V
  28. c c
  29. m m
  30. c c
  31. c c
  32. c c
  33. 10 - 15 10^{-15}
  34. C C
  35. C C
  36. × 10 1 6 \times 10^{1}6
  37. c = C / N A c=C/N_{A}
  38. × 10 1 6 \times 10^{1}6
  39. × 10 2 3 \times 10^{2}3
  40. c c
  41. c c
  42. c c

Molar_mass.html

  1. M ¯ \bar{M}
  2. x i x_{i}
  3. M i M_{i}
  4. M ¯ = i x i M i \bar{M}=\sum_{i}x_{i}M_{i}\,
  5. w i w_{i}
  6. 1 / M ¯ = i < m t p l > w i M i , 1/\bar{M}=\sum_{i}\frac{<}{m}tpl>{{w_{i}}}{{M_{i}}},
  7. p V = n R T pV=nRT
  8. ρ = n M V . \rho={{nM}\over{V}}.
  9. M = R T ρ p M={{RT\rho}\over{p}}
  10. M = w K f Δ T . M={{wK_{f}}\over{\Delta T}}.
  11. M = w K b Δ T . M={{wK_{b}}\over{\Delta T}}.

Molar_volume.html

  1. V m = M ρ V_{\rm m}={M\over\rho}
  2. V m = i = 1 N x i M i ρ mixture V_{\rm m}=\frac{\displaystyle\sum_{i=1}^{N}x_{i}M_{i}}{\rho_{\mathrm{mixture}}}
  3. V m = V n = R T P V_{\rm m}={V\over{n}}={{RT}\over{P}}
  4. V m = N A V cell Z V_{\rm m}={{N_{\rm A}V_{\rm cell}}\over{Z}}

Molecular_clock.html

  1. μ \mu
  2. μ \mu
  3. μ \mu

Molecular_dynamics.html

  1. O ( n 2 ) O(n^{2})
  2. O ( n log ( n ) ) O(n\log(n))
  3. O ( n ) O(n)
  4. X X
  5. V V
  6. F ( X ) = - U ( X ) = M V ˙ ( t ) F(X)=-\nabla U(X)=M\dot{V}(t)
  7. V ( t ) = X ˙ ( t ) . V(t)=\dot{X}(t).
  8. U ( X ) U(X)
  9. X X
  10. F F
  11. U ( X ) U(X)
  12. X X
  13. V V
  14. X X
  15. V V
  16. 10 10 10^{10}
  17. U ( r ) = 4 ε [ ( σ r ) 12 - ( σ r ) 6 ] U(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{% r}\right)^{6}\right]
  18. U i j ( r i j ) = z i z j 4 π ϵ 0 1 r i j + A l exp - r i j p l + C l r i j - n l + U_{ij}(r_{ij})=\frac{z_{i}z_{j}}{4\pi\epsilon_{0}}\frac{1}{r_{ij}}+A_{l}\exp% \frac{-r_{ij}}{p_{l}}+C_{l}r_{ij}^{-n_{l}}+\cdots

Moment-generating_function.html

  1. M X ( t ) := 𝔼 [ e t X ] , t , M_{X}(t):=\mathbb{E}\!\left[e^{tX}\right],\quad t\in\mathbb{R},
  2. M X ( 0 ) M_{X}(0)
  3. 𝐗 = ( X 1 , , X n ) \mathbf{X}=(X_{1},\ldots,X_{n})
  4. 𝐭 𝐗 = 𝐭 T 𝐗 \mathbf{t}\cdot\mathbf{X}=\mathbf{t}^{\mathrm{T}}\mathbf{X}
  5. M 𝐗 ( 𝐭 ) := 𝔼 ( e 𝐭 T 𝐗 ) . M_{\mathbf{X}}(\mathbf{t}):=\mathbb{E}\!\left(e^{\mathbf{t}^{\mathrm{T}}% \mathbf{X}}\right).
  6. e t X = 1 + t X + t 2 X 2 2 ! + t 3 X 3 3 ! + + t n X n n ! + . e^{t\,X}=1+t\,X+\frac{t^{2}\,X^{2}}{2!}+\frac{t^{3}\,X^{3}}{3!}+\cdots+\frac{t% ^{n}\,X^{n}}{n!}+\cdots.
  7. M X ( t ) = 𝔼 ( e t X ) \displaystyle M_{X}(t)=\mathbb{E}(e^{t\,X})
  8. P ( X = 1 ) = p \,P(X=1)=p
  9. 1 - p + p e t \,1-p+pe^{t}
  10. 1 - p + p e i t \,1-p+pe^{it}
  11. ( 1 - p ) k - 1 p (1-p)^{k-1}\,p\!
  12. p e t 1 - ( 1 - p ) e t \frac{pe^{t}}{1-(1-p)e^{t}}\!
  13. t < - ln ( 1 - p ) \forall t<-\ln(1-p)\!
  14. p e i t 1 - ( 1 - p ) e i t \frac{pe^{it}}{1-(1-p)\,e^{it}}\!
  15. ( 1 - p + p e t ) n \,(1-p+pe^{t})^{n}
  16. ( 1 - p + p e i t ) n \,(1-p+pe^{it})^{n}
  17. e λ ( e t - 1 ) \,e^{\lambda(e^{t}-1)}
  18. e λ ( e i t - 1 ) \,e^{\lambda(e^{it}-1)}
  19. e t b - e t a t ( b - a ) \,\frac{e^{tb}-e^{ta}}{t(b-a)}
  20. e i t b - e i t a i t ( b - a ) \,\frac{e^{itb}-e^{ita}}{it(b-a)}
  21. e a t - e ( b + 1 ) t ( b - a + 1 ) ( 1 - e t ) \,\frac{e^{at}-e^{(b+1)t}}{(b-a+1)(1-e^{t})}
  22. e a i t - e ( b + 1 ) i t ( b - a + 1 ) ( 1 - e i t ) \,\frac{e^{ait}-e^{(b+1)it}}{(b-a+1)(1-e^{it})}
  23. e t μ + 1 2 σ 2 t 2 \,e^{t\mu+\frac{1}{2}\sigma^{2}t^{2}}
  24. e i t μ - 1 2 σ 2 t 2 \,e^{it\mu-\frac{1}{2}\sigma^{2}t^{2}}
  25. ( 1 - 2 t ) - k / 2 \,(1-2t)^{-k/2}
  26. ( 1 - 2 i t ) - k / 2 \,(1-2it)^{-k/2}
  27. ( 1 - t θ ) - k \,(1-t\theta)^{-k}
  28. ( 1 - i t θ ) - k \,(1-it\theta)^{-k}
  29. ( 1 - t λ - 1 ) - 1 , ( t < λ ) \,(1-t\lambda^{-1})^{-1},\,(t<\lambda)
  30. ( 1 - i t λ - 1 ) - 1 \,(1-it\lambda^{-1})^{-1}
  31. e t T μ + 1 2 t T Σ t \,e^{t^{\mathrm{T}}\mu+\frac{1}{2}t^{\mathrm{T}}\Sigma t}
  32. e i t T μ - 1 2 t T Σ t \,e^{it^{\mathrm{T}}\mu-\frac{1}{2}t^{\mathrm{T}}\Sigma t}
  33. e t a \,e^{ta}
  34. e i t a \,e^{ita}
  35. e t μ 1 - b 2 t 2 \,\frac{e^{t\mu}}{1-b^{2}t^{2}}
  36. e i t μ 1 + b 2 t 2 \,\frac{e^{it\mu}}{1+b^{2}t^{2}}
  37. ( 1 - p ) r ( 1 - p e t ) r \,\frac{(1-p)^{r}}{(1-pe^{t})^{r}}
  38. ( 1 - p ) r ( 1 - p e i t ) r \,\frac{(1-p)^{r}}{(1-pe^{it})^{r}}
  39. e i t μ - θ | t | \,e^{it\mu-\theta|t|}
  40. M X ( t ) = - e t x d F ( x ) M_{X}(t)=\int_{-\infty}^{\infty}e^{tx}\,dF(x)
  41. M X ( t ) = i = 1 e t x i p i M_{X}(t)=\sum_{i=1}^{\infty}e^{tx_{i}}\,p_{i}
  42. M X ( t ) = - e t x f ( x ) d x M_{X}(t)=\int_{-\infty}^{\infty}e^{tx}f(x)\,dx
  43. M X ( t ) \displaystyle M_{X}(t)
  44. M S n ( t ) = M X 1 ( a 1 t ) M X 2 ( a 2 t ) M X n ( a n t ) . M_{S_{n}}(t)=M_{X_{1}}(a_{1}t)M_{X_{2}}(a_{2}t)\cdots M_{X_{n}}(a_{n}t)\,.
  45. M X ( t ) = E ( e t , X ) M_{X}(t)=E\left(e^{\langle t,X\rangle}\right)
  46. , \langle\cdot,\cdot\rangle
  47. M X ( t ) = M Y ( t ) , M_{X}(t)=M_{Y}(t),\,
  48. F X ( x ) = F Y ( x ) F_{X}(x)=F_{Y}(x)\,
  49. lim n i = 0 n t i m i i ! \lim_{n\rightarrow\infty}\sum_{i=0}^{n}\frac{t^{i}m_{i}}{i!}
  50. m n = E ( X n ) = M X ( n ) ( 0 ) = d n M X d t n ( 0 ) . m_{n}=E\left(X^{n}\right)=M_{X}^{(n)}(0)=\frac{d^{n}M_{X}}{dt^{n}}(0).
  51. φ X ( t ) \varphi_{X}(t)
  52. φ X ( t ) = M i X ( t ) = M X ( i t ) : \varphi_{X}(t)=M_{iX}(t)=M_{X}(it):
  53. G ( z ) = E [ z X ] . G(z)=E[z^{X}].\,
  54. G ( e t ) = E [ e t X ] = M X ( t ) . G(e^{t})=E[e^{tX}]=M_{X}(t).\,

Moment_(physics).html

  1. μ n = r n Q \mu_{n}=r^{n}\,Q
  2. Q Q
  3. μ n = r n ρ ( r ) d r \mu_{n}=\int r^{n}\,\rho(r)\,dr
  4. ρ \rho
  5. r n ρ ( r ) r^{n}\,\rho(r)
  6. r r
  7. τ = r F \mathbf{\tau}=rF
  8. 𝐫 × 𝐅 \mathbf{r}\times\mathbf{F}
  9. 𝐩 = q 𝐝 \mathbf{p}=q\,\mathbf{d}
  10. 𝐫 ρ ( 𝐫 ) d 3 r \int\mathbf{r}\,\rho(\mathbf{r})\,d^{3}r
  11. ρ ( 𝐫 ) \rho(\mathbf{r})
  12. I = r 2 m I=r^{2}m
  13. i r i 2 m i \sum_{i}r_{i}^{2}m_{i}
  14. r 2 ρ ( 𝐫 ) d 3 r \int r^{2}\rho(\mathbf{r})\,d^{3}r
  15. ρ ( 𝐫 ) \rho(\mathbf{r})
  16. Φ ( 𝐫 ) = ρ ( 𝐫 ) | 𝐫 - 𝐫 | d 3 r = l = 0 m = - l l ( 4 π 2 l + 1 ) q l m Y l m ( θ , ϕ ) r l + 1 \Phi(\mathbf{r})=\int\frac{\rho(\mathbf{r^{\prime}})}{|\mathbf{r}-\mathbf{r^{% \prime}}|}d^{3}r^{\prime}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\left(\frac{4\pi}{% 2l+1}\right)q_{lm}\,\frac{Y_{lm}(\theta,\phi)}{r^{l+1}}
  17. q l m q_{lm}
  18. q l m = ( r ) l ρ ( 𝐫 ) Y l m * ( θ , ϕ ) d 3 r q_{lm}=\int(r^{\prime})^{l}\,\rho(\mathbf{r^{\prime}})\,Y^{*}_{lm}(\theta^{% \prime},\phi^{\prime})\,d^{3}r^{\prime}
  19. 𝐫 \mathbf{r}^{\prime}
  20. ( r , ϕ , θ ) (r^{\prime},\phi^{\prime},\theta^{\prime})
  21. ρ \rho
  22. q l m q_{lm}
  23. q 00 q_{00}
  24. q 1 m q_{1m}
  25. q 2 m q_{2m}
  26. ρ \rho
  27. ( I = Σ m r 2 ) (I=\Sigma mr^{2})
  28. ( 𝐋 = 𝐫 × m 𝐯 ) (\mathbf{L}=\mathbf{r}\times m\mathbf{v})
  29. ( μ = I 𝐀 ) (\mathbf{\mu}=I\mathbf{A})
  30. ( 𝐩 = q 𝐝 ) (\mathbf{p}=q\mathbf{d})

Moment_of_inertia.html

  1. I = m r 2 d m I=\int_{m}r^{2}\mathrm{d}m
  2. I = L ω . I=\frac{L}{\omega}.
  3. τ = I α . \tau=I\alpha.
  4. I = m r 2 . I=mr^{2}.
  5. I = m k 2 , I=mk^{2},
  6. I = m r 2 . I=mr^{2}.
  7. s y m b o l τ = 𝐫 × 𝐅 symbol\tau=\mathbf{r}\times\mathbf{F}
  8. s y m b o l α symbol\alpha
  9. a = s y m b o l α × r a=symbol\alpha\times r
  10. F = m a F=ma
  11. s y m b o l τ = 𝐫 × 𝐅 = 𝐫 × ( m s y m b o l α × r ) = ( m r 2 ) s y m b o l α = I α e , symbol\tau=\mathbf{r}\times\mathbf{F}=\mathbf{r}\times(msymbol\alpha\times r)=% (mr^{2})symbol\alpha=I\alpha e,
  12. s y m b o l α symbol\alpha
  13. v = ω × 𝐫 v=\mathbf{ω}×\mathbf{r}
  14. ω \mathbf{ω}
  15. 𝐋 = 𝐫 × ( m 𝐯 ) = ( m r 2 ) s y m b o l ω = I ω e , \mathbf{L}=\mathbf{r}\times(m\mathbf{v})=(mr^{2})symbol\omega=I\omega e,
  16. E K = 1 2 m 𝐯 𝐯 = 1 2 ( m r 2 ) ω 2 = 1 2 I ω 2 . E\text{K}=\frac{1}{2}m\mathbf{v}\cdot\mathbf{v}=\frac{1}{2}(mr^{2})\omega^{2}=% \frac{1}{2}I\omega^{2}.
  17. m r mr
  18. 2 {}^{2}
  19. ω n \omega_{n}
  20. I P I_{P}
  21. ω n = m g r I P , \omega_{n}=\sqrt{\frac{mgr}{I_{P}}},
  22. m m
  23. g g
  24. r r
  25. P {}_{P}
  26. t t
  27. I P = m g r ω n 2 = m g r t 2 4 π 2 , I_{P}=\frac{mgr}{\omega_{n}^{2}}=\frac{mgrt^{2}}{4\pi^{2}},
  28. t t
  29. I C I_{C}
  30. I C = I P - m r 2 , I_{C}=I_{P}-mr^{2},
  31. m m
  32. r r
  33. P {}_{P}
  34. C {}_{C}
  35. K K
  36. I C I_{C}
  37. m m
  38. K = I C m . K=\sqrt{\frac{I_{C}}{m}}.
  39. L L
  40. L L
  41. ω n = g L = m g r I P , \omega_{n}=\sqrt{\frac{g}{L}}=\sqrt{\frac{mgr}{I_{P}}},
  42. L = g ω n 2 = I P m r . L=\frac{g}{\omega_{n}^{2}}=\frac{I_{P}}{mr}.
  43. L L
  44. L = g ω n 2 = 9.81 m / s 2 ( 3.14 rad / s ) 2 = 0.99 m . L=\frac{g}{\omega_{n}^{2}}=\frac{9.81\ \mathrm{m/s^{2}}}{(3.14\ \mathrm{rad/s}% )^{2}}=0.99\ \mathrm{m}.
  45. N N
  46. r r
  47. E K = i = 1 N ( 1 2 m i 𝐯 i 𝐯 i ) = i = 1 N ( 1 2 m i ( ω r i ) 2 ) = 1 2 ω 2 i = 1 N m i r i 2 . E\text{K}=\sum_{i=1}^{N}\left(\frac{1}{2}\,m_{i}\mathbf{v}_{i}\cdot\mathbf{v}_% {i}\right)=\sum_{i=1}^{N}\left(\frac{1}{2}\,m_{i}(\omega r_{i})^{2}\right)=% \frac{1}{2}\,\omega^{2}\sum_{i=1}^{N}m_{i}r_{i}^{2}.
  48. I P = i = 1 N m i r i 2 . I_{P}=\sum_{i=1}^{N}m_{i}r_{i}^{2}.
  49. I P = V ρ ( 𝐫 ) 𝐫 2 d V . I_{P}=\int_{V}\rho(\mathbf{r})\,\mathbf{r}^{2}\,dV.
  50. ρ ρ
  51. V V
  52. s s
  53. l l
  54. I C , rod = ρ x 2 d V = - / 2 / 2 ρ x 2 s d x = ρ s x 3 3 | - / 2 / 2 = ρ s 3 ( 3 8 + 3 8 ) = 1 12 m 2 , I_{C,\,\text{rod}}=\int\rho\,x^{2}dV=\int_{-\ell/2}^{\ell/2}\rho\,x^{2}sdx=% \rho s\frac{x^{3}}{3}\bigg|_{-\ell/2}^{\ell/2}=\frac{\rho s}{3}\left(\frac{% \ell^{3}}{8}+\frac{\ell^{3}}{8}\right)=\frac{1}{12}\,m\ell^{2},
  55. m = ρ s m=ρsℓ
  56. s s
  57. R R
  58. ρ ρ
  59. d V = s r d r d θ dV=srdrdθ
  60. I C , disc = ρ r 2 d V = 0 2 π 0 R ρ r 2 ( s r d r d θ ) = 2 π ρ s R 4 4 = 1 2 m R 2 , I_{C,\,\text{disc}}=\int\rho r^{2}dV=\int_{0}^{2\pi}\int_{0}^{R}\rho r^{2}(% srdrd\theta)=2\pi\rho s\frac{R^{4}}{4}=\frac{1}{2}mR^{2},
  61. m = π R 2 ρ s m=πR^{2}ρs
  62. I P = I C , rod + M rod ( L 2 ) 2 + I C , disc + M disc ( L + R ) 2 , I_{P}=I_{C,\,\text{rod}}+M\text{rod}\left(\frac{L}{2}\right)^{2}+I_{C,\,\text{% disc}}+M\text{disc}(L+R)^{2},
  63. x 2 + y 2 + z 2 = R 2 , x^{2}+y^{2}+z^{2}=R^{2},
  64. r ( z ) 2 = x 2 + y 2 = R 2 - z 2 . r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.
  65. I C , ball = - R R π ρ 2 r ( z ) 4 d z = - R R π ρ 2 ( R 2 - z 2 ) 2 d z = π ρ 2 ( R 4 z - 2 3 R 2 z 3 + 1 5 z 5 ) | - R R = π ρ ( 1 - 2 3 + 1 5 ) R 5 = 2 5 m R 2 , \begin{aligned}\displaystyle I_{C,\,\text{ball}}&\displaystyle=\int_{-R}^{R}% \frac{\pi\rho}{2}r(z)^{4}dz=\int_{-R}^{R}\frac{\pi\rho}{2}(R^{2}-z^{2})^{2}dz% \\ &\displaystyle=\frac{\pi\rho}{2}\left(R^{4}z-\frac{2}{3}R^{2}z^{3}+\frac{1}{5}% z^{5}\right)\bigg|_{-R}^{R}\\ &\displaystyle=\pi\rho\left(1-\frac{2}{3}+\frac{1}{5}\right)R^{5}\\ &\displaystyle=\frac{2}{5}mR^{2},\end{aligned}
  66. m = ( 4 / 3 ) π R 3 ρ m=(4/3)πR^{3}ρ
  67. k k
  68. n n
  69. P i , i = 1 , , n P_{i},i=1,...,n
  70. 𝐯 i \mathbf{v}_{i}
  71. Δ 𝐫 i = 𝐫 i - 𝐑 , 𝐯 i = s y m b o l ω × ( 𝐫 i - 𝐑 ) + 𝐕 , \Delta\mathbf{r}_{i}=\mathbf{r}_{i}-\mathbf{R},\quad\mathbf{v}_{i}=symbol% \omega\times(\mathbf{r}_{i}-\mathbf{R})+\mathbf{V},
  72. 𝐕 \mathbf{V}
  73. 𝐑 \mathbf{R}
  74. k k
  75. 𝐞 i \mathbf{e}_{i}
  76. 𝐑 \mathbf{R}
  77. 𝐫 i \mathbf{r}_{i}
  78. 𝐭 i = k × 𝐞 i \mathbf{t}_{i}=k×\mathbf{e}_{i}
  79. Δ r i 𝐞 i = 𝐫 i - 𝐑 , 𝐯 i = ω Δ r i 𝐭 i + 𝐕 , i = 1 , , n . \Delta r_{i}\mathbf{e}_{i}=\mathbf{r}_{i}-\mathbf{R},\quad\mathbf{v}_{i}=% \omega\Delta r_{i}\mathbf{t}_{i}+\mathbf{V},\quad i=1,\dots,n.
  80. 𝐋 = i = 1 n [ m i ( 𝐫 i - 𝐑 ) × 𝐯 i ] = i = 1 n [ m i Δ r i 𝐞 i × ( ω Δ r i 𝐭 i + 𝐕 ) ] = [ i = 1 n m i Δ r i 2 ] ω k + [ i = 1 n ( m i Δ r i 𝐞 i ) ] × 𝐕 . \begin{aligned}\displaystyle\mathbf{L}&\displaystyle=\sum_{i=1}^{n}\left[m_{i}% (\mathbf{r}_{i}-\mathbf{R})\times\mathbf{v}_{i}\right]\\ &\displaystyle=\sum_{i=1}^{n}\left[m_{i}\Delta r_{i}\mathbf{e}_{i}\times(% \omega\Delta r_{i}\mathbf{t}_{i}+\mathbf{V})\right]\\ &\displaystyle=\left[\sum_{i=1}^{n}m_{i}\Delta r_{i}^{2}\right]\omega\vec{k}+% \left[\sum_{i=1}^{n}\left(m_{i}\Delta r_{i}\mathbf{e}_{i}\right)\right]\times% \mathbf{V}.\end{aligned}
  81. Δ r i 𝐞 i = 𝐫 i - 𝐂 , i = 1 n ( m i Δ r i 𝐞 i ) = 0 , \begin{aligned}\displaystyle\Delta r_{i}\mathbf{e}_{i}&\displaystyle=\mathbf{r% }_{i}-\mathbf{C},\\ \displaystyle\sum_{i=1}^{n}\left(m_{i}\Delta r_{i}\mathbf{e}_{i}\right)&% \displaystyle=0,\end{aligned}
  82. C {}_{C}
  83. I C = i = 1 n m i Δ r i 2 , I_{C}=\sum_{i=1}^{n}m_{i}\Delta r_{i}^{2},
  84. 𝐋 = I C ω k . \mathbf{L}=I_{C}\omega\vec{k}.
  85. I I
  86. E K = 1 2 i = 1 n ( m i 𝐯 i 𝐯 i ) = 1 2 i = 1 n ( m i ( ω Δ r i 𝐭 i + 𝐕 ) ( ω Δ r i 𝐭 i + 𝐕 ) ) . E\text{K}=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}\mathbf{v}_{i}\cdot\mathbf{v}_{i% }\right)=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}(\omega\Delta r_{i}\mathbf{t}_{i}% +\mathbf{V})\cdot(\omega\Delta r_{i}\mathbf{t}_{i}+\mathbf{V})\right).
  87. E K = 1 2 ω 2 i = 1 n ( m i Δ r i 2 ( 𝐭 i 𝐭 i ) ) + ω 𝐕 ( i = 1 n m i Δ r i 𝐭 i ) + 1 2 ( i = 1 n m i ) 𝐕 𝐕 . E\text{K}=\frac{1}{2}\omega^{2}\sum_{i=1}^{n}\left(m_{i}\Delta r_{i}^{2}(% \mathbf{t}_{i}\cdot\mathbf{t}_{i})\right)+\omega\mathbf{V}\cdot\left(\sum_{i=1% }^{n}m_{i}\Delta r_{i}\mathbf{t}_{i}\right)+\frac{1}{2}\left(\sum_{i=1}^{n}m_{% i}\right)\mathbf{V}\cdot\mathbf{V}.
  88. C {}_{C}
  89. E K = 1 2 I C ω 2 + 1 2 M 𝐕 𝐕 . E\text{K}=\frac{1}{2}I_{C}\omega^{2}+\frac{1}{2}M\mathbf{V}\cdot\mathbf{V}.
  90. C {}_{C}
  91. P i , i = 1 , , N P_{i},i=1,...,N
  92. 𝐑 \mathbf{R}
  93. 𝐅 = i = 1 N m i 𝐀 i , s y m b o l τ = i = 1 N ( 𝐫 i - 𝐑 ) × m i 𝐀 i , \begin{aligned}\displaystyle\mathbf{F}&\displaystyle=\sum_{i=1}^{N}m_{i}% \mathbf{A}_{i},\\ \displaystyle symbol\tau&\displaystyle=\sum_{i=1}^{N}(\mathbf{r}_{i}-\mathbf{R% })\times m_{i}\mathbf{A}_{i},\end{aligned}
  94. 𝐫 i \mathbf{r}_{i}
  95. P P
  96. 𝐑 \mathbf{R}
  97. 𝐀 \mathbf{A}
  98. ω ω
  99. α α
  100. 𝐀 i = s y m b o l α × ( 𝐫 i - 𝐑 ) + s y m b o l ω \timessymbol ω × ( 𝐫 i - 𝐑 ) + 𝐀 . \mathbf{A}_{i}=symbol\alpha\times(\mathbf{r}_{i}-\mathbf{R})+symbol\omega% \timessymbol\omega\times(\mathbf{r}_{i}-\mathbf{R})+\mathbf{A}.
  101. k k
  102. 𝐞 i \mathbf{e}_{i}
  103. 𝐑 \mathbf{R}
  104. 𝐫 i \mathbf{r}_{i}
  105. 𝐭 i = k × 𝐞 i \mathbf{t}_{i}=k×\mathbf{e}_{i}
  106. 𝐀 i = α ( Δ r i 𝐭 i ) - ω 2 ( Δ r i 𝐞 i ) + 𝐀 . \mathbf{A}_{i}=\alpha(\Delta r_{i}\mathbf{t}_{i})-\omega^{2}(\Delta r_{i}% \mathbf{e}_{i})+\mathbf{A}.
  107. s y m b o l τ = i = 1 N ( [ m i Δ r i 𝐞 i ] × [ α ( Δ r i 𝐭 i ) - ω 2 ( Δ r i 𝐞 i ) + 𝐀 ] ) = ( i = 1 N m i Δ r i 2 ) α k + ( i = 1 N m i Δ r i 𝐞 i ) × 𝐀 , \begin{aligned}\displaystyle symbol\tau&\displaystyle=\sum_{i=1}^{N}\left(% \left[m_{i}\Delta r_{i}\mathbf{e}_{i}\right]\times\left[\alpha(\Delta r_{i}% \mathbf{t}_{i})-\omega^{2}(\Delta r_{i}\mathbf{e}_{i})+\mathbf{A}\right]\right% )\\ &\displaystyle=\left(\sum_{i=1}^{N}m_{i}\Delta r_{i}^{2}\right)\alpha\vec{k}+% \left(\sum_{i=1}^{N}m_{i}\Delta r_{i}\mathbf{e}_{i}\right)\times\mathbf{A},% \end{aligned}
  108. 𝐞 i × 𝐞 i = 0 \mathbf{e}_{i}×\mathbf{e}_{i}=0
  109. 𝐞 i × 𝐭 i = k \mathbf{e}_{i}×\mathbf{t}_{i}=k
  110. P P
  111. 𝐂 \mathbf{C}
  112. I I
  113. s y m b o l τ = I C α k . symbol\tau=I_{C}\alpha\vec{k}.
  114. I I
  115. P i , i = 1 , , n P_{i},i=1,...,n
  116. 𝐫 i \mathbf{r}_{i}
  117. 𝐯 i \mathbf{v}_{i}
  118. 𝐑 \mathbf{R}
  119. Δ 𝐫 i = 𝐫 i - 𝐑 \Delta\mathbf{r}_{i}=\mathbf{r}_{i}-\mathbf{R}
  120. 𝐯 i = s y m b o l ω × Δ 𝐫 i + 𝐕 R \mathbf{v}_{i}=symbol\omega\times\Delta\mathbf{r}_{i}+\mathbf{V}_{R}
  121. ω ω
  122. 𝐕 \mathbf{V}
  123. 𝐑 \mathbf{R}
  124. 𝐋 = i = 1 n ( m i Δ 𝐫 i × 𝐯 i ) = i = 1 n ( m i Δ 𝐫 i × ( s y m b o l ω × Δ 𝐫 i ) ) = - i = 1 n ( m i Δ 𝐫 i × ( Δ 𝐫 i \timessymbol ω ) ) , \mathbf{L}=\sum_{i=1}^{n}\left(m_{i}\Delta\mathbf{r}_{i}\times\mathbf{v}_{i}% \right)=\sum_{i=1}^{n}\left(m_{i}\Delta\mathbf{r}_{i}\times(symbol\omega\times% \Delta\mathbf{r}_{i})\right)=-\sum_{i=1}^{n}\left(m_{i}\Delta\mathbf{r}_{i}% \times(\Delta\mathbf{r}_{i}\timessymbol\omega)\right),
  125. 𝐕 \mathbf{V}
  126. B B
  127. 𝐛 \mathbf{b}
  128. [ B ] 𝐲 = 𝐛 × 𝐲 . [B]\mathbf{y}=\mathbf{b}\times\mathbf{y}.
  129. B B
  130. 𝐛 = \mathbf{b}=
  131. ( b (b
  132. [ B ] = [ 0 - b z b y b z 0 - b x - b y b x 0 ] . [B]=\begin{bmatrix}0&-b_{z}&b_{y}\\ b_{z}&0&-b_{x}\\ -b_{y}&b_{x}&0\end{bmatrix}.
  133. r r
  134. 𝐫 i - 𝐂 \mathbf{r}_{i}-\mathbf{C}
  135. 𝐋 = ( - i = 1 n m i [ Δ r i ] 2 ) s y m b o l ω = [ I C ] s y m b o l ω , \mathbf{L}=\left(-\sum_{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)symbol\omega=[I_% {C}]symbol\omega,
  136. I I
  137. [ I C ] = - i = 1 n m i [ Δ r i ] 2 , [I_{C}]=-\sum_{i=1}^{n}m_{i}[\Delta r_{i}]^{2},
  138. P i , i = 1 , , n P_{i},i=1,...,n
  139. i {}_{i}
  140. i {}_{i}
  141. E K = 1 2 i = 1 n ( m i 𝐯 i 𝐯 i ) = 1 2 i = 1 n ( m i ( s y m b o l ω × Δ 𝐫 i + 𝐕 C ) ( s y m b o l ω × Δ 𝐫 i + 𝐕 C ) ) , E\text{K}=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}\mathbf{v}_{i}\cdot\mathbf{v}_{i% }\right)=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}(symbol\omega\times\Delta\mathbf{% r}_{i}+\mathbf{V}_{C})\cdot(symbol\omega\times\Delta\mathbf{r}_{i}+\mathbf{V}_% {C})\right),
  142. E K = 1 2 i = 1 n ( m i ( s y m b o l ω × Δ 𝐫 i ) ( s y m b o l ω × Δ 𝐫 i ) ) + i = 1 n ( m i 𝐕 C ( s y m b o l ω × Δ 𝐫 i ) ) + 1 2 i = 1 n ( m i 𝐕 C 𝐕 C ) . E\text{K}=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}(symbol\omega\times\Delta\mathbf% {r}_{i})\cdot(symbol\omega\times\Delta\mathbf{r}_{i})\right)+\sum_{i=1}^{n}% \left(m_{i}\mathbf{V}_{C}\cdot(symbol\omega\times\Delta\mathbf{r}_{i})\right)+% \frac{1}{2}\sum_{i=1}^{n}\left(m_{i}\mathbf{V}_{C}\cdot\mathbf{V}_{C}\right).
  143. i {}_{i}
  144. E K = 1 2 i = 1 n ( m i ( [ Δ r i ] s y m b o l ω ) ( [ Δ r i ] s y m b o l ω ) ) + 1 2 ( i = 1 n m i ) 𝐕 C 𝐕 C . E\text{K}=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}([\Delta r_{i}]symbol\omega)% \cdot([\Delta r_{i}]symbol\omega)\right)+\frac{1}{2}\left(\sum_{i=1}^{n}m_{i}% \right)\mathbf{V}_{C}\cdot\mathbf{V}_{C}.
  145. E K = 1 2 i = 1 n ( m i ( s y m b o l ω T [ Δ r i ] T [ Δ r i ] s y m b o l ω ) ) + 1 2 ( i = 1 n m i ) 𝐕 C 𝐕 C . E\text{K}=\frac{1}{2}\sum_{i=1}^{n}\left(m_{i}(symbol\omega^{T}[\Delta r_{i}]^% {T}[\Delta r_{i}]symbol\omega)\right)+\frac{1}{2}\left(\sum_{i=1}^{n}m_{i}% \right)\mathbf{V}_{C}\cdot\mathbf{V}_{C}.
  146. E K = 1 2 s y m b o l ω ( - i = 1 n m i [ Δ r i ] 2 ) s y m b o l ω + 1 2 ( i = 1 n m i ) 𝐕 C 𝐕 C . E\text{K}=\frac{1}{2}symbol\omega\cdot\left(-\sum_{i=1}^{n}m_{i}[\Delta r_{i}]% ^{2}\right)symbol\omega+\frac{1}{2}\left(\sum_{i=1}^{n}m_{i}\right)\mathbf{V}_% {C}\cdot\mathbf{V}_{C}.
  147. E K = 1 2 s y m b o l ω [ I C ] s y m b o l ω + 1 2 M 𝐕 C 2 . E\text{K}=\frac{1}{2}symbol\omega\cdot[I_{C}]symbol\omega+\frac{1}{2}M\mathbf{% V}_{C}^{2}.
  148. C {}_{C}
  149. s y m b o l τ = i = 1 n ( ( 𝐫 𝐢 - 𝐑 ) × ( m i 𝐚 i ) ) , symbol\tau=\sum_{i=1}^{n}\left((\mathbf{r_{i}}-\mathbf{R})\times(m_{i}\mathbf{% a}_{i})\right),
  150. i {}_{i}
  151. i {}_{i}
  152. i {}_{i}
  153. 𝐚 i = s y m b o l α × ( 𝐫 i - 𝐑 ) + s y m b o l ω \timessymbol ω × ( 𝐫 i - 𝐑 ) + 𝐀 R . \mathbf{a}_{i}=symbol\alpha\times(\mathbf{r}_{i}-\mathbf{R})+symbol\omega% \timessymbol\omega\times(\mathbf{r}_{i}-\mathbf{R})+\mathbf{A}_{R}.
  154. i {}_{i}
  155. i {}_{i}
  156. s y m b o l τ = ( - i = 1 n m i [ Δ r i ] 2 ) s y m b o l α + s y m b o l ω × ( - i = 1 n m i [ Δ r i ] 2 ) s y m b o l ω symbol\tau=\left(-\sum_{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)symbol\alpha+% symbol\omega\times\left(-\sum_{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)symbol\omega
  157. Δ 𝐫 i × ( s y m b o l ω × ( s y m b o l ω × Δ 𝐫 i ) ) + s y m b o l ω × ( ( s y m b o l ω × Δ 𝐫 i ) × Δ 𝐫 i ) = 0 , \Delta\mathbf{r}_{i}\times(symbol\omega\times(symbol\omega\times\Delta\mathbf{% r}_{i}))+symbol\omega\times((symbol\omega\times\Delta\mathbf{r}_{i})\times% \Delta\mathbf{r}_{i})=0,
  158. s y m b o l τ = i = 1 n ( 𝐫 𝐢 - 𝐑 ) × ( m i 𝐚 i ) = i = 1 n s y m b o l Δ 𝐫 i × ( m i 𝐚 i ) = i = 1 n m i [ s y m b o l Δ 𝐫 i × 𝐚 i ] cross-product scalar multiplication = i = 1 n m i [ s y m b o l Δ 𝐫 i × ( 𝐚 tangential , i + 𝐚 centripetal , i + 𝐀 R ) ] = i = 1 n m i [ s y m b o l Δ 𝐫 i × ( 𝐚 tangential , i + 𝐚 centripetal , i + 0 ) ] 𝐑 is either at rest or moving at a constant velocity but not accelerated, or the origin of the fixed (world) coordinate reference system is placed at the centre of mass 𝐂 = i = 1 n m i [ s y m b o l Δ 𝐫 i × 𝐚 tangential , i + s y m b o l Δ 𝐫 i × 𝐚 centripetal , i ] cross-product distributivity over addition = i = 1 n m i [ s y m b o l Δ 𝐫 i × ( s y m b o l α \timessymbol Δ 𝐫 i ) + s y m b o l Δ 𝐫 i × ( s y m b o l ω × 𝐯 tangential , i ) ] s y m b o l τ = i = 1 n m i [ s y m b o l Δ 𝐫 i × ( s y m b o l α \timessymbol Δ 𝐫 i ) + s y m b o l Δ 𝐫 i × ( s y m b o l ω × ( s y m b o l ω \timessymbol Δ 𝐫 i ) ) ] \begin{aligned}\displaystyle symbol\tau&\displaystyle=\sum_{i=1}^{n}(\mathbf{r% _{i}}-\mathbf{R})\times(m_{i}\mathbf{a}_{i})\\ &\displaystyle=\sum_{i=1}^{n}symbol\Delta\mathbf{r}_{i}\times(m_{i}\mathbf{a}_% {i})\\ &\displaystyle=\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf{r}_{i}\times\mathbf{a}_% {i}]\;\ldots\,\text{ cross-product scalar multiplication}\\ &\displaystyle=\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf{r}_{i}\times(\mathbf{a}% _{\,\text{tangential},i}+\mathbf{a}_{\,\text{centripetal},i}+\mathbf{A}_{R})]% \\ &\displaystyle=\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf{r}_{i}\times(\mathbf{a}% _{\,\text{tangential},i}+\mathbf{a}_{\,\text{centripetal},i}+0)]\\ &\displaystyle\;\;\;\;\;\ldots\;\mathbf{R}\,\text{ is either at rest or moving% at a constant velocity but not accelerated, or }\\ &\displaystyle\;\;\;\;\;\;\;\;\;\;\;\,\text{the origin of the fixed (world) % coordinate reference system is placed at the centre of mass }\mathbf{C}\\ &\displaystyle=\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf{r}_{i}\times\mathbf{a}_% {\,\text{tangential},i}+symbol\Delta\mathbf{r}_{i}\times\mathbf{a}_{\,\text{% centripetal},i}]\;\ldots\,\text{ cross-product distributivity over addition}\\ &\displaystyle=\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf{r}_{i}\times(symbol% \alpha\timessymbol\Delta\mathbf{r}_{i})+symbol\Delta\mathbf{r}_{i}\times(% symbol\omega\times\mathbf{v}_{\,\text{tangential},i})]\\ \displaystyle symbol\tau&\displaystyle=\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf% {r}_{i}\times(symbol\alpha\timessymbol\Delta\mathbf{r}_{i})+symbol\Delta% \mathbf{r}_{i}\times(symbol\omega\times(symbol\omega\timessymbol\Delta\mathbf{% r}_{i}))]\\ \end{aligned}
  159. s y m b o l Δ 𝐫 i × ( s y m b o l ω × ( s y m b o l ω \timessymbol Δ 𝐫 i ) ) + s y m b o l ω × ( ( s y m b o l ω \timessymbol Δ 𝐫 i ) \timessymbol Δ 𝐫 i ) + ( s y m b o l ω \timessymbol Δ 𝐫 i ) × ( s y m b o l Δ 𝐫 i \timessymbol ω ) \displaystyle symbol\Delta\mathbf{r}_{i}\times(symbol\omega\times(symbol\omega% \timessymbol\Delta\mathbf{r}_{i}))+symbol\omega\times((symbol\omega% \timessymbol\Delta\mathbf{r}_{i})\timessymbol\Delta\mathbf{r}_{i})+(symbol% \omega\timessymbol\Delta\mathbf{r}_{i})\times(symbol\Delta\mathbf{r}_{i}% \timessymbol\omega)
  160. s y m b o l Δ 𝐫 i × ( s y m b o l ω × ( s y m b o l ω \timessymbol Δ 𝐫 i ) ) \displaystyle symbol\Delta\mathbf{r}_{i}\times(symbol\omega\times(symbol\omega% \timessymbol\Delta\mathbf{r}_{i}))
  161. s y m b o l τ \displaystyle symbol\tau
  162. 𝐮 \mathbf{u}\,
  163. - i = 1 n m i [ s y m b o l Δ 𝐫 i × ( s y m b o l Δ 𝐫 i × 𝐮 ) ] \displaystyle-\sum_{i=1}^{n}m_{i}[symbol\Delta\mathbf{r}_{i}\times(symbol% \Delta\mathbf{r}_{i}\times\mathbf{u})]
  164. s y m b o l τ \displaystyle symbol\tau
  165. s y m b o l τ = [ I C ] s y m b o l α + s y m b o l ω × [ I C ] s y m b o l ω , symbol\tau=[I_{C}]symbol\alpha+symbol\omega\times[I_{C}]symbol\omega,
  166. C {}_{C}
  167. R {}_{R}
  168. [ I R ] = - i = 1 n m i [ r i - R ] 2 . [I_{R}]=-\sum_{i=1}^{n}m_{i}[r_{i}-R]^{2}.
  169. 𝐑 = ( 𝐑 - 𝐂 ) + 𝐂 = 𝐝 + 𝐂 , \mathbf{R}=(\mathbf{R}-\mathbf{C})+\mathbf{C}=\mathbf{d}+\mathbf{C},
  170. [ I R ] = - i = 1 n m i [ r i - C - d ] 2 . [I_{R}]=-\sum_{i=1}^{n}m_{i}[r_{i}-C-d]^{2}.
  171. [ I R ] = ( - i = 1 n m i [ r i - C ] 2 ) + ( i = 1 n m i [ r i - C ] ) [ d ] + [ d ] ( i = 1 n m i [ r i - C ] ) + ( - i = 1 n m i ) [ d ] [ d ] . [I_{R}]=\left(-\sum_{i=1}^{n}m_{i}[r_{i}-C]^{2}\right)+\left(\sum_{i=1}^{n}m_{% i}[r_{i}-C]\right)[d]+[d]\left(\sum_{i=1}^{n}m_{i}[r_{i}-C]\right)+\left(-\sum% _{i=1}^{n}m_{i}\right)[d][d].
  172. C {}_{C}
  173. [ I R ] = [ I C ] - M [ d ] 2 , [I_{R}]=[I_{C}]-M[d]^{2},
  174. I L = 𝐒 [ - i = 1 N m i [ Δ r i ] 2 ] 𝐒 = 𝐒 [ I R ] 𝐒 = 𝐒 T [ I R ] 𝐒 , I_{L}=\mathbf{S}\cdot\left[-\sum_{i=1}^{N}m_{i}[\Delta r_{i}]^{2}\right]% \mathbf{S}=\mathbf{S}\cdot[I_{R}]\mathbf{S}=\mathbf{S}^{T}[I_{R}]\mathbf{S},
  175. R {}_{R}
  176. N N
  177. P i , i = 1 , , N P_{i},i=1,...,N
  178. i {}_{i}
  179. i {}_{i}
  180. Δ 𝐫 i = ( 𝐫 i - 𝐑 ) - ( 𝐒 ( 𝐫 i - 𝐑 ) ) 𝐒 = [ [ I ] - [ 𝐒𝐒 T ] ] ( Δ 𝐫 i ) , \Delta\mathbf{r}_{i}^{\perp}=(\mathbf{r}_{i}-\mathbf{R})-(\mathbf{S}\cdot(% \mathbf{r}_{i}-\mathbf{R}))\mathbf{S}=[[I]-[\mathbf{S}\mathbf{S}^{T}]](\Delta% \mathbf{r}_{i}),
  181. T {}^{T}
  182. - [ S ] 2 = [ I ] - [ 𝐒𝐒 T ] , -[S]^{2}=[I]-[\mathbf{S}\mathbf{S}^{T}],
  183. | Δ 𝐫 i | 2 = ( - [ S ] 2 ( Δ 𝐫 i ) ) ( - [ S ] 2 ( Δ 𝐫 i ) ) = - 𝐒 [ Δ r i ] [ Δ r i ] 𝐒 . |\Delta\mathbf{r}_{i}^{\perp}|^{2}=(-[S]^{2}(\Delta\mathbf{r}_{i}))\cdot(-[S]^% {2}(\Delta\mathbf{r}_{i}))=-\mathbf{S}\cdot[\Delta r_{i}][\Delta r_{i}]\mathbf% {S}.
  184. ( 𝐒 × ( 𝐒 × ( Δ 𝐫 i ) ) ) 𝐒 × ( 𝐒 × ( Δ 𝐫 i ) ) = ( 𝐒 × ( 𝐒 × ( Δ 𝐫 i ) ) ) × 𝐒 ( 𝐒 × ( Δ 𝐫 i ) ) , (\mathbf{S}\times(\mathbf{S}\times(\Delta\mathbf{r}_{i})))\cdot\mathbf{S}% \times(\mathbf{S}\times(\Delta\mathbf{r}_{i}))=(\mathbf{S}\times(\mathbf{S}% \times(\Delta\mathbf{r}_{i})))\times\mathbf{S}\cdot(\mathbf{S}\times(\Delta% \mathbf{r}_{i})),
  185. - ( Δ 𝐫 i ) × 𝐒 ( 𝐒 × ( Δ 𝐫 i ) ) = - 𝐒 [ Δ r i ] [ Δ r i ] 𝐒 , -(\Delta\mathbf{r}_{i})\times\mathbf{S}\cdot(\mathbf{S}\times(\Delta\mathbf{r}% _{i}))=-\mathbf{S}\cdot[\Delta r_{i}][\Delta r_{i}]\mathbf{S},
  186. i {}_{i}
  187. i {}_{i}
  188. I L = i = 1 N m i | Δ 𝐫 i | 2 = - i = 1 N m i 𝐒 [ Δ r i ] 2 𝐒 , I_{L}=\sum_{i=1}^{N}m_{i}|\Delta\mathbf{r}_{i}^{\perp}|^{2}=-\sum_{i=1}^{N}m_{% i}\mathbf{S}\cdot[\Delta r_{i}]^{2}\mathbf{S},
  189. I L = 𝐒 ( - i = 1 N m i [ Δ r i ] 2 ) 𝐒 = 𝐒 [ I R ] 𝐒 = 𝐒 T [ I R ] 𝐒 , I_{L}=\mathbf{S}\cdot(-\sum_{i=1}^{N}m_{i}[\Delta r_{i}]^{2})\mathbf{S}=% \mathbf{S}\cdot[I_{R}]\mathbf{S}=\mathbf{S}^{T}[I_{R}]\mathbf{S},
  190. R {}_{R}
  191. \otimes
  192. 𝐞 i 𝐞 j , i , j = 1 , 2 , 3 , \mathbf{e}_{i}\otimes\mathbf{e}_{j},\quad i,j=1,2,3,
  193. i {}_{i}
  194. 𝐈 = i = 1 3 j = 1 3 I i j 𝐞 i 𝐞 j . \mathbf{I}=\sum_{i=1}^{3}\sum_{j=1}^{3}I_{ij}\mathbf{e}_{i}\otimes\mathbf{e}_{% j}.
  195. P k , k = 1 , , N P_{k},k=1,...,N
  196. k {}_{k}
  197. k {}_{k}
  198. k {}_{k}
  199. k {}_{k}
  200. k {}_{k}
  201. 𝐈 = k = 1 N m k ( ( 𝐫 k 𝐫 k ) 𝐄 - 𝐫 k 𝐫 k ) , \mathbf{I}=\sum_{k=1}^{N}m_{k}((\mathbf{r}_{k}\cdot\mathbf{r}_{k})\mathbf{E}-% \mathbf{r}_{k}\otimes\mathbf{r}_{k}),
  202. 𝐄 = 𝐞 1 𝐞 1 + 𝐞 2 𝐞 2 + 𝐞 3 𝐞 3 . \mathbf{E}=\mathbf{e}_{1}\otimes\mathbf{e}_{1}+\mathbf{e}_{2}\otimes\mathbf{e}% _{2}+\mathbf{e}_{3}\otimes\mathbf{e}_{3}.
  203. 𝐈 = V ρ ( 𝐫 ) ( ( 𝐫 𝐫 ) 𝐄 - 𝐫 𝐫 ) d V , \mathbf{I}=\int_{V}\rho(\mathbf{r})\left(\left(\mathbf{r}\cdot\mathbf{r}\right% )\mathbf{E}-\mathbf{r}\otimes\mathbf{r}\right)\,dV,
  204. i j {}_{ij}
  205. j i {}_{ji}
  206. 𝐈 = V ρ ( 𝐫 ) ( r ^ ) 2 d V , \mathbf{I}=\int_{V}\rho(\mathbf{r})(\hat{r})^{2}\,dV,
  207. I n = 𝐧 𝐈 𝐧 , I_{n}=\mathbf{n}\cdot\mathbf{I}\cdot\mathbf{n},
  208. 12 {}_{12}
  209. I 12 = 𝐞 1 𝐈 𝐞 2 , I_{12}=\mathbf{e}_{1}\cdot\mathbf{I}\cdot\mathbf{e}_{2},
  210. [ I ] = [ I 11 I 12 I 13 I 21 I 22 I 23 I 31 I 32 I 33 ] = [ I x x I x y I x z I y x I y y I y z I z x I z y I z z ] . [I]=\begin{bmatrix}I_{11}&I_{12}&I_{13}\\ I_{21}&I_{22}&I_{23}\\ I_{31}&I_{32}&I_{33}\end{bmatrix}=\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\ I_{yx}&I_{yy}&I_{yz}\\ I_{zx}&I_{zy}&I_{zz}\end{bmatrix}.
  211. x x {}_{xx}
  212. x y {}_{xy}
  213. Δ 𝐫 i = ( 𝐫 i - 𝐑 ) - ( 𝐒 ( 𝐫 i - 𝐑 ) ) 𝐒 = [ [ I ] - [ 𝐒𝐒 T ] ] ( Δ 𝐫 i ) , \Delta\mathbf{r}_{i}^{\perp}=(\mathbf{r}_{i}-\mathbf{R})-(\mathbf{S}\cdot(% \mathbf{r}_{i}-\mathbf{R}))\mathbf{S}=[[I]-[\mathbf{S}\mathbf{S}^{T}]](\Delta% \mathbf{r}_{i}),
  214. T {}^{T}
  215. [ I | Δ 𝐫 | 2 - Δ 𝐫 Δ 𝐫 T ] . [I|\Delta\mathbf{r}|^{2}-\Delta\mathbf{r}\Delta\mathbf{r}^{T}].
  216. - [ R ] 2 = - [ 0 - z y z 0 - x - y x 0 ] 2 = [ y 2 + z 2 - x y - x z - y x x 2 + z 2 - y z - z x - z y x 2 + y 2 ] . -[R]^{2}=-\begin{bmatrix}0&-z&y\\ z&0&-x\\ -y&x&0\end{bmatrix}^{2}=\begin{bmatrix}y^{2}+z^{2}&-xy&-xz\\ -yx&x^{2}+z^{2}&-yz\\ -zx&-zy&x^{2}+y^{2}\end{bmatrix}.
  217. T {}^{T}
  218. - [ R ] 2 = | 𝐑 | 2 [ I ] - [ 𝐑𝐑 T ] = [ x 2 + y 2 + z 2 0 0 0 x 2 + y 2 + z 2 0 0 0 x 2 + y 2 + z 2 ] - [ x 2 x y x z y x y 2 y z z x z y z 2 ] , -[R]^{2}=|\mathbf{R}|^{2}[I]-[\mathbf{R}\mathbf{R}^{T}]=\begin{bmatrix}x^{2}+y% ^{2}+z^{2}&0&0\\ 0&x^{2}+y^{2}+z^{2}&0\\ 0&0&x^{2}+y^{2}+z^{2}\end{bmatrix}-\begin{bmatrix}x^{2}&xy&xz\\ yx&y^{2}&yz\\ zx&zy&z^{2}\end{bmatrix},
  219. | 𝐑 | 2 = 𝐑 𝐑 = tr [ 𝐑𝐑 T ] , |\mathbf{R}|^{2}=\mathbf{R}\cdot\mathbf{R}=\operatorname{tr}[\mathbf{R}\mathbf% {R}^{T}],
  220. C {}_{C}
  221. B {}^{B}
  222. 𝐱 = [ A ] 𝐲 , \mathbf{x}=[A]\mathbf{y},
  223. [ I C ] = [ A ] [ I C B ] [ A T ] . [I_{C}]=[A][I_{C}^{B}][A^{T}].
  224. C {}_{C}
  225. B {}^{B}
  226. [ I C B ] = [ Q ] [ Λ ] [ Q T ] , [I_{C}^{B}]=[Q][\Lambda][Q^{T}],
  227. [ Λ ] = [ I 1 0 0 0 I 2 0 0 0 I 3 ] . [\Lambda]=\begin{bmatrix}I_{1}&0&0\\ 0&I_{2}&0\\ 0&0&I_{3}\end{bmatrix}.
  228. 1 {}_{1}
  229. 2 {}_{2}
  230. 3 {}_{3}
  231. 𝐱 T [ Λ ] 𝐱 = 1 , \mathbf{x}^{T}[\Lambda]\mathbf{x}=1,
  232. I 1 x 2 + I 2 y 2 + I 3 z 2 = 1 , I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,
  233. x 2 ( 1 / I 1 ) 2 + y 2 ( 1 / I 2 ) 2 + z 2 ( 1 / I 3 ) 2 = 1 , \frac{x^{2}}{(1/\sqrt{I_{1}})^{2}}+\frac{y^{2}}{(1/\sqrt{I_{2}})^{2}}+\frac{z^% {2}}{(1/\sqrt{I_{3}})^{2}}=1,
  234. a = 1 I 1 , b = 1 I 2 , c = 1 I 3 . a=\frac{1}{\sqrt{I_{1}}},\quad b=\frac{1}{\sqrt{I_{2}}},\quad c=\frac{1}{\sqrt% {I_{3}}}.
  235. n {}_{n}
  236. 𝐱 T [ Λ ] 𝐱 = | 𝐱 | 2 𝐧 T [ Λ ] 𝐧 = | 𝐱 | 2 I n = 1. \mathbf{x}^{T}[\Lambda]\mathbf{x}=|\mathbf{x}|^{2}\mathbf{n}^{T}[\Lambda]% \mathbf{n}=|\mathbf{x}|^{2}I_{n}=1.
  237. | 𝐱 | = 1 I n . |\mathbf{x}|=\frac{1}{\sqrt{I_{n}}}.

Money_supply.html

  1. M * V = P * Q M*V=P*Q
  2. M M
  3. V V
  4. P P
  5. Q Q

Moneyness.html

  1. M ( S , K , τ , r , σ ) , M(S,K,\tau,r,\sigma),
  2. ln ( F / K ) . \ln\left(F/K\right).
  3. ln ( F / K ) = ln ( S / K ) + r T . \ln\left(F/K\right)=\ln(S/K)+rT.
  4. ln ( F / K ) / τ . \ln\left(F/K\right)\Big/\sqrt{\tau}.
  5. m = ln ( F / K ) σ τ . m=\frac{\ln\left(F/K\right)}{\sigma\sqrt{\tau}}.
  6. d ± = ln ( F / K ) ± ( σ 2 / 2 ) τ σ τ . d_{\pm}=\frac{\ln\left(F/K\right)\pm(\sigma^{2}/2)\tau}{\sigma\sqrt{\tau}}.
  7. m = ln ( F / K ) σ τ = 1 2 ( d - + d + ) , m=\frac{\ln(F/K)}{\sigma\sqrt{\tau}}=\tfrac{1}{2}\left(d_{-}+d_{+}\right),
  8. d - < m < d + , d_{-}<m<d_{+},
  9. σ τ / 2 \sigma\sqrt{\tau}/2
  10. N ( d - ) < N ( m ) < N ( d + ) = Δ . N(d_{-})<N(m)<N(d_{+})=\Delta.

Monge_array.html

  1. i , j , k , \scriptstyle i,\,j,\,k,\,\ell
  2. 1 i < k m and 1 j < n 1\leq i<k\leq m\,\text{ and }1\leq j<\ell\leq n
  3. A [ i , j ] + A [ k , ] A [ i , ] + A [ k , j ] . A[i,j]+A[k,\ell]\leq A[i,\ell]+A[k,j].\,
  4. [ 10 17 13 28 23 17 22 16 29 23 24 28 22 34 24 11 13 6 17 7 45 44 32 37 23 36 33 19 21 6 75 66 51 53 34 ] \begin{bmatrix}10&17&13&28&23\\ 17&22&16&29&23\\ 24&28&22&34&24\\ 11&13&6&17&7\\ 45&44&32&37&23\\ 36&33&19&21&6\\ 75&66&51&53&34\end{bmatrix}
  5. [ 17 23 11 7 ] \begin{bmatrix}17&23\\ 11&7\end{bmatrix}
  6. A [ i , j ] + A [ i + 1 , j + 1 ] A [ i , j + 1 ] + A [ i + 1 , j ] A[i,j]+A[i+1,j+1]\leq A[i,j+1]+A[i+1,j]
  7. 1 i < m 1\leq i<m
  8. 1 j < n 1\leq j<n
  9. f ( x ) = arg min i { 1 , , m } A [ x , i ] f(x)=\arg\min_{i\in\{1,\ldots,m\}}A[x,i]
  10. f ( j ) f ( j + 1 ) f(j)\leq f(j+1)
  11. 1 j < n 1\leq j<n
  12. A [ i , i ] + A [ r , s ] A [ i , s ] + A [ r , i ] A[i,i]+A[r,s]\leq A[i,s]+A[r,i]
  13. 1 i < r , s n 1\leq i<r,s\leq n

Monic_polynomial.html

  1. x n + c n - 1 x n - 1 + + c 2 x 2 + c 1 x + c 0 x^{n}+c_{n-1}x^{n-1}+\cdots+c_{2}x^{2}+c_{1}x+c_{0}
  2. a x 2 + b x + c = 0 \ ax^{2}+bx+c=0
  3. a 0 a\neq 0
  4. x 2 + p x + q = 0 \ x^{2}+px+q=0
  5. 2 x 2 + 3 x + 1 = 0 2x^{2}+3x+1=0
  6. x 2 + 3 2 x + 1 2 = 0. x^{2}+\frac{3}{2}x+\frac{1}{2}=0.
  7. x = 1 2 ( - p ± p 2 - 4 q ) . x=\frac{1}{2}\left(-p\pm\sqrt{p^{2}-4q}\right).
  8. 2 x 2 + 3 x + 1 = 0 \ 2x^{2}+3x+1=0
  9. x 2 + 5 x + 6 = 0 \ x^{2}+5x+6=0
  10. x 2 + 7 x + 8 = 0 \ x^{2}+7x+8=0
  11. C := { b B : p ( x ) A [ x ] , which is monic and such that p ( b ) = 0 } . C:=\{b\in B:\exists\,p(x)\in A[x]\,,\hbox{ which is monic and such that }p(b)=% 0\}\,.
  12. A = A=\mathbb{Z}
  13. B = B=\mathbb{C}
  14. p ( x , y ) = 2 x y 2 + x 2 - y 2 + 3 x + 5 y - 8 \ p(x,y)=2xy^{2}+x^{2}-y^{2}+3x+5y-8
  15. p ( x , y ) = 1 x 2 + ( 2 y 2 + 3 ) x + ( - y 2 + 5 y - 8 ) p(x,y)=1\cdot x^{2}+(2y^{2}+3)\cdot x+(-y^{2}+5y-8)

Monodromy.html

  1. p : X ~ X p:\tilde{X}\to X
  2. F = p - 1 ( x ) F=p^{-1}(x)
  3. x ~ F \tilde{x}\in F
  4. γ ~ \tilde{\gamma}
  5. x ~ γ \tilde{x}\cdot\gamma
  6. γ ~ ( 1 ) \tilde{\gamma}(1)
  7. x ~ \tilde{x}
  8. x ~ \tilde{x}
  9. p * ( π 1 ( X ~ , x ~ ) ) p_{*}(\pi_{1}(\tilde{X},\tilde{x}))
  10. X ~ \tilde{X}
  11. x ~ \tilde{x}
  12. M 1 M p + 1 = i d M_{1}...M_{p+1}=id
  13. p : X ~ X p:\tilde{X}\to X
  14. ( M , ) (M,\mathcal{F})
  15. \mathcal{F}

Monolayer.html

  1. t = 3 × 10 - 4 Pa × s P t=\frac{3\times 10^{-4}\,\mathrm{Pa}\times\mathrm{s}}{P}
  2. Π = γ o - γ \Pi=\gamma^{o}-\gamma
  3. γ o \gamma^{o}
  4. Γ - 1 \Gamma^{-1}
  5. Π A = R T \Pi A=RT
  6. A A
  7. γ = γ o - m C \gamma=\gamma_{o}-mC
  8. Π = Γ R T \Pi=\Gamma RT
  9. Γ = Γ max C a + C \Gamma=\Gamma_{\max}\frac{C}{a+C}
  10. Π = Γ max R T ( 1 + C a ) \Pi=\Gamma_{\max}RT\left(1+\frac{C}{a}\right)

Monotone_convergence_theorem.html

  1. { a n } \{a_{n}\}
  2. sup n { a n } \sup\limits_{n}\{a_{n}\}
  3. { a n } \{a_{n}\}
  4. c = sup n { a n } c=\sup_{n}\{a_{n}\}
  5. ε > 0 \varepsilon>0
  6. N N
  7. a N > c - ε a_{N}>c-\varepsilon
  8. c - ε c-\varepsilon
  9. { a n } \{a_{n}\}
  10. c c
  11. sup n { a n } \sup_{n}\{a_{n}\}
  12. { a n } \{a_{n}\}
  13. n > N , | c - a n | | c - a N | < ε n>N,|c-a_{n}|\leq|c-a_{N}|<\varepsilon
  14. { a n } \{a_{n}\}
  15. sup n { a n } . \sup_{n}\{a_{n}\}.
  16. { a n } \{a_{n}\}
  17. lim j k a j , k = k lim j a j , k . \lim_{j\to\infty}\sum_{k}a_{j,k}=\sum_{k}\lim_{j\to\infty}a_{j,k}.
  18. ( 1 + 1 / n ) n = k = 0 n ( n k ) / n k = k = 0 n 1 k ! × n n × n - 1 n × × n - k + 1 n , (1+1/n)^{n}=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}/n^{k}=\sum_{k=0}^{n}% \frac{1}{k!}\times\frac{n}{n}\times\frac{n-1}{n}\times\cdots\times\frac{n-k+1}% {n},
  19. ( n k ) / n k = 1 k ! × n n × n - 1 n × × n - k + 1 n ; {\left({{n}\atop{k}}\right)}/n^{k}=\frac{1}{k!}\times\frac{n}{n}\times\frac{n-% 1}{n}\times\cdots\times\frac{n-k+1}{n};
  20. ( 1 + 1 / n ) n (1+1/n)^{n}
  21. 1 k ! \frac{1}{k!}
  22. f 1 , f 2 , f_{1},f_{2},\ldots
  23. 0 f k ( x ) f k + 1 ( x ) . 0\leq f_{k}(x)\leq f_{k+1}(x).\,
  24. ( f n ) (f_{n})
  25. f ( x ) := lim k f k ( x ) . f(x):=\lim_{k\to\infty}f_{k}(x).\,
  26. lim k f k d μ = f d μ . \lim_{k\to\infty}\int f_{k}\,\mathrm{d}\mu=\int f\,\mathrm{d}\mu.
  27. ( f k ) (f_{k})
  28. ( f k ( x ) ) (f_{k}(x))
  29. x N x\notin N
  30. f k d μ = X N f k d μ , and f d μ = X N f d μ , \int f_{k}\,\mathrm{d}\mu=\int_{X\setminus N}f_{k}\,\mathrm{d}\mu,\ \,\text{% and}\ \int f\,\mathrm{d}\mu=\int_{X\setminus N}f\,\mathrm{d}\mu,
  31. f - 1 ( I ) = { x X f ( x ) I } . f^{-1}(I)=\{x\in X\mid f(x)\in I\}.
  32. k , f k ( x ) f ( x ) \forall k,f_{k}(x)\leq f(x)
  33. f ( x ) I f k ( x ) I , k . f(x)\in I\Leftrightarrow f_{k}(x)\in I,~{}\forall k\in\mathbb{N}.
  34. { x X f ( x ) I } = k { x X f k ( x ) I } . \{x\in X\mid f(x)\in I\}=\bigcap_{k\in\mathbb{N}}\{x\in X\mid f_{k}(x)\in I\}.
  35. f k f_{k}
  36. f d μ \int f\,\mathrm{d}\mu
  37. f d μ lim k f k d μ . \int f\,\mathrm{d}\mu\geq\lim_{k}\int f_{k}\,\mathrm{d}\mu.
  38. f d μ = sup { g d μ g S F , g f } , \int f\,\mathrm{d}\mu=\sup\left\{\int g\,\mathrm{d}\mu\mid g\in SF,\ g\leq f% \right\},
  39. f k ( x ) f ( x ) f_{k}(x)\leq f(x)
  40. { g d μ g S F , g f k } { g d μ g S F , g f } . \left\{\int g\,\mathrm{d}\mu\mid g\in SF,\ g\leq f_{k}\right\}\subseteq\left\{% \int g\,\mathrm{d}\mu\mid g\in SF,\ g\leq f\right\}.
  41. f d μ lim k f k d μ , \int f\,\mathrm{d}\mu\geq\lim_{k}\int f_{k}\,\mathrm{d}\mu,
  42. f d μ lim k f k d μ . \int f\,\mathrm{d}\mu\leq\lim_{k}\int f_{k}\,\mathrm{d}\mu.
  43. lim k g k d μ = f d μ . \lim_{k}\int g_{k}\,\mathrm{d}\mu=\int f\,\mathrm{d}\mu.
  44. k k\in\mathbb{N}
  45. g k d μ lim j f j d μ \int g_{k}\,\mathrm{d}\mu\leq\lim_{j}\int f_{j}\,\mathrm{d}\mu
  46. lim j f j ( x ) g k ( x ) \lim_{j}f_{j}(x)\geq g_{k}(x)\,
  47. lim j f j d μ g k d μ . \lim_{j}\int f_{j}\,\mathrm{d}\mu\geq\int g_{k}\,\mathrm{d}\mu.
  48. g k g_{k}
  49. g k g_{k}
  50. f j f_{j}
  51. B n = { x B : f n ( x ) 1 - ϵ } . B_{n}=\{x\in B:f_{n}(x)\geq 1-\epsilon\}.\,
  52. n n\in\mathbb{N}
  53. μ ( B n ) ( 1 - ϵ ) = ( 1 - ϵ ) 1 B n d μ f n d μ \mu(B_{n})(1-\epsilon)=\int(1-\epsilon)1_{B_{n}}\,\mathrm{d}\mu\leq\int f_{n}% \,\mathrm{d}\mu
  54. lim j f j ( x ) g k ( x ) \lim_{j}f_{j}(x)\geq g_{k}(x)
  55. B n B_{n}
  56. n B n = B . \bigcup_{n}B_{n}=B.
  57. g k d μ = 1 B d μ = μ ( B ) = μ ( n B n ) . \int g_{k}\,\mathrm{d}\mu=\int 1_{B}\,\mathrm{d}\mu=\mu(B)=\mu\left(\bigcup_{n% }B_{n}\right).
  58. μ ( n B n ) = lim n μ ( B n ) lim n ( 1 - ϵ ) - 1 f n d μ . \mu\left(\bigcup_{n}B_{n}\right)=\lim_{n}\mu(B_{n})\leq\lim_{n}(1-\epsilon)^{-% 1}\int f_{n}\,\mathrm{d}\mu.

Monotonicity_criterion.html

  1. x x
  2. x x
  3. x x
  4. x x
  5. z > x > y z>x>y
  6. x > y > z x>y>z
  7. x x
  8. x x
  9. x x
  10. y y
  11. z z

Morlet_wavelet.html

  1. κ σ \kappa_{\sigma}
  2. Ψ σ ( t ) = c σ π - 1 4 e - 1 2 t 2 ( e i σ t - κ σ ) \Psi_{\sigma}(t)=c_{\sigma}\pi^{-\frac{1}{4}}e^{-\frac{1}{2}t^{2}}(e^{i\sigma t% }-\kappa_{\sigma})
  3. κ σ = e - 1 2 σ 2 \kappa_{\sigma}=e^{-\frac{1}{2}\sigma^{2}}
  4. c σ c_{\sigma}
  5. c σ = ( 1 + e - σ 2 - 2 e - 3 4 σ 2 ) - 1 2 c_{\sigma}=\left(1+e^{-\sigma^{2}}-2e^{-\frac{3}{4}\sigma^{2}}\right)^{-\frac{% 1}{2}}
  6. Ψ ^ σ ( ω ) = c σ π - 1 4 ( e - 1 2 ( σ - ω ) 2 - κ σ e - 1 2 ω 2 ) \hat{\Psi}_{\sigma}(\omega)=c_{\sigma}\pi^{-\frac{1}{4}}\left(e^{-\frac{1}{2}(% \sigma-\omega)^{2}}-\kappa_{\sigma}e^{-\frac{1}{2}\omega^{2}}\right)
  7. ω Ψ \omega_{\Psi}
  8. Ψ ^ σ ( ω ) \hat{\Psi}_{\sigma}(\omega)
  9. ( ω Ψ - σ ) 2 - 1 = ( ω Ψ 2 - 1 ) e - σ ω Ψ (\omega_{\Psi}-\sigma)^{2}-1=(\omega_{\Psi}^{2}-1)e^{-\sigma\omega_{\Psi}}
  10. σ \sigma
  11. σ > 5 \sigma>5
  12. σ \sigma
  13. σ \sigma
  14. κ σ \kappa_{\sigma}
  15. σ > 5 κ σ < 10 - 5 \sigma>5\quad\Rightarrow\quad\kappa_{\sigma}<10^{-5}\,
  16. σ > 5 \sigma>5
  17. ω Ψ σ \omega_{\Psi}\simeq\sigma

Motivation.html

  1. Motivation = Expectancy × Value 1 + Impulsiveness × Delay \mathrm{Motivation}=\frac{\mbox{Expectancy × Value}~{}}{\mbox{1 + % Impulsiveness × Delay}~{}}
  2. M o t i v a t i o n Motivation
  3. E x p e c t a n c y Expectancy
  4. V a l u e Value
  5. I m p u l s i v e n e s s Impulsiveness
  6. D e l a y Delay
  7. MPS = {\,\text{MPS}}=
  8. Autonomy × Feedback × Skill Variety+Task Identity+Task Significance 3 {\,\text{Autonomy}}\,\times\,{\,\text{Feedback}}\,\times\frac{\,\text{Skill % Variety+Task Identity+Task Significance }}{\,\text{3}}

Motorcycle_speedway.html

  1. ( Total points Total rides ) × 4 \left(\frac{\hbox{Total points}}{\hbox{Total rides}}\right)\times 4

Möbius_transformation.html

  1. f ( z ) = a z + b c z + d f(z)=\frac{az+b}{cz+d}
  2. 𝐂 ^ = 𝐂 { } \widehat{\mathbf{C}}=\mathbf{C}\cup\{\infty\}
  3. 𝐂 ^ \widehat{\mathbf{C}}
  4. 𝐂 ^ \widehat{\mathbf{C}}
  5. 𝐂𝐏 1 \mathbf{C}\mathbf{P}^{1}
  6. 𝐂𝐏 1 \mathbf{C}\mathbf{P}^{1}
  7. Aut ( 𝐂 ^ ) \operatorname{Aut}(\widehat{\mathbf{C}})\,
  8. f ( z ) = a z + b c z + d f(z)=\frac{az+b}{cz+d}
  9. f ( - d / c ) = and f ( ) = a / c ; f(-d/c)=\infty\,\text{ and }f(\infty)=a/c;
  10. f ( ) = . f(\infty)=\infty.
  11. Aut ( 𝐂 ^ ) \operatorname{Aut}(\widehat{\mathbf{C}})
  12. f 1 ( z ) = z + d / c f_{1}(z)=z+d/c\quad
  13. f 2 ( z ) = 1 / z f_{2}(z)=1/z\quad
  14. f 3 ( z ) = b c - a d c 2 z f_{3}(z)=\frac{bc-ad}{c^{2}}z\quad
  15. f 4 ( z ) = z + a / c f_{4}(z)=z+a/c\quad
  16. f 4 f 3 f 2 f 1 ( z ) = f ( z ) = a z + b c z + d . f_{4}\circ f_{3}\circ f_{2}\circ f_{1}(z)=f(z)=\frac{az+b}{cz+d}.
  17. g 1 g 2 g 3 g 4 ( z ) = f - 1 ( z ) = d z - b - c z + a g_{1}\circ g_{2}\circ g_{3}\circ g_{4}(z)=f^{-1}(z)=\frac{dz-b}{-cz+a}
  18. z 1 , z 2 , z 3 , z 4 z_{1},z_{2},z_{3},z_{4}
  19. w 1 , w 2 , w 3 , w 4 w_{1},w_{2},w_{3},w_{4}
  20. ( z 1 - z 3 ) ( z 2 - z 4 ) ( z 2 - z 3 ) ( z 1 - z 4 ) = ( w 1 - w 3 ) ( w 2 - w 4 ) ( w 2 - w 3 ) ( w 1 - w 4 ) . \frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})}=\frac{(w_{1}-w_{% 3})(w_{2}-w_{4})}{(w_{2}-w_{3})(w_{1}-w_{4})}.
  21. z 1 , z 2 , z 3 , z 4 z_{1},z_{2},z_{3},z_{4}
  22. z 1 , z 2 , z 3 , z_{1},z_{2},z_{3},\infty
  23. ( z 1 - z 3 ) ( z 2 - z 3 ) . \frac{(z_{1}-z_{3})}{(z_{2}-z_{3})}.
  24. z * = e 2 i θ z - z 0 ¯ + z 0 . z^{*}=e^{2i\theta}\overline{z-z_{0}}+z_{0}.
  25. z * = r 2 z - z 0 ¯ + z 0 z^{*}=\frac{r^{2}}{\overline{z-z_{0}}}+z_{0}
  26. = ( a b c d ) \mathfrak{H}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  27. f ( z ) = a z + b c z + d . f(z)=\frac{az+b}{cz+d}.
  28. π : GL ( 2 , 𝐂 ) Aut ( 𝐂 ^ ) \pi\colon\operatorname{GL}(2,\mathbf{C})\to\operatorname{Aut}(\widehat{\mathbf% {C}})
  29. \mathfrak{H}
  30. \mathfrak{H}
  31. Aut ( 𝐂 ^ ) PGL ( 2 , 𝐂 ) . \operatorname{Aut}(\widehat{\mathbf{C}})\cong\operatorname{PGL}(2,\mathbf{C}).
  32. [ z 1 : z 2 ] z 1 / z 2 . [z_{1}:z_{2}]\leftrightarrow z_{1}/z_{2}.
  33. \mathfrak{H}
  34. Aut ( 𝐂 ^ ) PSL ( 2 , 𝐂 ) . \operatorname{Aut}(\widehat{\mathbf{C}})\cong\operatorname{PSL}(2,\mathbf{C}).
  35. f 1 ( z ) = ( z - z 1 ) ( z 2 - z 3 ) ( z - z 3 ) ( z 2 - z 1 ) f_{1}(z)=\frac{(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}
  36. 1 = ( z 2 - z 3 - z 1 ( z 2 - z 3 ) z 2 - z 1 - z 3 ( z 2 - z 1 ) ) \mathfrak{H}_{1}=\begin{pmatrix}z_{2}-z_{3}&-z_{1}(z_{2}-z_{3})\\ z_{2}-z_{1}&-z_{3}(z_{2}-z_{1})\end{pmatrix}
  37. 1 \mathfrak{H}_{1}
  38. 2 \mathfrak{H}_{2}
  39. \mathfrak{H}
  40. = 2 - 1 1 . \mathfrak{H}=\mathfrak{H}_{2}^{-1}\mathfrak{H}_{1}.
  41. w = a z + b c z + d w=\frac{az+b}{cz+d}
  42. c w z - a z + d w - b = 0 \,cwz-az+dw-b=0
  43. ( z ) \mathfrak{H}(z)
  44. ( z 1 , z 2 , z 3 ) (z_{1},z_{2},z_{3})
  45. ( w 1 , w 2 , w 3 ) (w_{1},w_{2},w_{3})
  46. ( z i , w i ) (z_{i},w_{i})
  47. det ( z w z w 1 z 1 w 1 z 1 w 1 1 z 2 w 2 z 2 w 2 1 z 3 w 3 z 3 w 3 1 ) \det\begin{pmatrix}zw&z&w&1\\ z_{1}w_{1}&z_{1}&w_{1}&1\\ z_{2}w_{2}&z_{2}&w_{2}&1\\ z_{3}w_{3}&z_{3}&w_{3}&1\end{pmatrix}\,
  48. a = det ( z 1 w 1 w 1 1 z 2 w 2 w 2 1 z 3 w 3 w 3 1 ) a=\det\begin{pmatrix}z_{1}w_{1}&w_{1}&1\\ z_{2}w_{2}&w_{2}&1\\ z_{3}w_{3}&w_{3}&1\end{pmatrix}\,
  49. b = det ( z 1 w 1 z 1 w 1 z 2 w 2 z 2 w 2 z 3 w 3 z 3 w 3 ) b=\det\begin{pmatrix}z_{1}w_{1}&z_{1}&w_{1}\\ z_{2}w_{2}&z_{2}&w_{2}\\ z_{3}w_{3}&z_{3}&w_{3}\end{pmatrix}\,
  50. c = det ( z 1 w 1 1 z 2 w 2 1 z 3 w 3 1 ) c=\det\begin{pmatrix}z_{1}&w_{1}&1\\ z_{2}&w_{2}&1\\ z_{3}&w_{3}&1\end{pmatrix}\,
  51. d = det ( z 1 w 1 z 1 1 z 2 w 2 z 2 1 z 3 w 3 z 3 1 ) d=\det\begin{pmatrix}z_{1}w_{1}&z_{1}&1\\ z_{2}w_{2}&z_{2}&1\\ z_{3}w_{3}&z_{3}&1\end{pmatrix}
  52. = ( a b c d ) \,\mathfrak{H}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  53. \mathfrak{H}
  54. ( z 1 - z 2 ) ( z 1 - z 3 ) ( z 2 - z 3 ) ( w 1 - w 2 ) ( w 1 - w 3 ) ( w 2 - w 3 ) (z_{1}-z_{2})(z_{1}-z_{3})(z_{2}-z_{3})(w_{1}-w_{2})(w_{1}-w_{3})(w_{2}-w_{3})
  55. tr = a + d \operatorname{tr}\,\mathfrak{H}=a+d
  56. tr 𝔊 𝔊 - 1 = tr , \operatorname{tr}\,\mathfrak{GHG}^{-1}=\operatorname{tr}\,\mathfrak{H},
  57. \mathfrak{H}
  58. , \mathfrak{H},\mathfrak{H}^{\prime}
  59. det = det = 1 \det\mathfrak{H}=\det\mathfrak{H}^{\prime}=1
  60. tr 2 = tr 2 . \operatorname{tr}^{2}\,\mathfrak{H}=\operatorname{tr}^{2}\,\mathfrak{H}^{% \prime}.
  61. \mathfrak{H}
  62. det = a d - b c = 1 \det{\mathfrak{H}}=ad-bc=1
  63. tr 2 = ( a + d ) 2 = 4 \operatorname{tr}^{2}\mathfrak{H}=(a+d)^{2}=4
  64. \mathfrak{H}
  65. \mathfrak{H}
  66. 𝐂 ^ = 𝐂 { } \widehat{\mathbf{C}}=\mathbf{C}\cup\{\infty\}
  67. ( 1 1 0 1 ) \begin{pmatrix}1&1\\ 0&1\end{pmatrix}
  68. 𝐂 ^ \widehat{\mathbf{C}}
  69. { ( 1 b 0 1 ) b 𝐂 } ; \left\{\begin{pmatrix}1&b\\ 0&1\end{pmatrix}\mid b\in\mathbf{C}\right\};
  70. ( λ 0 0 λ - 1 ) \begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}
  71. \mathfrak{H}
  72. 0 tr 2 < 4. 0\leq\operatorname{tr}^{2}\mathfrak{H}<4.\,
  73. λ = e i α \lambda=e^{i\alpha}
  74. ( cos α - sin α sin α cos α ) \begin{pmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{pmatrix}
  75. \mathfrak{H}
  76. n \mathfrak{H}^{n}
  77. tr = 0 \operatorname{tr}\mathfrak{H}=0
  78. ( 0 - 1 1 0 ) . \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}.
  79. 1 / z , 1/z,
  80. 1 - z 1-z
  81. z / ( z - 1 ) z/(z-1)
  82. \mathfrak{H}
  83. tr 2 > 4. \operatorname{tr}^{2}\mathfrak{H}>4.\,
  84. tr 2 \operatorname{tr}^{2}\mathfrak{H}
  85. | λ | 1 |\lambda|\neq 1
  86. ( i 0 0 - i ) \begin{pmatrix}i&0\\ 0&-i\end{pmatrix}
  87. k = e ± i θ 1 k=e^{\pm i\theta}\neq 1
  88. ( e i θ / 2 0 0 e - i θ / 2 ) \begin{pmatrix}e^{i\theta/2}&0\\ 0&e^{-i\theta/2}\end{pmatrix}
  89. ( 1 a 0 1 ) \begin{pmatrix}1&a\\ 0&1\end{pmatrix}
  90. k 𝐑 + k\in\mathbf{R}^{+}
  91. k = e ± θ 1 k=e^{\pm\theta}\neq 1
  92. ( e θ / 2 0 0 e - θ / 2 ) \begin{pmatrix}e^{\theta/2}&0\\ 0&e^{-\theta/2}\end{pmatrix}
  93. | k | 1 |k|\neq 1
  94. k = λ 2 , λ - 2 k=\lambda^{2},\lambda^{-2}
  95. ( λ 0 0 λ - 1 ) \begin{pmatrix}\lambda&0\\ 0&\lambda^{-1}\end{pmatrix}
  96. γ 1 , γ 2 \gamma_{1},\gamma_{2}
  97. f ( z ) = a z + b c z + d f(z)=\frac{az+b}{cz+d}
  98. c γ 2 - ( a - d ) γ - b = 0 , c\gamma^{2}-(a-d)\gamma-b=0\ ,
  99. γ 1 , 2 = ( a - d ) ± ( a - d ) 2 + 4 b c 2 c = ( a - d ) ± ( a + d ) 2 - 4 ( a d - b c ) 2 c . \gamma_{1,2}=\frac{(a-d)\pm\sqrt{(a-d)^{2}+4bc}}{2c}=\frac{(a-d)\pm\sqrt{(a+d)% ^{2}-4(ad-bc)}}{2c}.
  100. ( a - d ) 2 + 4 c b = ( a - d ) 2 + 4 a d - 4 = ( a + d ) 2 - 4 = tr 2 - 4. (a-d)^{2}+4cb=(a-d)^{2}+4ad-4=(a+d)^{2}-4=\operatorname{tr}^{2}\mathfrak{H}-4.
  101. γ = - b a - d . \gamma=-\frac{b}{a-d}.
  102. z α z + β . z\mapsto\alpha z+\beta.\,
  103. z z + β . z\mapsto z+\beta.
  104. χ ( 𝐂 ^ ) = 2. \chi(\hat{\mathbf{C}})=2.
  105. ( 1 + x ) / ( 1 - x ) (1+x)/(1-x)
  106. z k z z\mapsto kz\,
  107. g ( z ) = z - γ 1 z - γ 2 g(z)=\frac{z-\gamma_{1}}{z-\gamma_{2}}
  108. g f g - 1 gfg^{-1}
  109. g f g - 1 ( z ) = k z gfg^{-1}(z)=kz
  110. f ( z ) - γ 1 f ( z ) - γ 2 = k z - γ 1 z - γ 2 . \frac{f(z)-\gamma_{1}}{f(z)-\gamma_{2}}=k\frac{z-\gamma_{1}}{z-\gamma_{2}}.
  111. ( k ; γ 1 , γ 2 ) = ( γ 1 - k γ 2 ( k - 1 ) γ 1 γ 2 1 - k k γ 1 - γ 2 ) \mathfrak{H}(k;\gamma_{1},\gamma_{2})=\begin{pmatrix}\gamma_{1}-k\gamma_{2}&(k% -1)\gamma_{1}\gamma_{2}\\ 1-k&k\gamma_{1}-\gamma_{2}\end{pmatrix}
  112. ( k ; γ , ) = ( k ( 1 - k ) γ 0 1 ) . \mathfrak{H}(k;\gamma,\infty)=\begin{pmatrix}k&(1-k)\gamma\\ 0&1\end{pmatrix}.
  113. f ( γ 1 ) = k f^{\prime}(\gamma_{1})=k\,
  114. f ( γ 2 ) = 1 / k . f^{\prime}(\gamma_{2})=1/k.\,
  115. ( k ; γ 1 , γ 2 ) = ( 1 / k ; γ 2 , γ 1 ) . \mathfrak{H}(k;\gamma_{1},\gamma_{2})=\mathfrak{H}(1/k;\gamma_{2},\gamma_{1}).
  116. g f g - 1 gfg^{-1}
  117. g f g - 1 ( z ) = z + β . gfg^{-1}(z)=z+\beta\,.
  118. 1 f ( z ) - γ = 1 z - γ + β . \frac{1}{f(z)-\gamma}=\frac{1}{z-\gamma}+\beta.
  119. ( β ; γ ) = ( 1 + γ β - β γ 2 β 1 - γ β ) \mathfrak{H}(\beta;\gamma)=\begin{pmatrix}1+\gamma\beta&-\beta\gamma^{2}\\ \beta&1-\gamma\beta\end{pmatrix}
  120. ( β ; ) = ( 1 β 0 1 ) \mathfrak{H}(\beta;\infty)=\begin{pmatrix}1&\beta\\ 0&1\end{pmatrix}
  121. f ( γ ) = 1. f^{\prime}(\gamma)=1.\,
  122. e ρ + α i = k . e^{\rho+\alpha i}=k.\;
  123. \mathfrak{H}
  124. = n \mathfrak{H}^{\prime}=\mathfrak{H}^{n}
  125. γ 1 = γ 1 , γ 2 = γ 2 , k = k n \gamma_{1}^{\prime}=\gamma_{1},\gamma_{2}^{\prime}=\gamma_{2},k^{\prime}=k^{n}
  126. z = - d c z_{\infty}=-\frac{d}{c}
  127. \mathfrak{H}
  128. \mathfrak{H}
  129. Z = a c Z_{\infty}=\frac{a}{c}
  130. γ 1 + γ 2 = z + Z . \gamma_{1}+\gamma_{2}=z_{\infty}+Z_{\infty}.
  131. \mathfrak{H}
  132. z z_{\infty}
  133. = ( Z - γ 1 γ 2 1 - z ) , Z = γ 1 + γ 2 - z . \mathfrak{H}=\begin{pmatrix}Z_{\infty}&-\gamma_{1}\gamma_{2}\\ 1&-z_{\infty}\end{pmatrix},\;\;Z_{\infty}=\gamma_{1}+\gamma_{2}-z_{\infty}.
  134. z z_{\infty}
  135. γ 1 , γ 2 \gamma_{1},\gamma_{2}
  136. z = k γ 1 - γ 2 1 - k z_{\infty}=\frac{k\gamma_{1}-\gamma_{2}}{1-k}
  137. k = γ 2 - z γ 1 - z = Z - γ 1 Z - γ 2 = a - c γ 1 a - c γ 2 , k=\frac{\gamma_{2}-z_{\infty}}{\gamma_{1}-z_{\infty}}=\frac{Z_{\infty}-\gamma_% {1}}{Z_{\infty}-\gamma_{2}}=\frac{a-c\gamma_{1}}{a-c\gamma_{2}},
  138. k = ( a + d ) + ( a - d ) 2 + 4 b c ( a + d ) - ( a - d ) 2 + 4 b c . k=\frac{(a+d)+\sqrt{(a-d)^{2}+4bc}}{(a+d)-\sqrt{(a-d)^{2}+4bc}}.
  139. λ 1 λ 2 \lambda_{1}\over\lambda_{2}
  140. = ( a b c d ) \mathfrak{H}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  141. det ( λ I 2 - ) = λ 2 - tr λ + det = λ 2 - ( a + d ) λ + ( a d - b c ) \det(\lambda I_{2}-\mathfrak{H})=\lambda^{2}-\operatorname{tr}\mathfrak{H}\,% \lambda+\det\mathfrak{H}=\lambda^{2}-(a+d)\lambda+(ad-bc)
  142. λ i = ( a + d ) ± ( a - d ) 2 + 4 b c 2 = ( a + d ) ± ( a + d ) 2 - 4 ( a d - b c ) 2 = c γ i + d . \lambda_{i}=\frac{(a+d)\pm\sqrt{(a-d)^{2}+4bc}}{2}=\frac{(a+d)\pm\sqrt{(a+d)^{% 2}-4(ad-bc)}}{2}=c\gamma_{i}+d\ .
  143. Q ( x 0 , x 1 , x 2 , x 3 ) = x 0 2 - x 1 2 - x 2 2 - x 3 2 . Q(x_{0},x_{1},x_{2},x_{3})=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.
  144. X = [ x 0 + x 1 x 2 + i x 3 x 2 - i x 3 x 0 - x 1 ] . X=\begin{bmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\ x_{2}-ix_{3}&x_{0}-x_{1}\end{bmatrix}.
  145. PSL ( 2 , 𝐂 ) S O + ( 1 , 3 ) . \operatorname{PSL}(2,\mathbf{C})\cong SO^{+}(1,3).
  146. X = ξ ξ ¯ T = ξ ξ * . X=\xi\bar{\xi}^{T}=\xi\xi^{*}.
  147. x 1 2 + x 2 2 + x 3 2 = 1 x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1
  148. ( 1 , x 1 1 - x 3 , x 2 1 - x 3 , 0 ) . \left(1,\frac{x_{1}}{1-x_{3}},\frac{x_{2}}{1-x_{3}},0\right).
  149. ζ = x 1 + i x 2 1 - x 3 , \zeta=\frac{x_{1}+ix_{2}}{1-x_{3}},
  150. x 1 \displaystyle x_{1}
  151. ( z , w ) ( x 0 , x 1 , x 2 , x 3 ) = ( z z ¯ + w w ¯ , z w ¯ + w z ¯ , i - 1 ( z w ¯ - w z ¯ ) , z z ¯ - w w ¯ ) (z,w)\mapsto(x_{0},x_{1},x_{2},x_{3})=(z\bar{z}+w\bar{w},z\bar{w}+w\bar{z},i^{% -1}(z\bar{w}-w\bar{z}),z\bar{z}-w\bar{w})
  152. z z ¯ + w w ¯ = 1. z\bar{z}+w\bar{w}=1.
  153. [ x 0 + x 1 x 2 + i x 3 x 2 - i x 3 x 0 - x 1 ] = 2 [ z w ] [ z ¯ w ¯ ] . \begin{bmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\ x_{2}-ix_{3}&x_{0}-x_{1}\end{bmatrix}=2\begin{bmatrix}z\\ w\end{bmatrix}\begin{bmatrix}\bar{z}&\bar{w}\end{bmatrix}.
  154. f ( z ) = z + i i z + 1 f(z)=\frac{z+i}{iz+1}
  155. \mathcal{M}
  156. 0 := { z u z - v ¯ v z + u ¯ : | u | 2 + | v | 2 = 1 } , \mathcal{M}_{0}:=\left\{z\mapsto\frac{uz-\bar{v}}{vz+\bar{u}}:|u|^{2}+|v|^{2}=% 1\right\},
  157. PSL ( 2 , 𝐂 ) \mathcal{M}\cong\operatorname{PSL}(2,\mathbf{C})
  158. n = 2 n=2
  159. 0 1 - 1 0. 0\mapsto 1\mapsto\infty\mapsto-1\mapsto 0.

Multilinear_map.html

  1. f : V 1 × × V n W , f\colon V_{1}\times\cdots\times V_{n}\to W\,\text{,}
  2. V 1 , , V n V_{1},\ldots,V_{n}
  3. W W\!
  4. i i\!
  5. v i v_{i}\!
  6. f ( v 1 , , v n ) f(v_{1},\ldots,v_{n})
  7. v i v_{i}\!
  8. 3 \mathbb{R}^{3}
  9. F : m n F\colon\mathbb{R}^{m}\to\mathbb{R}^{n}
  10. k k\!
  11. F F\!
  12. p p\!
  13. k k\!
  14. D k f ( p ) : m × × m n D^{k}\!f(p)\colon\mathbb{R}^{m}\times\cdots\times\mathbb{R}^{m}\to\mathbb{R}^{n}
  15. f : V 1 × × V n W , f\colon V_{1}\times\cdots\times V_{n}\to W\,\text{,}
  16. V i V_{i}\!
  17. d i d_{i}\!
  18. W W\!
  19. d d\!
  20. { 𝐞 i 1 , , 𝐞 i d i } \{\,\textbf{e}_{i1},\ldots,\,\textbf{e}_{id_{i}}\}
  21. V i V_{i}\!
  22. { 𝐛 1 , , 𝐛 d } \{\,\textbf{b}_{1},\ldots,\,\textbf{b}_{d}\}
  23. W W\!
  24. A j 1 j n k A_{j_{1}\cdots j_{n}}^{k}
  25. f ( 𝐞 1 j 1 , , 𝐞 n j n ) = A j 1 j n 1 𝐛 1 + + A j 1 j n d 𝐛 d . f(\,\textbf{e}_{1j_{1}},\ldots,\,\textbf{e}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1% }\,\,\textbf{b}_{1}+\cdots+A_{j_{1}\cdots j_{n}}^{d}\,\,\textbf{b}_{d}.
  26. { A j 1 j n k 1 j i d i , 1 k d } \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}
  27. f f\!
  28. 𝐯 i = j = 1 d i v i j 𝐞 i j \,\textbf{v}_{i}=\sum_{j=1}^{d_{i}}v_{ij}\,\textbf{e}_{ij}\!
  29. 1 i n 1\leq i\leq n\!
  30. f ( 𝐯 1 , , 𝐯 n ) = j 1 = 1 d 1 j n = 1 d n k = 1 d A j 1 j n k v 1 j 1 v n j n 𝐛 k . f(\,\textbf{v}_{1},\ldots,\,\textbf{v}_{n})=\sum_{j_{1}=1}^{d_{1}}\cdots\sum_{% j_{n}=1}^{d_{n}}\sum_{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_% {n}}\,\textbf{b}_{k}.
  31. f : R 2 × R 2 × R 2 R f\colon R^{2}\times R^{2}\times R^{2}\to R
  32. V i = R 2 , d i = 2 V_{i}=R^{2},d_{i}=2
  33. W = R , d = 1 W=R,d=1
  34. V i V_{i}
  35. { 𝐞 i 1 , , 𝐞 i d i } = { 𝐞 1 , 𝐞 2 } = { ( 1 , 0 ) , ( 0 , 1 ) } \{\,\textbf{e}_{i1},\ldots,\,\textbf{e}_{id_{i}}\}=\{\,\textbf{e}_{1},\,% \textbf{e}_{2}\}=\{(1,0),(0,1)\}
  36. f ( 𝐞 1 i , 𝐞 2 j , 𝐞 3 k ) = f ( 𝐞 i , 𝐞 j , 𝐞 k ) = A i j k f(\,\textbf{e}_{1i},\,\textbf{e}_{2j},\,\textbf{e}_{3k})=f(\,\textbf{e}_{i},\,% \textbf{e}_{j},\,\textbf{e}_{k})=A_{ijk}
  37. i , j , k { 1 , 2 } i,j,k\in\{1,2\}
  38. A i j k A_{ijk}
  39. V i V_{i}
  40. { 𝐞 1 , 𝐞 1 , 𝐞 1 } , { 𝐞 1 , 𝐞 1 , 𝐞 2 } , { 𝐞 1 , 𝐞 2 , 𝐞 1 } , { 𝐞 1 , 𝐞 2 , 𝐞 2 } , { 𝐞 2 , 𝐞 1 , 𝐞 1 } , { 𝐞 2 , 𝐞 1 , 𝐞 2 } , { 𝐞 2 , 𝐞 2 , 𝐞 1 } , { 𝐞 2 , 𝐞 2 , 𝐞 2 } , \{\,\textbf{e}_{1},\,\textbf{e}_{1},\,\textbf{e}_{1}\},\{\,\textbf{e}_{1},\,% \textbf{e}_{1},\,\textbf{e}_{2}\},\{\,\textbf{e}_{1},\,\textbf{e}_{2},\,% \textbf{e}_{1}\},\{\,\textbf{e}_{1},\,\textbf{e}_{2},\,\textbf{e}_{2}\},\{\,% \textbf{e}_{2},\,\textbf{e}_{1},\,\textbf{e}_{1}\},\{\,\textbf{e}_{2},\,% \textbf{e}_{1},\,\textbf{e}_{2}\},\{\,\textbf{e}_{2},\,\textbf{e}_{2},\,% \textbf{e}_{1}\},\{\,\textbf{e}_{2},\,\textbf{e}_{2},\,\textbf{e}_{2}\},
  41. 𝐯 i V i = R 2 \,\textbf{v}_{i}\in V_{i}=R^{2}
  42. 𝐯 i = j = 1 2 v i j 𝐞 i j = v i 1 × 𝐞 1 + v i 2 × 𝐞 2 = v i 1 × ( 1 , 0 ) + v i 2 × ( 0 , 1 ) \,\textbf{v}_{i}=\sum_{j=1}^{2}v_{ij}\,\textbf{e}_{ij}=v_{i1}\times\,\textbf{e% }_{1}+v_{i2}\times\,\textbf{e}_{2}=v_{i1}\times(1,0)+v_{i2}\times(0,1)\!
  43. 𝐯 i R 2 \,\textbf{v}_{i}\in R^{2}
  44. f ( 𝐯 1 , 𝐯 2 , 𝐯 3 ) = i = 1 2 j = 1 2 k = 1 2 A i j k v 1 i v 2 j v 3 k f(\,\textbf{v}_{1},\,\textbf{v}_{2},\,\textbf{v}_{3})=\sum_{i=1}^{2}\sum_{j=1}% ^{2}\sum_{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k}
  45. f ( ( a , b ) , ( c , d ) , ( e , f ) ) = a c e × f ( 𝐞 1 , 𝐞 1 , 𝐞 1 ) + a c f × f ( 𝐞 1 , 𝐞 1 , 𝐞 2 ) + a d e × f ( 𝐞 1 , 𝐞 2 , 𝐞 1 ) + a d f × f ( 𝐞 1 , 𝐞 2 , 𝐞 2 ) + b c e × f ( 𝐞 2 , 𝐞 1 , 𝐞 1 ) + b c f × f ( 𝐞 2 , 𝐞 1 , 𝐞 2 ) + b d e × f ( 𝐞 2 , 𝐞 2 , 𝐞 1 ) + b d f × f ( 𝐞 2 , 𝐞 2 , 𝐞 2 ) f((a,b),(c,d),(e,f))=ace\times f(\,\textbf{e}_{1},\,\textbf{e}_{1},\,\textbf{e% }_{1})+acf\times f(\,\textbf{e}_{1},\,\textbf{e}_{1},\,\textbf{e}_{2})+ade% \times f(\,\textbf{e}_{1},\,\textbf{e}_{2},\,\textbf{e}_{1})+adf\times f(\,% \textbf{e}_{1},\,\textbf{e}_{2},\,\textbf{e}_{2})+bce\times f(\,\textbf{e}_{2}% ,\,\textbf{e}_{1},\,\textbf{e}_{1})+bcf\times f(\,\textbf{e}_{2},\,\textbf{e}_% {1},\,\textbf{e}_{2})+bde\times f(\,\textbf{e}_{2},\,\textbf{e}_{2},\,\textbf{% e}_{1})+bdf\times f(\,\textbf{e}_{2},\,\textbf{e}_{2},\,\textbf{e}_{2})
  46. f : V 1 × × V n W , f\colon V_{1}\times\cdots\times V_{n}\to W\,\text{,}
  47. F : V 1 V n W , F\colon V_{1}\otimes\cdots\otimes V_{n}\to W\,\text{,}
  48. V 1 V n V_{1}\otimes\cdots\otimes V_{n}\!
  49. V 1 , , V n V_{1},\ldots,V_{n}
  50. f f\!
  51. F F\!
  52. F ( v 1 v n ) = f ( v 1 , , v n ) . F(v_{1}\otimes\cdots\otimes v_{n})=f(v_{1},\ldots,v_{n}).
  53. a i a_{i}
  54. D ( A ) = D ( a 1 , , a n ) D(A)=D(a_{1},\ldots,a_{n})\,
  55. D ( a 1 , , c a i + a i , , a n ) = c D ( a 1 , , a i , , a n ) + D ( a 1 , , a i , , a n ) D(a_{1},\ldots,ca_{i}+a_{i}^{\prime},\ldots,a_{n})=cD(a_{1},\ldots,a_{i},% \ldots,a_{n})+D(a_{1},\ldots,a_{i}^{\prime},\ldots,a_{n})\,
  56. e ^ j \hat{e}_{j}
  57. a i a_{i}
  58. a i = j = 1 n A ( i , j ) e ^ j a_{i}=\sum_{j=1}^{n}A(i,j)\hat{e}_{j}
  59. D ( A ) = D ( j = 1 n A ( 1 , j ) e ^ j , a 2 , , a n ) = j = 1 n A ( 1 , j ) D ( e ^ j , a 2 , , a n ) D(A)=D\left(\sum_{j=1}^{n}A(1,j)\hat{e}_{j},a_{2},\ldots,a_{n}\right)=\sum_{j=% 1}^{n}A(1,j)D(\hat{e}_{j},a_{2},\ldots,a_{n})
  60. a i a_{i}
  61. D ( A ) = 1 k i n A ( 1 , k 1 ) A ( 2 , k 2 ) A ( n , k n ) D ( e ^ k 1 , , e ^ k n ) D(A)=\sum_{1\leq k_{i}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D(\hat{e}_{k_% {1}},\dots,\hat{e}_{k_{n}})
  62. 1 i n 1\leq i\leq n
  63. 1 k i n = 1 k 1 n 1 k i n 1 k n n \sum_{1\leq k_{i}\leq n}=\sum_{1\leq k_{1}\leq n}\ldots\sum_{1\leq k_{i}\leq n% }\ldots\sum_{1\leq k_{n}\leq n}\,
  64. D D
  65. e ^ k 1 , , e ^ k n \hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}
  66. D ( A ) = A 1 , 1 A 2 , 1 D ( e ^ 1 , e ^ 1 ) + A 1 , 1 A 2 , 2 D ( e ^ 1 , e ^ 2 ) + A 1 , 2 A 2 , 1 D ( e ^ 2 , e ^ 1 ) + A 1 , 2 A 2 , 2 D ( e ^ 2 , e ^ 2 ) D(A)=A_{1,1}A_{2,1}D(\hat{e}_{1},\hat{e}_{1})+A_{1,1}A_{2,2}D(\hat{e}_{1},\hat% {e}_{2})+A_{1,2}A_{2,1}D(\hat{e}_{2},\hat{e}_{1})+A_{1,2}A_{2,2}D(\hat{e}_{2},% \hat{e}_{2})\,
  67. e ^ 1 = [ 1 , 0 ] \hat{e}_{1}=[1,0]
  68. e ^ 2 = [ 0 , 1 ] \hat{e}_{2}=[0,1]
  69. D ( e ^ 1 , e ^ 1 ) = D ( e ^ 2 , e ^ 2 ) = 0 D(\hat{e}_{1},\hat{e}_{1})=D(\hat{e}_{2},\hat{e}_{2})=0
  70. D ( e ^ 2 , e ^ 1 ) = - D ( e ^ 1 , e ^ 2 ) = - D ( I ) D(\hat{e}_{2},\hat{e}_{1})=-D(\hat{e}_{1},\hat{e}_{2})=-D(I)
  71. D ( I ) = 1 D(I)=1
  72. D ( A ) = A 1 , 1 A 2 , 2 - A 1 , 2 A 2 , 1 D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}\,

Multiplicative_inverse.html

  1. z = a + b i z=a+bi
  2. z ¯ = a - b i \bar{z}=a-bi
  3. z z ¯ = z 2 z\bar{z}=\|z\|^{2}
  4. 1 z = z ¯ z z ¯ = z ¯ z 2 = a - b i a 2 + b 2 = a a 2 + b 2 - b a 2 + b 2 i . \frac{1}{z}=\frac{\bar{z}}{z\bar{z}}=\frac{\bar{z}}{\|z\|^{2}}=\frac{a-bi}{a^{% 2}+b^{2}}=\frac{a}{a^{2}+b^{2}}-\frac{b}{a^{2}+b^{2}}i.
  5. 1 / z = z ¯ 1/z=\bar{z}
  6. i i
  7. i i
  8. i i
  9. i i
  10. i i
  11. i i
  12. z = r ( c o s φ + i s i n φ ) z=r(cos φ+isin φ)
  13. 1 z = 1 r ( cos ( - φ ) + i sin ( - φ ) ) . \frac{1}{z}=\frac{1}{r}\left(\cos(-\varphi)+i\sin(-\varphi)\right).
  14. d d x x - 1 = ( - 1 ) x ( - 1 ) - 1 = - x - 2 = - 1 x 2 . \frac{d}{dx}x^{-1}=(-1)x^{(-1)-1}=-x^{-2}=-\frac{1}{x^{2}}.
  15. 1 x d x = x 0 0 + C \int\frac{1}{x}\,dx=\frac{x^{0}}{0}\ +C
  16. 1 a 1 x d x = ln a , \int_{1}^{a}\frac{1}{x}\,dx=\ln a,
  17. 1 x d x = ln x + C . \int\frac{1}{x}\,dx=\ln x+C.
  18. d d x e x = e x \frac{d}{dx}e^{x}=e^{x}
  19. y = e x y=e^{x}
  20. x = ln y x=\ln y
  21. d y d x = y d y y = d x 1 y d y = 1 d x 1 y d y = x + C = ln y + C . \frac{dy}{dx}=y\quad\Rightarrow\quad\frac{dy}{y}=dx\quad\Rightarrow\quad\int% \frac{1}{y}\,dy=\int 1\,dx\quad\Rightarrow\quad\int\frac{1}{y}\,dy=x+C=\ln y+C.
  22. f ( x ) = 1 / x - b f(x)=1/x-b
  23. x 0 x_{0}
  24. x n + 1 = x n - f ( x n ) f ( x n ) = x n - 1 / x n - b - 1 / x n 2 = 2 x n - b x n 2 = x n ( 2 - b x n ) . x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}=x_{n}-\frac{1/x_{n}-b}{-1/x_{% n}^{2}}=2x_{n}-bx_{n}^{2}=x_{n}(2-bx_{n}).
  25. f ( 1 / e ) f(1/e)
  26. f ( x ) = x x f(x)=x^{x}
  27. ϕ = 1 / ϕ + 1 \phi=1/\phi+1
  28. - ϕ = - 1 / ϕ - 1 -\phi=-1/\phi-1
  29. f ( n ) = n + ( n 2 + 1 ) , n N , n > 0 f(n)=n+\sqrt{(n^{2}+1)},n\in N,n>0
  30. f ( 2 ) f(2)
  31. 2 + 5 2+\sqrt{5}
  32. 1 / ( 2 + 5 ) 1/(2+\sqrt{5})
  33. - 2 + 5 -2+\sqrt{5}
  34. 4 4
  35. a x \displaystyle ax

Multiset.html

  1. ( A , m ) \left(A,m\right)
  2. A A
  3. m : A 1 m\colon A\to\mathbb{N}_{\geq 1}
  4. A A
  5. 1 = { 1 , 2 , 3 , } \mathbb{N}_{\geq 1}=\left\{1,2,3,\dots\right\}
  6. A A
  7. a a
  8. A A
  9. a a
  10. m ( a ) m\!\left(a\right)
  11. U U
  12. A A
  13. m U : U m_{U}\colon U\to\mathbb{N}
  14. U U
  15. = { 0 , 1 , 2 , 3 , } \mathbb{N}=\left\{0,1,2,3,\dots\right\}
  16. m m
  17. U U
  18. A A
  19. 1 A 1_{A}
  20. m m
  21. { ( a , m ( a ) ) : a A } \left\{\left(a,m\left(a\right)\right):a\in A\right\}
  22. { a , a , b } \left\{a,a,b\right\}
  23. ( { a , b } , { ( a , 2 ) , ( b , 1 ) } ) \left(\left\{a,b\right\},\left\{\left(a,2\right),\left(b,1\right)\right\}\right)
  24. { a , b } \left\{a,b\right\}
  25. ( { a , b } , { ( a , 1 ) , ( b , 1 ) } ) \left(\left\{a,b\right\},\left\{\left(a,1\right),\left(b,1\right)\right\}\right)
  26. ( a i ) \left(a_{i}\right)
  27. i i
  28. { a i } \left\{a_{i}\right\}
  29. x x
  30. i i
  31. a i = x a_{i}=x
  32. 𝟏 A : X { 0 , 1 } \mathbf{1}_{A}:X\to\{0,1\}\,
  33. 𝟏 A ( x ) = { 1 if x A , 0 if x A . \mathbf{1}_{A}(x)=\begin{cases}1&\,\text{if }x\in A,\\ 0&\,\text{if }x\notin A.\end{cases}
  34. 𝟏 A B ( x ) = min { 𝟏 A ( x ) , 𝟏 B ( x ) } . \mathbf{1}_{A\cap B}(x)=\min\{\mathbf{1}_{A}(x),\mathbf{1}_{B}(x)\}.
  35. 𝟏 A B ( x ) = max { 𝟏 A ( x ) , 𝟏 B ( x ) } . \mathbf{1}_{A\cup B}(x)=\max\{{\mathbf{1}_{A}(x),\mathbf{1}_{B}(x)}\}.
  36. A B x 𝟏 A ( x ) 𝟏 B ( x ) . A\subseteq B\Leftrightarrow\forall x\mathbf{1}_{A}(x)\leq\mathbf{1}_{B}(x).
  37. 𝟏 A × B ( x , y ) = 𝟏 A ( x ) 𝟏 B ( y ) . \mathbf{1}_{A\times B}(x,y)=\mathbf{1}_{A}(x)\cdot\mathbf{1}_{B}(y).
  38. | A | = x X 𝟏 A ( x ) . |A|=\sum_{x\in X}\mathbf{1}_{A}(x).
  39. 𝟏 A B ( x ) = 𝟏 A ( x ) + 𝟏 B ( x ) . \mathbf{1}_{A\uplus B}(x)=\mathbf{1}_{A}(x)+\mathbf{1}_{B}(x).
  40. 𝟏 A B ( x ) = max ( 0 , 𝟏 A ( x ) - 𝟏 B ( x ) ) . \mathbf{1}_{A\setminus B}(x)=\max(0,\mathbf{1}_{A}(x)-\mathbf{1}_{B}(x)).
  41. 𝟏 n A ( x ) = n × 𝟏 A ( x ) . \mathbf{1}_{n\otimes A}(x)=n\times\mathbf{1}_{A}(x).\,
  42. { 1 , 1 , 1 , 3 } { 1 , 1 , 2 } = { 1 , 1 } \{1,1,1,3\}\cap\{1,1,2\}=\{1,1\}\,
  43. { 1 , 1 } { 1 , 2 } = { 1 , 1 , 2 } \{1,1\}\cup\{1,2\}=\{1,1,2\}\,
  44. { 1 , 1 } { 1 , 1 , 1 , 2 } \{1,1\}\subseteq\{1,1,1,2\}\,
  45. | { 1 , 1 } | = 2 \left|\{1,1\}\right|=2\,
  46. { 1 , 1 } × { 1 , 2 } = { ( 1 , 1 ) , ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 2 ) } \{1,1\}\times\{1,2\}=\{(1,1),(1,1),(1,2),(1,2)\}\,
  47. { 1 , 1 } { 1 , 2 } = { 1 , 1 , 1 , 2 } \{1,1\}\uplus\{1,2\}=\{1,1,1,2\}\,
  48. 120 = 2 3 3 1 5 1 120=2^{3}3^{1}5^{1}\,
  49. λ λ
  50. A A
  51. A - λ I A-λI
  52. A A
  53. ( ( n k ) ) \textstyle\left(\!\!{n\choose k}\!\!\right)
  54. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  55. ( ( n k ) ) = ( n + k - 1 k ) = ( n + k - 1 ) ! k ! ( n - 1 ) ! = n ( n + 1 ) ( n + 2 ) ( n + k - 1 ) k ! , \left(\!\!{n\choose k}\!\!\right)={n+k-1\choose k}=\frac{(n+k-1)!}{k!\,(n-1)!}% ={n(n+1)(n+2)\cdots(n+k-1)\over k!},
  56. ( ( n k ) ) = n k ¯ k ! , \left(\!\!{n\choose k}\!\!\right)={n^{\overline{k}}\over k!},
  57. ( n k ) = n k ¯ k ! . {n\choose k}={n^{\underline{k}}\over k!}.
  58. \bullet\bullet\bullet\bullet\bullet\bullet\mid\bullet\bullet\mid\bullet\bullet% \bullet\mid\bullet\bullet\bullet\bullet\bullet\bullet\bullet
  59. ( 4 + 18 - 1 4 - 1 ) = ( 4 + 18 - 1 18 ) = 1330 , {4+18-1\choose 4-1}={4+18-1\choose 18}=1330,
  60. ( ( 4 18 ) ) = ( 21 18 ) = 21 ! 18 ! 3 ! = ( 21 3 ) = ( ( 19 3 ) ) , \left(\!\!{4\choose 18}\!\!\right)={21\choose 18}=\frac{21!}{18!\,3!}={21% \choose 3}=\left(\!\!{19\choose 3}\!\!\right),
  61. = \color P u r p l e 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 𝟏𝟗 𝟐𝟎 𝟐𝟏 𝟏 𝟐 𝟑 \color P u r p l e 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 , =\frac{{\color{Purple}{\mathfrak{4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10% \cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\mathbf{% \cdot 19\cdot 20\cdot 21}}{\mathbf{1\cdot 2\cdot 3}{\color{Purple}{\mathfrak{% \cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13% \cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}},
  62. = 1 2 3 4 5 16 17 18 19 𝟐𝟎 𝟐𝟏 1 2 3 4 5 16 17 18 × 1 𝟐 𝟑 , =\frac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf{\cdot\;% 19\cdot 20\cdot 21}}{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18% \,\times\,\mathbf{1\cdot 2\cdot 3\quad}},
  63. = 19 20 21 1 2 3 . =\frac{19\cdot 20\cdot 21}{1\cdot 2\cdot 3}.
  64. ( n k ) = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) k ! {n\choose k}={n(n-1)(n-2)\cdots(n-k+1)\over k!}
  65. ( ( n k ) ) = ( - 1 ) k ( - n k ) . \left(\!\!{n\choose k}\!\!\right)=(-1)^{k}{-n\choose k}.
  66. ( ( n k ) ) = ( ( n k - 1 ) ) + ( ( n - 1 k ) ) for n , k > 0 \left(\!\!{n\choose k}\!\!\right)=\left(\!\!{n\choose k-1}\!\!\right)+\left(\!% \!{n-1\choose k}\!\!\right)\quad\mbox{for }~{}n,k>0
  67. ( ( n 0 ) ) = 1 , n 𝒩 , and ( ( 0 k ) ) = 0 , k > 0. \left(\!\!{n\choose 0}\!\!\right)=1,\quad n\in\mathcal{N},\quad\mbox{and}~{}% \quad\left(\!\!{0\choose k}\!\!\right)=0,\quad k>0.
  68. ( ( n k - 1 ) ) \left(\!\!{n\choose k-1}\!\!\right)
  69. ( ( n - 1 k ) ) . \left(\!\!{n-1\choose k}\!\!\right).
  70. ( ( n k ) ) = ( ( n k - 1 ) ) + ( ( n - 1 k ) ) . \left(\!\!{n\choose k}\!\!\right)=\left(\!\!{n\choose k-1}\!\!\right)+\left(\!% \!{n-1\choose k}\!\!\right).
  71. ( 1 + x ) n = k = 0 n ( n k ) x k . (1+x)^{n}=\sum_{k=0}^{n}{n\choose k}\cdot x^{k}.
  72. n = 0 N k = 0 K ( K k ) ( N - K n - k ) x k y n - k . \sum_{n=0}^{N}\sum_{k=0}^{K}{K\choose k}\cdot{N-K\choose n-k}\cdot x^{k}\cdot y% ^{n-k}.
  73. ( K k ) ( N - K n - k ) . {K\choose k}\cdot{N-K\choose n-k}.
  74. ( 1 - x ) - n = k = 0 ( - n k ) ( - x ) k (1-x)^{-n}=\sum_{k=0}^{\infty}{-n\choose k}\cdot(-x)^{k}
  75. g A ( t ) = log ( i e t A i ) . g_{A}(t)=\log\left(\sum_{i}e^{t\cdot A_{i}}\right).
  76. g A + B ( t ) = log ( i j e t ( A i + B j ) ) = log ( i j e t A i e t B j ) g_{A+B}(t)=\log\left(\sum_{i}\sum_{j}e^{t\cdot(A_{i}+B_{j})}\right)=\log\left(% \sum_{i}\sum_{j}e^{t\cdot A_{i}}\cdot e^{t\cdot B_{j}}\right)
  77. g A + B ( t ) = log ( i e t A i j e t B j ) = log ( i e t A i ) + log ( j e t B j ) g_{A+B}(t)=\log\left(\sum_{i}e^{t\cdot A_{i}}\cdot\sum_{j}e^{t\cdot B_{j}}% \right)=\log\left(\sum_{i}e^{t\cdot A_{i}}\right)+\log\left(\sum_{j}e^{t\cdot B% _{j}}\right)
  78. g A + B ( t ) = g A ( t ) + g B ( t ) g_{A+B}(t)=g_{A}(t)+g_{B}(t)
  79. g A B ( t ) = log ( i j e t A i B j ) g_{A\cdot B}(t)=\log\left(\sum_{i}\sum_{j}e^{t\cdot A_{i}\cdot B_{j}}\right)\,
  80. g k A ( t ) = log ( i e t ( k A i ) ) = log ( i e ( t k ) A i ) = g A ( k t ) . g_{k\cdot A}(t)=\log\left(\sum_{i}e^{t\cdot(k\cdot A_{i})}\right)=\log\left(% \sum_{i}e^{(t\cdot k)\cdot A_{i}}\right)=g_{A}\left(k\cdot t\right).
  81. g k × A ( t ) = k g A ( t ) . g_{k\times A}(t)=k\cdot g_{A}(t).
  82. lim k k - 1 ( k 2 × { + 1 , - 1 } ) . \lim_{k\rightarrow\infty}k^{-1}\cdot(k^{2}\times\{+1,-1\}).
  83. lim k g k - 1 ( k 2 × { + 1 , - 1 } ) ( t ) = lim k d ( k 2 log ( e + t k - 1 + e - t k - 1 ) ) d t = lim k d ( k 2 log ( 2 ) + 2 - 1 t 2 + ) d t = t . \begin{aligned}\displaystyle\lim_{k\rightarrow\infty}g^{\prime}_{k^{-1}\cdot(k% ^{2}\times\{+1,-1\})}(t)&\displaystyle=\lim_{k\rightarrow\infty}\frac{d(k^{2}% \cdot\log(e^{+t\cdot k^{-1}}+e^{-t\cdot k^{-1}}))}{dt}\\ &\displaystyle=\lim_{k\rightarrow\infty}\frac{d(k^{2}\cdot\log(2)+2^{-1}\cdot t% ^{2}+\cdots)}{dt}=t.\end{aligned}

Multistatic_radar.html

  1. N N
  2. M M
  3. N M NM

Multivalued_function.html

  1. log ( a + b i ) \log(a+bi)
  2. a a
  3. b b
  4. log a 2 + b 2 + i arg ( a + b i ) + 2 π n i \log{\sqrt{a^{2}+b^{2}}}+i\arg(a+bi)+2\pi ni
  5. n n
  6. tan ( π 4 ) = tan ( 5 π 4 ) = tan ( - 3 π 4 ) = tan ( ( 2 n + 1 ) π 4 ) = = 1. \tan\left({\textstyle\frac{\pi}{4}}\right)=\tan\left({\textstyle\frac{5\pi}{4}% }\right)=\tan\left({\textstyle\frac{-3\pi}{4}}\right)=\tan\left({\textstyle% \frac{(2n+1)\pi}{4}}\right)=\cdots=1.

Municipal_bond.html

  1. r m = r c ( 1 - t ) r_{m}=r_{c}(1-t)\,
  2. r m = ( 10 % ) ( 100 % - 38 % ) = 6.2 % r_{m}=(10\%)(100\%-38\%)=6.2\%\,
  3. t = 1 - r m r c . t=1-\frac{r_{m}}{r_{c}}.
  4. r c = r m ( 1 - t ) r_{c}=\frac{r_{m}}{(1-t)}

Naive_Bayes_classifier.html

  1. 𝐱 = ( x 1 , , x n ) \mathbf{x}=(x_{1},\dots,x_{n})
  2. n n
  3. p ( C k | x 1 , , x n ) p(C_{k}|x_{1},\dots,x_{n})\,
  4. k k
  5. n n
  6. p ( C k | 𝐱 ) = p ( C k ) p ( 𝐱 | C k ) p ( 𝐱 ) . p(C_{k}|\mathbf{x})=\frac{p(C_{k})\ p(\mathbf{x}|C_{k})}{p(\mathbf{x})}.\,
  7. posterior = prior × likelihood evidence . \mbox{posterior}~{}=\frac{\mbox{prior}~{}\times\mbox{likelihood}~{}}{\mbox{% evidence}~{}}.\,
  8. C C
  9. F i F_{i}
  10. p ( C k , x 1 , , x n ) p(C_{k},x_{1},\dots,x_{n})\,
  11. p ( C k , x 1 , , x n ) \displaystyle p(C_{k},x_{1},\dots,x_{n})
  12. F i F_{i}
  13. F j F_{j}
  14. j i j\neq i
  15. C C
  16. p ( x i | C k , x j ) = p ( x i | C k ) p(x_{i}|C_{k},x_{j})=p(x_{i}|C_{k})\,
  17. p ( x i | C k , x j , x k ) = p ( x i | C k ) p(x_{i}|C_{k},x_{j},x_{k})=p(x_{i}|C_{k})\,
  18. p ( x i | C k , x j , x k , x l ) = p ( x i | C k ) p(x_{i}|C_{k},x_{j},x_{k},x_{l})=p(x_{i}|C_{k})\,
  19. i j , k , l i\neq j,k,l
  20. p ( C k | x 1 , , x n ) p ( C k , x 1 , , x n ) p ( C k ) p ( x 1 | C k ) p ( x 2 | C k ) p ( x 3 | C k ) p ( C k ) i = 1 n p ( x i | C k ) . \begin{aligned}\displaystyle p(C_{k}|x_{1},\dots,x_{n})&\displaystyle% \varpropto p(C_{k},x_{1},\dots,x_{n})\\ &\displaystyle\varpropto p(C_{k})\ p(x_{1}|C_{k})\ p(x_{2}|C_{k})\ p(x_{3}|C_{% k})\ \cdots\\ &\displaystyle\varpropto p(C_{k})\prod_{i=1}^{n}p(x_{i}|C_{k})\,.\end{aligned}
  21. C C
  22. p ( C k | x 1 , , x n ) = 1 Z p ( C k ) i = 1 n p ( x i | C k ) p(C_{k}|x_{1},\dots,x_{n})=\frac{1}{Z}p(C_{k})\prod_{i=1}^{n}p(x_{i}|C_{k})
  23. Z = p ( 𝐱 ) Z=p(\mathbf{x})
  24. x 1 , , x n x_{1},\dots,x_{n}
  25. y ^ = C k \hat{y}=C_{k}
  26. k k
  27. y ^ = argmax k { 1 , , K } p ( C k ) i = 1 n p ( x i | C k ) . \hat{y}=\underset{k\in\{1,\dots,K\}}{\operatorname{argmax}}\ p(C_{k})% \displaystyle\prod_{i=1}^{n}p(x_{i}|C_{k}).
  28. x x
  29. x x
  30. μ c \mu_{c}
  31. x x
  32. σ c 2 \sigma^{2}_{c}
  33. x x
  34. p ( x = v | c ) p(x=v|c)
  35. v v
  36. μ c \mu_{c}
  37. σ c 2 \sigma^{2}_{c}
  38. p ( x = v | c ) = 1 2 π σ c 2 e - ( v - μ c ) 2 2 σ c 2 p(x=v|c)=\frac{1}{\sqrt{2\pi\sigma^{2}_{c}}}\,e^{-\frac{(v-\mu_{c})^{2}}{2% \sigma^{2}_{c}}}
  39. ( p 1 , , p n ) (p_{1},\dots,p_{n})
  40. p i p_{i}
  41. i i
  42. K K
  43. 𝐱 = ( x 1 , , x n ) \mathbf{x}=(x_{1},\dots,x_{n})
  44. x i x_{i}
  45. i i
  46. 𝐱 \mathbf{x}
  47. p ( 𝐱 | C k ) = ( i x i ) ! i x i ! i p k i x i p(\mathbf{x}|C_{k})=\frac{(\sum_{i}x_{i})!}{\prod_{i}x_{i}!}\prod_{i}{p_{ki}}^% {x_{i}}
  48. log p ( C k | 𝐱 ) log ( p ( C k ) i = 1 n p k i x i ) = log p ( C k ) + i = 1 n x i log p k i = b + 𝐰 k 𝐱 \begin{aligned}\displaystyle\log p(C_{k}|\mathbf{x})&\displaystyle\varpropto% \log\left(p(C_{k})\prod_{i=1}^{n}{p_{ki}}^{x_{i}}\right)\\ &\displaystyle=\log p(C_{k})+\sum_{i=1}^{n}x_{i}\cdot\log p_{ki}\\ &\displaystyle=b+\mathbf{w}_{k}^{\top}\mathbf{x}\end{aligned}
  49. b = log p ( C k ) b=\log p(C_{k})
  50. w k i = log p k i w_{ki}=\log p_{ki}
  51. x i x_{i}
  52. i i
  53. C k C_{k}
  54. p ( 𝐱 | C k ) = i = 1 n p k i x i ( 1 - p k i ) ( 1 - x i ) p(\mathbf{x}|C_{k})=\prod_{i=1}^{n}p_{ki}^{x_{i}}(1-p_{ki})^{(1-x_{i})}
  55. p k i p_{ki}
  56. C k C_{k}
  57. w i w_{i}
  58. D = L U D=L\uplus U
  59. L L
  60. U U
  61. L L
  62. P ( C | x ) P(C|x)
  63. x x
  64. D D
  65. P ( D | θ ) P(D|\theta)
  66. θ \theta
  67. p ( C , 𝐱 ) p(C,\mathbf{x})
  68. p ( C | 𝐱 ) p(C|\mathbf{x})
  69. C 1 C_{1}
  70. p ( C 1 | 𝐱 ) p(C_{1}|\mathbf{x})
  71. p ( C 2 | 𝐱 ) p(C_{2}|\mathbf{x})
  72. log p ( C 1 | 𝐱 ) p ( C 2 | 𝐱 ) = log p ( C 1 | 𝐱 ) - log p ( C 2 | 𝐱 ) > 0 \log\frac{p(C_{1}|\mathbf{x})}{p(C_{2}|\mathbf{x})}=\log p(C_{1}|\mathbf{x})-% \log p(C_{2}|\mathbf{x})>0
  73. b + 𝐰 x > 0 b+\mathbf{w}^{\top}x>0
  74. b + 𝐰 x b+\mathbf{w}^{\top}x
  75. p o s t e r i o r ( m a l e ) = P ( m a l e ) p ( h e i g h t | m a l e ) p ( w e i g h t | m a l e ) p ( f o o t s i z e | m a l e ) e v i d e n c e posterior(male)=\frac{P(male)\,p(height|male)\,p(weight|male)\,p(footsize|male% )}{evidence}
  76. p o s t e r i o r ( f e m a l e ) = P ( f e m a l e ) p ( h e i g h t | f e m a l e ) p ( w e i g h t | f e m a l e ) p ( f o o t s i z e | f e m a l e ) e v i d e n c e posterior(female)=\frac{P(female)\,p(height|female)\,p(weight|female)\,p(% footsize|female)}{evidence}
  77. e v i d e n c e = P ( m a l e ) p ( h e i g h t | m a l e ) p ( w e i g h t | m a l e ) p ( f o o t s i z e | m a l e ) evidence=P(male)\,p(height|male)\,p(weight|male)\,p(footsize|male)
  78. + P ( f e m a l e ) p ( h e i g h t | f e m a l e ) p ( w e i g h t | f e m a l e ) p ( f o o t s i z e | f e m a l e ) +P(female)\,p(height|female)\,p(weight|female)\,p(footsize|female)
  79. P ( m a l e ) = 0.5 P(male)=0.5
  80. p ( height | male ) = 1 2 π σ 2 exp ( - ( 6 - μ ) 2 2 σ 2 ) 1.5789 p(\mbox{height}~{}|\mbox{male}~{})=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(% \frac{-(6-\mu)^{2}}{2\sigma^{2}}\right)\approx 1.5789
  81. μ = 5.855 \mu=5.855
  82. σ 2 = 3.5033 10 - 2 \sigma^{2}=3.5033\cdot 10^{-2}
  83. p ( weight | male ) = 5.9881 10 - 6 p(\mbox{weight}~{}|\mbox{male}~{})=5.9881\cdot 10^{-6}
  84. p ( foot size | male ) = 1.3112 10 - 3 p(\mbox{foot size}~{}|\mbox{male}~{})=1.3112\cdot 10^{-3}
  85. posterior numerator (male) = their product = 6.1984 10 - 9 \mbox{posterior numerator (male)}~{}=\mbox{their product}~{}=6.1984\cdot 10^{-9}
  86. P ( female ) = 0.5 P(\mbox{female}~{})=0.5
  87. p ( height | female ) = 2.2346 10 - 1 p(\mbox{height}~{}|\mbox{female}~{})=2.2346\cdot 10^{-1}
  88. p ( weight | female ) = 1.6789 10 - 2 p(\mbox{weight}~{}|\mbox{female}~{})=1.6789\cdot 10^{-2}
  89. p ( foot size | female ) = 2.8669 10 - 1 p(\mbox{foot size}~{}|\mbox{female}~{})=2.8669\cdot 10^{-1}
  90. posterior numerator (female) = their product = 5.3778 10 - 4 \mbox{posterior numerator (female)}~{}=\mbox{their product}~{}=5.3778\cdot 10^% {-4}
  91. p ( w i | C ) p(w_{i}|C)\,
  92. w i w_{i}
  93. p ( D | C ) = i p ( w i | C ) p(D|C)=\prod_{i}p(w_{i}|C)\,
  94. p ( C | D ) p(C|D)\,
  95. p ( D | C ) = p ( D C ) p ( C ) p(D|C)={p(D\cap C)\over p(C)}
  96. p ( C | D ) = p ( D C ) p ( D ) p(C|D)={p(D\cap C)\over p(D)}
  97. p ( C | D ) = p ( C ) p ( D ) p ( D | C ) p(C|D)={p(C)\over p(D)}\,p(D|C)
  98. p ( D | S ) = i p ( w i | S ) p(D|S)=\prod_{i}p(w_{i}|S)\,
  99. p ( D | ¬ S ) = i p ( w i | ¬ S ) p(D|\neg S)=\prod_{i}p(w_{i}|\neg S)\,
  100. p ( S | D ) = p ( S ) p ( D ) i p ( w i | S ) p(S|D)={p(S)\over p(D)}\,\prod_{i}p(w_{i}|S)
  101. p ( ¬ S | D ) = p ( ¬ S ) p ( D ) i p ( w i | ¬ S ) p(\neg S|D)={p(\neg S)\over p(D)}\,\prod_{i}p(w_{i}|\neg S)
  102. p ( S | D ) p ( ¬ S | D ) = p ( S ) i p ( w i | S ) p ( ¬ S ) i p ( w i | ¬ S ) {p(S|D)\over p(\neg S|D)}={p(S)\,\prod_{i}p(w_{i}|S)\over p(\neg S)\,\prod_{i}% p(w_{i}|\neg S)}
  103. p ( S | D ) p ( ¬ S | D ) = p ( S ) p ( ¬ S ) i p ( w i | S ) p ( w i | ¬ S ) {p(S|D)\over p(\neg S|D)}={p(S)\over p(\neg S)}\,\prod_{i}{p(w_{i}|S)\over p(w% _{i}|\neg S)}
  104. ln p ( S | D ) p ( ¬ S | D ) = ln p ( S ) p ( ¬ S ) + i ln p ( w i | S ) p ( w i | ¬ S ) \ln{p(S|D)\over p(\neg S|D)}=\ln{p(S)\over p(\neg S)}+\sum_{i}\ln{p(w_{i}|S)% \over p(w_{i}|\neg S)}
  105. p ( S | D ) > p ( ¬ S | D ) p(S|D)>p(\neg S|D)
  106. ln p ( S | D ) p ( ¬ S | D ) > 0 \ln{p(S|D)\over p(\neg S|D)}>0

Napier's_bones.html

  1. 485 16364 96431 485\frac{16364}{96431}

NE.html

  1. n e n_{e}

Near_and_far_field.html

  1. λ λ
  2. 1 r 1∕r
  3. r r
  4. D D
  5. r r
  6. λ λ
  7. r λ r≪λ
  8. r 2 λ r≫2λ
  9. r = λ r=λ
  10. r = 2 λ r=2λ
  11. D D
  12. d f = 2 D 2 λ , d_{\rm f}={{2D^{2}}\over{\lambda}},
  13. D D
  14. λ λ
  15. D D
  16. d f D , d_{\rm f}\gg D,
  17. d f λ , d_{\rm f}\gg\lambda,
  18. D D
  19. D D
  20. D λ 1 D∕λ≫1
  21. D D
  22. D λ 1 D∕λ≫1
  23. D D
  24. λ λ
  25. S > 1 S>1
  26. 𝐄 \mathbf{E}
  27. 𝐇 \mathbf{H}
  28. 𝐄 \mathbf{E}
  29. 𝐇 \mathbf{H}
  30. c = 1 c=1
  31. 𝐄 \mathbf{E}
  32. 𝐇 \mathbf{H}
  33. 𝐄 \mathbf{E}
  34. 𝐇 \mathbf{H}
  35. 1 2 π 1∕2π
  36. λ 2 π λ∕2π
  37. 0.159 × λ 0.159 ×λ
  38. λ 2 π λ∕2π
  39. 𝐄 \mathbf{E}
  40. 𝐇 \mathbf{H}
  41. 𝐄 \mathbf{E}
  42. 𝐇 \mathbf{H}
  43. 𝐄 \mathbf{E}
  44. 𝐇 \mathbf{H}
  45. 𝐄 \mathbf{E}
  46. 𝐇 \mathbf{H}
  47. 𝐄 \mathbf{E}
  48. 𝐇 \mathbf{H}
  49. 𝐄 \mathbf{E}
  50. 𝐇 \mathbf{H}
  51. 𝐄 \mathbf{E}
  52. 𝐁 \mathbf{B}
  53. 1 r 1∕r
  54. 1 r 1∕r
  55. r r
  56. r r
  57. r r
  58. 1 r 1∕r
  59. r r
  60. r r
  61. r r
  62. r r
  63. Z 0 = def μ 0 c 0 = μ 0 ε 0 = 1 ε 0 c 0 Z_{0}\ \overset{\underset{\mathrm{def}}{}}{=}\ \mu_{0}c_{0}=\sqrt{\frac{\mu_{0% }}{\varepsilon_{0}}}=\frac{1}{\varepsilon_{0}c_{0}}
  64. Z 0 120 π 377 Ω Z_{0}\approx 120\pi\approx 377\ \Omega
  65. r λ r∕λ
  66. | Z W | 240 π 2 r λ 2370 r λ |Z_{W}|\approx 240\pi^{2}\frac{r}{\lambda}\approx 2370\frac{r}{\lambda}
  67. r λ r∕λ
  68. | Z W | 60 λ r |Z_{W}|\approx 60\frac{\lambda}{r}

Necessity_and_sufficiency.html

  1. \Rightarrow
  2. \Leftarrow
  3. \Leftrightarrow
  4. \Rightarrow
  5. \star
  6. \star
  7. \star
  8. \star
  9. \star
  10. \star
  11. \Rightarrow
  12. x x
  13. x x
  14. \Leftrightarrow

Negation.html

  1. p p^{\prime}\!
  2. p ¯ \bar{p}
  3. ! p !p\!
  4. ¬ ( a b ) ( ¬ a ¬ b ) \neg(a\vee b)\equiv(\neg a\wedge\neg b)
  5. ¬ ( a b ) ( ¬ a ¬ b ) \neg(a\wedge b)\equiv(\neg a\vee\neg b)
  6. \in
  7. \land
  8. \land
  9. \in
  10. \in

Negative_feedback.html

  1. MV ( t ) = K p ( e ( t ) + 1 T i 0 t e ( τ ) d τ + T d d d t e ( t ) ) \mathrm{MV(t)}=K_{p}\left(\,{e(t)}+\frac{1}{T_{i}}\int_{0}^{t}{e(\tau)}\,{d% \tau}+T_{d}\frac{d}{dt}e(t)\right)
  2. T i T_{i}
  3. T d T_{d}
  4. O I = A 1 + β A 1 β , \frac{O}{I}=\frac{A}{1+\beta A}\approx\frac{1}{\beta}\ ,
  5. O = A I 1 + β A + D 1 + β A , O=\frac{AI}{1+\beta A}+\frac{D}{1+\beta A}\ ,
  6. Error signal = I - β O = I ( 1 - β O I ) = I 1 + β A - β D 1 + β A . \,\text{Error signal}=I-\beta O=I\left(1-\beta\frac{O}{I}\right)=\frac{I}{1+% \beta A}-\frac{\beta D}{1+\beta A}\ .
  7. V out = R 1 + R 2 R 1 V in = 1 β V in V_{\,\text{out}}=\frac{R_{\,\text{1}}+R_{\,\text{2}}}{R_{\,\text{1}}}V_{\,% \text{in}}\!=\frac{1}{\beta}V_{\,\text{in}}\,

Negative_feedback_amplifier.html

  1. V o u t = A O L V i n V_{out}=A_{OL}\cdot V^{\prime}_{in}
  2. V i n = V i n - β V o u t V^{\prime}_{in}=V_{in}-\beta\cdot V_{out}
  3. V o u t = A O L ( V i n - β V o u t ) V_{out}=A_{OL}(V_{in}-\beta\cdot V_{out})
  4. V o u t ( 1 + β A O L ) = V i n A O L V_{out}(1+\beta\cdot A_{OL})=V_{in}\cdot A_{OL}
  5. A fb = V out V in = A O L 1 + β A O L A_{\mathrm{fb}}=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{A_{OL}}{1+\beta% \cdot A_{OL}}
  6. A O L ( f ) = A 0 1 + j f / f C , A_{OL}(f)=\frac{A_{0}}{1+jf/f_{C}}\ ,
  7. A f b ( f ) = A O L 1 + β A O L A_{fb}(f)=\frac{A_{OL}}{1+\beta A_{OL}}
  8. = A 0 / ( 1 + j f / f C ) 1 + β A 0 / ( 1 + j f / f C ) =\frac{A_{0}/(1+jf/f_{C})}{1+\beta A_{0}/(1+jf/f_{C})}
  9. = A 0 1 + j f / f C + β A 0 =\frac{A_{0}}{1+jf/f_{C}+\beta A_{0}}
  10. = A 0 ( 1 + β A 0 ) ( 1 + j f ( 1 + β A 0 ) f C ) . =\frac{A_{0}}{(1+\beta A_{0})\left(1+j\frac{f}{(1+\beta A_{0})f_{C}}\right)}\ .
  11. x O = a 11 x S + a 12 x j , x_{O}=a_{11}x_{S}+a_{12}x_{j}\ ,
  12. x i = a 21 x S + a 22 x j , x_{i}=a_{21}x_{S}+a_{22}x_{j}\ ,
  13. x j = P x i . x_{j}=Px_{i}\ .
  14. x O x S = a 11 + a 12 a 21 P 1 - P a 22 . \frac{x_{O}}{x_{S}}=a_{11}+\frac{a_{12}a_{21}P}{1-Pa_{22}}\ .
  15. 1 R f + R 2 \frac{1}{R_{f}+R_{2}}
  16. - R 2 R 2 + R f -\frac{R_{2}}{R_{2}+R_{f}}
  17. R 2 R 2 + R f \frac{R_{2}}{R_{2}+R_{f}}
  18. R 2 / / R f R_{2}//R_{f}
  19. A O L = β i B i S = g m R C ( β β + 1 ) ( R 1 R 22 + r π 2 + R C β + 1 ) . A_{OL}=\frac{\beta i_{B}}{i_{S}}=g_{m}R_{C}\left(\frac{\beta}{\beta+1}\right)% \left(\frac{R_{1}}{R_{22}+\frac{r_{\pi 2}+R_{C}}{\beta+1}}\right)\ .
  20. A F B = A O L 1 + β F B A O L A_{FB}=\frac{A_{OL}}{1+{\beta}_{FB}A_{OL}}
  21. A F B = A O L 1 + R 2 R 2 + R f A O L , A_{FB}=\frac{A_{OL}}{1+\frac{R_{2}}{R_{2}+R_{f}}A_{OL}}\ ,
  22. V x = I x R i n + β v o u t , V_{x}=I_{x}R_{in}+\beta v_{out}\ ,
  23. R i n ( f b ) = V x I x = ( 1 + β A v ) R i n . R_{in}(fb)=\frac{V_{x}}{I_{x}}=\left(1+\beta A_{v}\right)R_{in}\ .
  24. I x = V i n R i n + β i o u t . I_{x}=\frac{V_{in}}{R_{in}}+\beta i_{out}\ .
  25. R i n ( f b ) = V x I x = R i n ( 1 + β A i ) . R_{in}(fb)=\frac{V_{x}}{I_{x}}=\frac{R_{in}}{\left(1+\beta A_{i}\right)}\ .
  26. R i n = R 1 1 + β F B A O L , R_{in}=\frac{R_{1}}{1+{\beta}_{FB}A_{OL}}\ ,
  27. v L i S = A F B ( R C 2 R L ) . \frac{v_{L}}{i_{S}}=A_{FB}(R_{C2}\parallel R_{L})\ .
  28. i L i S = A F B R C 2 R C 2 + R L . \frac{i_{L}}{i_{S}}=A_{FB}\frac{R_{C2}}{R_{C2}+R_{L}}\ .

Negative_mass.html

  1. P s y s = m v + ( - m ) v = [ m + ( - m ) ] v = 0 × v = 0. P_{sys}=mv+(-m)v=[m+(-m)]v=0\times v=0.
  2. E k , s y s = 1 2 m v 2 + 1 2 ( - m ) v 2 = 1 2 [ m + ( - m ) ] v 2 = 1 2 ( 0 ) v 2 = 0 E_{k,sys}={1\over 2}mv^{2}+{1\over 2}(-m)v^{2}={1\over 2}[m+(-m)]v^{2}={1\over 2% }(0)v^{2}=0
  3. E = m c 2 E=mc^{2}
  4. - E = - m c 2 -E=-mc^{2}
  5. R μ ν ( + ) - 1 2 g μ ν R ( + ) g μ ν ( + ) = 8 π G c 4 [ T μ ν ( + ) + φ T μ ν ( - ) ] R_{\mu\nu}^{(+)}-{1\over 2}g_{\mu\nu}\,R^{(+)}g_{\mu\nu}^{(+)}={8\pi G\over c^% {4}}[T_{\mu\nu}^{(+)}+\varphi T_{\mu\nu}^{(-)}]
  6. R μ ν ( - ) - 1 2 g μ ν R ( - ) g μ ν ( - ) = - 8 π G c 4 [ ϕ T μ ν ( + ) + T μ ν ( - ) ] R_{\mu\nu}^{(-)}-{1\over 2}g_{\mu\nu}\,R^{(-)}g_{\mu\nu}^{(-)}=-{8\pi G\over c% ^{4}}[\phi T_{\mu\nu}^{(+)}+T_{\mu\nu}^{(-)}]

Negative_number.html

  1. + 3 +3
  2. 0 3 = 3. 0−3=−3.
  3. 5 8 = 3 5−8=−3
  4. 8 5 = 3 8−5=3
  5. 8 8
  6. 5 5
  7. 8 > 5 8>5
  8. 8 8
  9. 5 5
  10. + 3 +3
  11. y z y−z
  12. x −x
  13. ( x ) −(−x)
  14. 5 −5
  15. 7 + 5 7+−5
  16. 7 + ( 5 ) 7+(−5)
  17. 7 5 7–5
  18. ( 3 ) + ( 5 ) = 8 (−3)+(−5)=−8
  19. 8 + ( 3 ) = 8 3 = 5 8+(−3)=8−3=5
  20. ( 2 ) + 7 = 7 2 = 5 (−2)+7=7−2=5
  21. 8 8
  22. 3 3
  23. 5 5
  24. ( 8 ) + 3 = 3 8 = 5 (−8)+3=3−8=−5
  25. 2 + ( 7 ) = 2 7 = 5 2+(−7)=2−7=−5
  26. 5 8 = 3 5−8=−3
  27. 5 8 = 5 + ( 8 ) = 3 5−8=5+(−8)=−3
  28. ( 3 ) 5 = ( 3 ) + ( 5 ) = 8 (−3)−5=(−3)+(−5)=−8
  29. 3 ( 5 ) = 3 + 5 = 8 3−(−5)=3+5=8
  30. ( 5 ) ( 8 ) = ( 5 ) + 8 = 3 (−5)−(−8)=(−5)+8=3
  31. ( 2 ) × 3 = 6 (−2)×3=−6
  32. ( 2 ) × ( 3 ) = 6 (−2)×(−3)=6
  33. 2 −2
  34. 6 −6
  35. ( 2 ) × 3 = ( 2 ) + ( 2 ) + ( 2 ) = 6 (−2)×3=(−2)+(−2)+(−2)=−6
  36. ( 2 (−2
  37. ) × ( 3 )×(−3
  38. ) = + 6 )=+6
  39. ( 2 ) × ( 3 ) + 2 × ( 3 ) = ( 2 + 2 ) × ( 3 ) = 0 × ( 3 ) = 0 (−2)×(−3) + 2×(−3)=(−2+2)×(−3)=0×(−3)=0
  40. 2 × ( 3 ) = 6 2×(−3)=−6
  41. ( 2 ) × ( 3 ) (−2)×(−3)
  42. 6 6
  43. 8 ÷ ( 2 ) = 4 8÷(−2)=−4
  44. ( 8 ) ÷ 2 = 4 (−8)÷2=−4
  45. ( 8 ) ÷ ( 2 ) = 4 (−8)÷(−2)=4
  46. 3 −3
  47. 3 3
  48. 3 + ( 3 ) = 0 3+(−3)=0
  49. x + ( x ) = 0 x+(−x)=0
  50. x x
  51. 3 −3
  52. + 3 +3
  53. ( x ) = x −(−x)=x
  54. 3 −3
  55. 3 3
  56. 3 3
  57. 0
  58. 0
  59. x + y = 0 , x+y\prime=0,
  60. x + y = 0. x+y\,\,=0.
  61. ( a ± b ) ( c ± d ) (a\pm b)(c\pm d)

Negative_resistance.html

  1. v / i \scriptstyle v/i\,
  2. Δ v / Δ i \scriptstyle\Delta v/\Delta i
  3. Δ v / Δ i < 0 \scriptstyle\Delta v/\Delta i\;<\;0
  4. i \scriptstyle i\,
  5. v \scriptstyle v\,
  6. v \scriptstyle v\,
  7. R static = v i R_{\mathrm{static}}={v\over i}\,
  8. i \scriptstyle i\,
  9. v \scriptstyle v\,
  10. R static < 0 ) . \scriptstyle R\text{static}\;<\;0).
  11. r diff = d v d i r_{\mathrm{diff}}=\frac{dv}{di}\,
  12. G static = 1 R static = i v G_{\mathrm{static}}={1\over R_{\mathrm{static}}}={i\over v}\,
  13. g diff = 1 r diff = d i d v g_{\mathrm{diff}}={1\over r_{\mathrm{diff}}}={di\over dv}\,
  14. R static = v / i > 0 \scriptstyle R\text{static}\;=\;v/i\;>\;0
  15. P = v i > 0 \scriptstyle P\;=\;vi\;>\;0
  16. v \scriptstyle v
  17. i \scriptstyle i
  18. i \scriptstyle i
  19. v / i < 0 \scriptstyle v/i\;<\;0
  20. v \scriptstyle v
  21. i \scriptstyle i
  22. P = v i < 0 \scriptstyle P\;=\;vi\;<\;0
  23. r diff = Δ v / Δ i < 0 \scriptstyle r\text{diff}\;=\;\Delta v/\Delta i\;<\;0
  24. P = v i > 0 \scriptstyle P\;=\;vi\;>\;0
  25. Δ i \scriptstyle\Delta i\,
  26. Δ v \scriptstyle\Delta v\,
  27. P AC = Δ v Δ i < 0 \scriptstyle P\text{AC}\;=\;\Delta v\Delta i\;<\;0
  28. Δ i \scriptstyle\Delta i\,
  29. Δ v \scriptstyle\Delta v\,
  30. r diff \scriptstyle r\text{diff}\,
  31. R static \scriptstyle R\text{static}\,
  32. R static \scriptstyle R\text{static}\,
  33. R static = v / i \scriptstyle R\text{static}\;=\;v/i
  34. R static < 0 \scriptstyle R\text{static}\;<\;0
  35. P = i 2 R static \scriptstyle P\;=\;i^{2}R\text{static}
  36. P 0 \scriptstyle P\;\geq\;0
  37. R static 0 \scriptstyle R\text{static}\;\geq\;0
  38. R static = v / i < 0 \scriptstyle R\text{static}\;=\;v/i\;<\;0
  39. R static = v i < 0 R_{\mathrm{static}}=\frac{v}{i}<0\,
  40. P = i v = i 2 R s t a t i c P=iv=i^{2}R_{static}\,
  41. R static < 0 \scriptstyle R\text{static}\;<\;0
  42. V , I : | v | > V or | i | > I R static = v / i 0 \exists V,I:|v|>V\,\text{ or }|i|>I\Rightarrow R_{\mathrm{static}}=v/i\geq 0\,
  43. P m a x = I V P_{max}=IV\,
  44. r diff = d v d i < 0 r_{\mathrm{diff}}=\frac{dv}{di}<0\,
  45. PVR = i 1 / i 2 \,\text{PVR}=i_{1}/i_{2}\,
  46. V b \scriptstyle V_{b}
  47. Δ v / Δ i = - r \scriptstyle\Delta v/\Delta i\,=\,-r
  48. R \scriptstyle R\,
  49. r \scriptstyle r\,
  50. v o = - r R - r v i = r r - R v i v_{o}=\frac{-r}{R-r}v_{i}=\frac{r}{r-R}v_{i}\,
  51. G v = r r - R G_{v}=\frac{r}{r-R}\,
  52. r - R \scriptstyle r-R\,
  53. r \scriptstyle r\,
  54. v o \scriptstyle v_{o}
  55. v i \scriptstyle v_{i}
  56. G v \scriptstyle G_{v}\,
  57. R \scriptstyle R\,
  58. r \scriptstyle r\,
  59. v ( t ) = V bias + Δ v ( t ) v(t)=V\text{bias}+\Delta v(t)\,
  60. i ( t ) = I bias + Δ i ( t ) i(t)=I\text{bias}+\Delta i(t)\,
  61. Δ v \scriptstyle\Delta v\,
  62. Δ i \scriptstyle\Delta i\,
  63. P AC = Δ v Δ i = r diff | Δ i | 2 < 0 P\text{AC}=\Delta v\Delta i=r\text{diff}|\Delta i|^{2}<0\,
  64. | P AC | I bias V bias |P\text{AC}|\leq I\text{bias}V\text{bias}\,
  65. v 1 , v 2 , i 1 , a n d i 2 \scriptstyle v_{1},\;v_{2},\;i_{1},\;and\;i_{2}
  66. P A C ( r m s ) 1 8 ( v 2 - v 1 ) ( i 1 - i 2 ) P_{AC(rms)}\leq\frac{1}{8}(v_{2}-v_{1})(i_{1}-i_{2})\,
  67. Z N ( j ω ) \scriptstyle Z\text{N}(j\omega)\;
  68. Z L ( j ω ) \scriptstyle Z\text{L}(j\omega)\,
  69. V I \scriptstyle V_{I}\,
  70. V R \scriptstyle V_{R}\,
  71. Γ \scriptstyle\Gamma\,
  72. | Γ | | V R V I | > 1 |\Gamma|\equiv\bigg|\frac{V_{R}}{V_{I}}\bigg|>1\,
  73. Γ Z N - Z L Z N + Z L \Gamma\equiv\frac{Z_{N}-Z_{L}}{Z_{N}+Z_{L}}\,
  74. Z N ( j ω ) = R N + j X N \scriptstyle Z_{N}(j\omega)\,=\,R_{N}\,+\,jX_{N}
  75. Z L ( j ω ) = R L + j X L \scriptstyle Z_{L}(j\omega)\,=\,R_{L}\,+\,jX_{L}
  76. R N < 0 \scriptstyle R_{N}\,<\,0
  77. R L > 0 \scriptstyle R_{L}\,>\,0\,
  78. | Γ | > 0 \scriptstyle|\Gamma|\,>\,0\,
  79. | Γ | = 1 \scriptstyle|\Gamma|\,=\,1\,
  80. Z L ( j ω ) \scriptstyle Z_{L}(j\omega)\,
  81. R N \scriptstyle R_{N}\,
  82. Z L ( j ω ) + Z N ( j ω ) = 0 \scriptstyle Z_{L}(j\omega)\;+\;Z_{N}(j\omega)\;=\;0\,
  83. Z L + Z N = R L + R N = R L - r > 0 Z_{L}+Z_{N}=R_{L}+R_{N}=R_{L}-r>0\,
  84. G N = 1 / R N \scriptstyle G_{N}\;=\;1/R_{N}
  85. Y L ( j ω ) + Y N ( j ω ) = 0 \scriptstyle Y_{L}(j\omega)\;+\;Y_{N}(j\omega)\;=\;0
  86. Y L + Y N = G L + G N = 1 R L + 1 R N = 1 R L + 1 - r > 0 Y_{L}+Y_{N}=G_{L}+G_{N}={1\over R_{L}}+{1\over R_{N}}={1\over R_{L}}+{1\over-r% }>0\,
  87. 1 R L > 1 r {1\over R_{L}}>{1\over r}\,
  88. Z L ( j ω ) \scriptstyle Z_{L}(j\omega)
  89. R N = - r \scriptstyle R_{N}\;=\;-r
  90. X N = 0 \scriptstyle X_{N}\;=\;0
  91. V = V S - I R V=V_{S}-IR\,
  92. V S \scriptstyle V_{S}
  93. R L \scriptstyle R_{L}\,
  94. R L \scriptstyle R_{L}\,
  95. R L \scriptstyle R_{L}\,
  96. R L < r \scriptstyle R_{L}\;<\;r
  97. R L > r \scriptstyle R_{L}\;>\;r
  98. R L = r \scriptstyle R_{L}\;=\;r
  99. r = R L \scriptstyle r\;=\;R_{L}
  100. R L > r \scriptstyle R_{L}\;>\;r
  101. R L < r \scriptstyle R_{L}\;<\;r
  102. i = v - A v R 1 + v R in i={{v-Av}\over R_{1}}+{v\over R\text{in}}\,
  103. R = v i = R 1 1 + R 1 / R in - A R={v\over i}={R_{1}\over{1+R_{1}/R\text{in}-A}}\,
  104. A > 1 + R 1 / R in A>1+R_{1}/R\text{in}\,
  105. R i \scriptstyle R_{i}\,
  106. A \scriptstyle A\,
  107. β ( j ω ) \scriptstyle\beta(j\omega)\,
  108. R if = R i 1 - A β R\text{if}=\frac{R\text{i}}{1-A\beta}\,
  109. A β \scriptstyle A\beta\,
  110. R i f \scriptstyle R_{if}\,
  111. Δ v Δ i = v i = R if < 0 {\Delta v\over\Delta i}={v\over i}=R\text{if}<0\,
  112. R if \scriptstyle R\text{if}
  113. r loss \scriptstyle r\text{loss}
  114. R if = - r loss \scriptstyle R\text{if}\;=\;-r\text{loss}
  115. | R if | < r loss \scriptstyle|R\text{if}|\;<\;r\text{loss}
  116. R 1 \scriptstyle R\text{1}
  117. v o = v ( R 1 + R 1 ) / R 1 = 2 v v_{o}=v(R_{1}+R_{1})/R_{1}=2v\,
  118. v \scriptstyle v\,
  119. Z \scriptstyle Z
  120. i = v - v o Z = v - 2 v Z = - v Z i=\frac{v-v_{o}}{Z}=\frac{v-2v}{Z}=-\frac{v}{Z}\,
  121. z in = v i = - Z z\text{in}=\frac{v}{i}=-Z\,\!
  122. Z \scriptstyle Z
  123. Z \scriptstyle Z
  124. R \scriptstyle R
  125. V S / 2 < v < - V S / 2 \scriptstyle V\text{S}/2\;<\;v\;<\;-V\text{S}/2
  126. - R \scriptstyle-R
  127. Z \scriptstyle Z
  128. Z C ( j ω ) \scriptstyle Z\text{C}(j\omega)
  129. i = - C d v d t Z C = - 1 / j ω C i=-C{dv\over dt}\qquad\qquad Z_{C}=-1/j\omega C\,
  130. C > 0 \scriptstyle C\;>\;0
  131. Z L ( j ω ) \scriptstyle Z\text{L}(j\omega)
  132. v = - L d i d t Z L = - j ω L v=-L{di\over dt}\qquad\qquad Z_{L}=-j\omega L\,
  133. r < R \scriptstyle r\;<\;R
  134. r \scriptstyle r
  135. R \scriptstyle R
  136. V b \scriptstyle V\text{b}
  137. d v / d i = - r \scriptstyle dv/di\;=\;-r
  138. R \scriptstyle R
  139. L C \scriptstyle LC
  140. i ( t ) \scriptstyle i(t)
  141. d 2 i d t 2 + R - r L d i d t + 1 L C i = 0 \frac{d^{2}i}{dt^{2}}+\frac{R-r}{L}\frac{di}{dt}+\frac{1}{LC}i=0\,
  142. i ( t ) = i 0 e α t cos ( ω t + ϕ ) i(t)=i_{0}e^{\alpha t}\cos(\omega t+\phi)\,
  143. α = r - R 2 L ω = 1 L C - ( r - R 2 L ) 2 \alpha=\frac{r-R}{2L}\quad\omega=\sqrt{\frac{1}{LC}-\Big(\frac{r-R}{2L}\Big)^{% 2}}\,
  144. i ( t ) \scriptstyle i(t)
  145. I bias \scriptstyle I\text{bias}
  146. i ( t ) = i 0 \scriptstyle i(t)\;=\;i_{0}
  147. R \scriptstyle R
  148. r \scriptstyle r
  149. r < R α < 0 r<R\Rightarrow\alpha<0\,
  150. R \scriptstyle R
  151. r = R α = 0 r=R\Rightarrow\alpha=0\,
  152. r > R α > 0 r>R\Rightarrow\alpha>0\,
  153. R = r / 3 \scriptstyle R\;=\;r/3
  154. i 0 \scriptstyle i_{0}
  155. r \scriptstyle r
  156. r \scriptstyle r
  157. R - r \scriptstyle R\;-\;r
  158. α \scriptstyle\alpha
  159. r = R \scriptstyle r\;=\;R
  160. R \scriptstyle R
  161. r \scriptstyle r
  162. R \scriptstyle R
  163. r \scriptstyle r
  164. Z N = R N ( I , ω ) + j X N ( I , ω ) \scriptstyle Z_{N}=R_{N}(I,\omega)+jX_{N}(I,\omega)\,
  165. Z L = R L ( ω ) + j X L ( ω ) \scriptstyle Z_{L}=R_{L}(\omega)+jX_{L}(\omega)\,
  166. ( Z N + Z L ) I = 0 \scriptstyle(Z_{N}+Z_{L})I=0\,
  167. Z N + Z L \scriptstyle Z_{N}+Z_{L}\,
  168. R N - R L R_{N}\leq-R_{L}\,
  169. X N = - X L X_{N}=-X_{L}\,
  170. Γ N = V 2 / V 1 \scriptstyle\Gamma_{N}=V_{2}/V_{1}\,
  171. Γ L = V 1 / V 2 \scriptstyle\Gamma_{L}=V_{1}/V_{2}\,
  172. | Γ N Γ L | 1 |\Gamma_{N}\Gamma_{L}|\geq 1\,
  173. Z 0 \scriptstyle Z_{0}
  174. P in = V I 2 / R 1 P\text{in}=V_{I}^{2}/R_{1}\,
  175. P out = V R 2 / R 1 P\text{out}=V_{R}^{2}/R_{1}\,
  176. G P \scriptstyle G_{P}
  177. G P = P out P in = V R 2 V I 2 = | Γ | 2 G\text{P}={P\text{out}\over P\text{in}}={V_{R}^{2}\over V_{I}^{2}}=|\Gamma|^{2}\,
  178. | Γ | 2 = | Z N - Z 1 Z N + Z 1 | 2 |\Gamma|^{2}=\Bigg|{Z_{N}-Z_{1}\over Z_{N}+Z_{1}}\Bigg|^{2}\,
  179. | Γ | 2 = | R N + j X N - ( R 1 + j X 1 ) R N + j X N + R 1 + j X 1 | 2 |\Gamma|^{2}=\Bigg|{R_{N}+jX_{N}-(R_{1}+jX_{1})\over R_{N}+jX_{N}+R_{1}+jX_{1}% }\Bigg|^{2}\,
  180. R N \scriptstyle R\text{N}
  181. X 1 = - X N \scriptstyle X_{1}\;=\;-X_{N}
  182. G P = | Γ | 2 = ( r + R 1 ) 2 + 4 X N 2 ( r - R 1 ) 2 G\text{P}=|\Gamma|^{2}={(r+R_{1})^{2}+4X_{N}^{2}\over(r-R_{1})^{2}}\,
  183. R 1 < r \scriptstyle R_{1}\;<\;r
  184. R 1 > r \scriptstyle R_{1}\;>\;r
  185. R 1 \scriptstyle R\text{1}
  186. r \scriptstyle r

Negligible_set.html

  1. A k I k A\subset\bigcup_{k}I_{k}
  2. k | I k | < ϵ . \sum_{k}|I_{k}|<\epsilon.

Nernst_equation.html

  1. E red = E red + R T z F ln a Ox a Red E\text{red}=E^{\ominus}\text{red}+\frac{RT}{zF}\ln\frac{a\text{Ox}}{a\text{Red}}
  2. Ox + z e - = Red \,\text{Ox}+ze^{-}=\,\text{Red}
  3. E cell = E cell - R T z F ln Q r E\text{cell}=E^{\ominus}\text{cell}-\frac{RT}{zF}\ln Q_{r}
  4. R R
  5. T T
  6. a a
  7. F F
  8. z z
  9. R T / F RT/F
  10. E = E 0 + 0.05916 z log 10 a Ox a Red . E=E^{0}+\frac{0.05916}{z}\log_{10}\frac{a\text{Ox}}{a\text{Red}}.
  11. E = R T z F ln [ ion outside cell ] [ ion inside cell ] = 2.3026 R T z F log 10 [ ion outside cell ] [ ion inside cell ] . E=\frac{RT}{zF}\ln\frac{[\,\text{ion outside cell}]}{[\,\text{ion inside cell}% ]}=2.3026\frac{RT}{zF}\log_{10}\frac{[\,\text{ion outside cell}]}{[\,\text{ion% inside cell}]}.
  12. E m = R T F ln ( i N P M i + [ M i + ] out + j M P A j - [ A j - ] in i N P M i + [ M i + ] in + j M P A j - [ A j - ] out ) E_{m}=\frac{RT}{F}\ln{\left(\frac{\sum_{i}^{N}P_{M^{+}_{i}}[M^{+}_{i}]_{% \mathrm{out}}+\sum_{j}^{M}P_{A^{-}_{j}}[A^{-}_{j}]_{\mathrm{in}}}{\sum_{i}^{N}% P_{M^{+}_{i}}[M^{+}_{i}]_{\mathrm{in}}+\sum_{j}^{M}P_{A^{-}_{j}}[A^{-}_{j}]_{% \mathrm{out}}}\right)}
  13. E m E_{m}
  14. P ion P_{\mathrm{ion}}
  15. [ i o n ] out [ion]_{\mathrm{out}}
  16. [ i o n ] in [ion]_{\mathrm{in}}
  17. R R
  18. T T
  19. F F
  20. V m = R T F ln ( r P K [ K ] o + P N a [ N a ] o r P K [ K ] i + P N a [ N a ] i ) V_{m}=\frac{RT}{F}\ln{\left(\frac{rP_{K}[K]_{o}+P_{Na}[Na]_{o}}{rP_{K}[K]_{i}+% P_{Na}[Na]_{i}}\right)}
  21. V m = R T F ln ( P K [ K ] o + P N a [ N a ] o + P C l [ C l ] i P K [ K ] i + P N a [ N a ] i + P C l [ C l ] o ) V_{m}=\frac{RT}{F}\ln{\left(\frac{P_{K}[K]_{o}+P_{Na}[Na]_{o}+P_{Cl}[Cl]_{i}}{% P_{K}[K]_{i}+P_{Na}[Na]_{i}+P_{Cl}[Cl]_{o}}\right)}
  22. Ox + e - Red \,\text{Ox}+e^{-}\rightleftharpoons\,\text{Red}\,
  23. μ c \mu_{c}
  24. [ Red ] [ Ox ] = exp ( - [ barrier for losing an electron ] / k T ) exp ( - [ barrier for gaining an electron ] / k T ) = exp ( μ c / k T ) . \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}=\frac{\exp\left(-[\mbox{barrier for % losing an electron}~{}]/kT\right)}{\exp\left(-[\mbox{barrier for gaining an % electron}~{}]/kT\right)}=\exp\left(\mu_{c}/kT\right).
  25. μ c = k T ln [ Red ] [ Ox ] . \mu_{c}=kT\ln\frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.
  26. μ c 0 \mu_{c}\neq 0
  27. μ c = μ c 0 + k T ln [ Red ] [ Ox ] . \mu_{c}=\mu_{c}^{0}+kT\ln\frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.
  28. e e
  29. k / e = R / F k/e=R/F
  30. Ox + e - Red \mathrm{Ox}+e^{-}\rightarrow\mathrm{Red}
  31. E \displaystyle E
  32. R = k N A R=kN_{A}
  33. F = e N A F=eN_{A}
  34. S = def k ln Ω , S\ \stackrel{\mathrm{def}}{=}\ k\ln\Omega,
  35. Ω \Omega
  36. Δ S = n R ln ( V 2 / V 1 ) \Delta S=nR\ln(V_{2}/V_{1})
  37. V 2 / V 1 V_{2}/V_{1}
  38. S = k ln ( constant × V ) = - k ln ( constant × c ) . S=k\ln\ (\mathrm{constant}\times V)=-k\ln\ (\mathrm{constant}\times c).
  39. Δ S = S 2 - S 1 = - k ln c 2 c 1 , \Delta S=S_{2}-S_{1}=-k\ln\frac{c_{2}}{c_{1}},
  40. S 2 = S 1 - k ln c 2 c 1 . S_{2}=S_{1}-k\ln\frac{c_{2}}{c_{1}}.
  41. c 1 c_{1}
  42. c 2 c_{2}
  43. S ( A ) = S 0 ( A ) - k ln [ A ] , S(A)=S^{0}(A)-k\ln[A],\,
  44. S 0 S^{0}
  45. a A + b B y Y + z Z aA+bB\rightarrow yY+zZ
  46. Δ S rxn = [ y S ( Y ) + z S ( Z ) ] - [ a S ( A ) + b S ( B ) ] = Δ S rxn 0 - k ln [ Y ] y [ Z ] z [ A ] a [ B ] b . \Delta S_{\mathrm{rxn}}=[yS(Y)+zS(Z)]-[aS(A)+bS(B)]=\Delta S^{0}_{\mathrm{rxn}% }-k\ln\frac{[Y]^{y}[Z]^{z}}{[A]^{a}[B]^{b}}.
  47. Q = j a j ν j i a i ν i [ Z ] z [ Y ] y [ A ] a [ B ] b . Q=\frac{\prod_{j}a_{j}^{\nu_{j}}}{\prod_{i}a_{i}^{\nu_{i}}}\approx\frac{[Z]^{z% }[Y]^{y}}{[A]^{a}[B]^{b}}.
  48. a j ν j a_{j}^{\nu_{j}}
  49. E = μ c / e E=\mu_{c}/e
  50. Δ G \Delta G
  51. Δ G = - n F E \Delta G=-nFE
  52. F F
  53. Δ G \Delta G
  54. G = H - T S G=H-TS
  55. Δ G = Δ H - T Δ S = Δ G 0 + k T ln Q , \Delta G=\Delta H-T\Delta S=\Delta G^{0}+kT\ln Q,\,
  56. E = E 0 - k T n e ln Q . E=E^{0}-\frac{kT}{ne}\ln Q.
  57. Ox + n e - Red , \mathrm{Ox}+ne^{-}\rightarrow\mathrm{Red},
  58. Q = [ Red ] [ Ox ] Q=\frac{[\mathrm{Red}]}{[\mathrm{Ox}]}
  59. E \displaystyle E
  60. E 0 E^{0}
  61. E 0 E^{0^{\prime}}
  62. 0 = E o - R T n F ln K ln K = n F E o R T \begin{aligned}\displaystyle 0&\displaystyle=E^{o}-\frac{RT}{nF}\ln K\\ \displaystyle\ln K&\displaystyle=\frac{nFE^{o}}{RT}\end{aligned}
  63. log 10 K = n E o 59.2 mV at T = 298 K . \log_{10}K=\frac{nE^{o}}{59.2\,\text{ mV}}\quad\,\text{at }T=298\,\text{ K}.
  64. R = N < s u b > A k R=N<sub>Ak

Nernst_heat_theorem.html

  1. lim T 0 Δ S = 0 \lim_{T\to 0}\Delta S=0
  2. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S

Network_Time_Protocol.html

  1. δ = ( t 3 - t 0 ) - ( t 2 - t 1 ) \delta={(t_{3}-t_{0})-(t_{2}-t_{1})}
  2. θ = ( t 1 - t 0 ) + ( t 2 - t 3 ) 2 \theta={(t_{1}-t_{0})+(t_{2}-t_{3})\over 2}

Neutron_moderator.html

  1. E E
  2. E = 1 2 m v 2 = 3 2 k B T E=\frac{1}{2}mv^{2}=\frac{3}{2}k_{B}T
  3. ξ \xi
  4. A A
  5. ξ = ln E 0 E = 1 + ( A - 1 ) 2 2 A ln ( A - 1 A + 1 ) \xi=\ln\frac{E_{0}}{E}=1+\frac{(A-1)^{2}}{2A}\ln\left(\frac{A-1}{A+1}\right)
  6. ξ 2 A + 1 \xi\simeq\frac{2}{A+1}
  7. n n
  8. E 0 E_{0}
  9. E 1 E_{1}
  10. n = 1 ξ ( ln E 0 - ln E 1 ) n=\frac{1}{\xi}(\ln E_{0}-\ln E_{1})
  11. Σ s \Sigma_{s}
  12. ξ \xi
  13. Σ a \Sigma_{a}
  14. ξ Σ s Σ a \frac{\xi\Sigma_{s}}{\Sigma_{a}}
  15. ξ \xi
  16. E 0 E_{0}
  17. E E
  18. n n
  19. Σ s \Sigma_{s}
  20. Σ a \Sigma_{a}
  21. ξ \xi

Neutropenia.html

  1. ( % n e u t r o p h i l s + % b a n d s ) × ( W B C ) ( 100 ) (\%neutrophils+\%bands)\times(WBC)\over(100)

Newton's_law_of_cooling.html

  1. d Q d t = h A ( T ( t ) - T env ) = h A Δ T ( t ) {\frac{dQ}{dt}=h\cdot A\cdot(T(t)-T_{\,\text{env}})=h\cdot A\Delta T(t)\quad}
  2. Q Q
  3. h h
  4. A A
  5. T T
  6. T env T_{\,\text{env}}
  7. Δ T ( t ) = T ( t ) - T env \Delta T(t)=T(t)-T_{\,\text{env}}
  8. Bi = h L C k b \mathrm{Bi}=\frac{hL_{C}}{\ k_{b}}
  9. L C = V body A surface \mathit{L_{C}}=\frac{V_{\rm body}}{A_{\rm surface}}
  10. C C
  11. T T
  12. Q = C T Q=CT
  13. C C
  14. C = d Q / d T C=dQ/dT
  15. d Q / d t = C ( d T / d t ) dQ/dt=C(dT/dt)
  16. d Q / d t dQ/dt
  17. T ( t ) T(t)
  18. t t
  19. T e n v T_{env}
  20. d T ( t ) d t = - r ( T ( t ) - T env ) = - r Δ T ( t ) \frac{dT(t)}{dt}=-r(T(t)-T_{\mathrm{env}})=-r\Delta T(t)\quad
  21. r = h A / C r=hA/C
  22. t i m e - 1 time^{-1}
  23. t 0 t_{0}
  24. r = 1 / t 0 = - ( d T ( t ) / d t ) / Δ T r=1/t_{0}=-(dT(t)/dt)/\Delta T
  25. t 0 = C / h A t_{0}=C/hA
  26. C C
  27. c p c_{p}
  28. m m
  29. t 0 t_{0}
  30. m c p / h A mc_{p}/hA
  31. T ( t ) = T env + ( T ( 0 ) - T env ) e - r t . T(t)=T_{\mathrm{env}}+(T(0)-T_{\mathrm{env}})\ e^{-rt}.\quad
  32. Δ T ( t ) \Delta T(t)\quad
  33. T ( t ) - T env , T(t)-T_{\mathrm{env}}\ ,\quad
  34. Δ T ( 0 ) \Delta T(0)\quad
  35. Δ T ( t ) = Δ T ( 0 ) e - r t = Δ T ( 0 ) e - t / t 0 . \Delta T(t)=\Delta T(0)\ e^{-rt}=\Delta T(0)\ e^{-t/t_{0}}.\quad
  36. Δ T ( t ) \Delta T(t)
  37. d T ( t ) d t = d Δ T ( t ) d t = - 1 t 0 Δ T ( t ) \frac{dT(t)}{dt}=\frac{d\Delta T(t)}{dt}=-\frac{1}{t_{0}}\Delta T(t)\quad

Newton's_law_of_universal_gravitation.html

  1. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  2. 𝐅 12 = - G m 1 m 2 | 𝐫 12 | 2 𝐫 ^ 12 \mathbf{F}_{12}=-G{m_{1}m_{2}\over{|\mathbf{r}_{12}|}^{2}}\,\mathbf{\hat{r}}_{% 12}
  3. 𝐫 ^ 12 = def 𝐫 2 - 𝐫 1 | 𝐫 2 - 𝐫 1 | \mathbf{\hat{r}}_{12}\ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf{r}_{2}-% \mathbf{r}_{1}}{|\mathbf{r}_{2}-\mathbf{r}_{1}|}
  4. 𝐠 ( 𝐫 ) = - G m 1 | 𝐫 | 2 𝐫 ^ \mathbf{g}(\mathbf{r})=-G{m_{1}\over{{|\mathbf{r}|}^{2}}}\,\mathbf{\hat{r}}
  5. 𝐅 ( 𝐫 ) = m 𝐠 ( 𝐫 ) . \mathbf{F}(\mathbf{r})=m\mathbf{g}(\mathbf{r}).
  6. 𝐠 ( 𝐫 ) = - V ( 𝐫 ) . \mathbf{g}(\mathbf{r})=-\nabla V(\mathbf{r}).
  7. V ( r ) = - G m 1 r . V(r)=-G\frac{m_{1}}{r}.
  8. V 𝐠 ( 𝐫 ) d 𝐀 = - 4 π G M e n c \int\!\!\!\!\int_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\bigcirc% \,\,\mathbf{g(r)}\cdot d\mathbf{A}=-4\pi GM_{enc}
  9. V \partial V
  10. M e n c M_{enc}
  11. R R
  12. M M
  13. | 𝐠 ( 𝐫 ) | = { 0 , if r < R G M r 2 , if r R |\mathbf{g(r)}|=\begin{cases}0,&\mbox{if }~{}r<R\\ \\ \dfrac{GM}{r^{2}},&\mbox{if }~{}r\geq R\end{cases}
  14. R R
  15. M M
  16. | 𝐠 ( 𝐫 ) | = { G M r R 3 , if r < R G M r 2 , if r R |\mathbf{g(r)}|=\begin{cases}\dfrac{GMr}{R^{3}},&\mbox{if }~{}r<R\\ \\ \dfrac{GM}{r^{2}},&\mbox{if }~{}r\geq R\end{cases}
  17. Φ c 2 = G M sun r orbit c 2 10 - 8 , ( v Earth c ) 2 = ( 2 π r orbit ( 1 yr ) c ) 2 10 - 8 \frac{\Phi}{c^{2}}=\frac{GM_{\mathrm{sun}}}{r_{\mathrm{orbit}}c^{2}}\sim 10^{-% 8},\quad\left(\frac{v_{\mathrm{Earth}}}{c}\right)^{2}=\left(\frac{2\pi r_{% \mathrm{orbit}}}{(1\ \mathrm{yr})c}\right)^{2}\sim 10^{-8}
  18. F = G m 1 m 2 r 2 + B m 1 m 2 r 3 F=G\frac{m_{1}m_{2}}{r^{2}}+B\frac{m_{1}m_{2}}{r^{3}}
  19. F ( r ) = k m 1 m 2 r 2 exp ( - α r ) F(r)=k\frac{m_{1}m_{2}}{r^{2}}\exp(-\alpha r)
  20. F ( r ) = k m 1 m 2 r 2 ( 1 + α r 3 ) F(r)=k\frac{m_{1}m_{2}}{r^{2}}\left(1+{\alpha\over{r^{3}}}\right)

Newton_(unit).html

  1. F = m a F=ma
  2. 𝖥 = 𝖬𝖫 𝖳 2 {\mathsf{F}}=\frac{\mathsf{ML}}{{\mathsf{T}}^{2}}
  3. 1 / 9.81 {1}/{9.81}

Newton_metre.html

  1. 1 N m = 1 kg m 2 s 2 , 1 J = 1 kg m 2 s 2 1\,\mathrm{N}\!\cdot\!\mathrm{m}=1\frac{\mathrm{kg}\,\mathrm{m}^{2}}{\mathrm{s% }^{2}}\quad,\quad 1\,\mathrm{J}=1\frac{\mathrm{kg}\,\mathrm{m}^{2}}{\mathrm{s}% ^{2}}

Newton_polynomial.html

  1. ( x 0 , y 0 ) , , ( x k , y k ) (x_{0},y_{0}),\ldots,(x_{k},y_{k})
  2. N ( x ) := j = 0 k a j n j ( x ) N(x):=\sum_{j=0}^{k}a_{j}n_{j}(x)
  3. n j ( x ) := i = 0 j - 1 ( x - x i ) n_{j}(x):=\prod_{i=0}^{j-1}(x-x_{i})
  4. n 0 ( x ) 1 n_{0}(x)\equiv 1
  5. a j := [ y 0 , , y j ] a_{j}:=[y_{0},\ldots,y_{j}]
  6. [ y 0 , , y j ] [y_{0},\ldots,y_{j}]
  7. N ( x ) = [ y 0 ] + [ y 0 , y 1 ] ( x - x 0 ) + + [ y 0 , , y k ] ( x - x 0 ) ( x - x 1 ) ( x - x k - 1 ) . N(x)=[y_{0}]+[y_{0},y_{1}](x-x_{0})+\cdots+[y_{0},\ldots,y_{k}](x-x_{0})(x-x_{% 1})\cdots(x-x_{k-1}).
  8. x 0 , x 1 , , x k x_{0},x_{1},\dots,x_{k}
  9. h = x i + 1 - x i h=x_{i+1}-x_{i}
  10. i = 0 , 1 , , k - 1 i=0,1,\dots,k-1
  11. x = x 0 + s h x=x_{0}+sh
  12. x - x i x-x_{i}
  13. ( s - i ) h (s-i)h
  14. N ( x ) \displaystyle N(x)
  15. x k , x k - 1 , , x 0 {x}_{k},{x}_{k-1},\dots,{x}_{0}
  16. N ( x ) = [ y k ] + [ y k , y k - 1 ] ( x - x k ) + + [ y k , , y 0 ] ( x - x k ) ( x - x k - 1 ) ( x - x 1 ) N(x)=[y_{k}]+[{y}_{k},{y}_{k-1}](x-{x}_{k})+\cdots+[{y}_{k},\ldots,{y}_{0}](x-% {x}_{k})(x-{x}_{k-1})\cdots(x-{x}_{1})
  17. x k , x k - 1 , , x 0 {x}_{k},\;{x}_{k-1},\;\dots,\;{x}_{0}
  18. x k + s h {x}_{k}+sh
  19. x i = x k - ( k - i ) h {x}_{i}={x}_{k}-(k-i)h
  20. N ( x ) = [ y k ] + [ y k , y k - 1 ] s h + + [ y k , , y 0 ] s ( s + 1 ) ( s + k - 1 ) h k = i = 0 k ( - 1 ) i ( - s i ) i ! h i [ y k , , y k - i ] \begin{aligned}\displaystyle N(x)&\displaystyle=[{y}_{k}]+[{y}_{k},{y}_{k-1}]% sh+\cdots+[{y}_{k},\ldots,{y}_{0}]s(s+1)\cdots(s+k-1){h}^{k}\\ &\displaystyle=\sum_{i=0}^{k}{(-1)}^{i}{-s\choose i}i!{h}^{i}[{y}_{k},\ldots,{% y}_{k-i}]\end{aligned}
  21. p n ( z ) p_{n}(z)
  22. p n ( z ) = ( z n ) = z ( z - 1 ) ( z - n + 1 ) n ! p_{n}(z)={z\choose n}=\frac{z(z-1)\cdots(z-n+1)}{n!}
  23. n j ( x ) := i = 0 j - 1 ( x - x i ) j = 0 , , k . n_{j}(x):=\prod_{i=0}^{j-1}(x-x_{i})\qquad j=0,\ldots,k.
  24. Π k \Pi_{k}
  25. [ 1 0 1 x 1 - x 0 1 x 2 - x 0 ( x 2 - x 0 ) ( x 2 - x 1 ) 1 x k - x 0 j = 0 k - 1 ( x k - x j ) ] [ a 0 a k ] = [ y 0 y k ] \begin{bmatrix}1&&\ldots&&0\\ 1&x_{1}-x_{0}&&&\\ 1&x_{2}-x_{0}&(x_{2}-x_{0})(x_{2}-x_{1})&&\vdots\\ \vdots&\vdots&&\ddots&\\ 1&x_{k}-x_{0}&\ldots&\ldots&\prod_{j=0}^{k-1}(x_{k}-x_{j})\end{bmatrix}\begin{% bmatrix}a_{0}\\ \\ \vdots\\ \\ a_{k}\end{bmatrix}=\begin{bmatrix}y_{0}\\ \\ \vdots\\ \\ y_{k}\end{bmatrix}
  26. i = 0 j a i n i ( x j ) = y j j = 0 , , k . \sum_{i=0}^{j}a_{i}n_{i}(x_{j})=y_{j}\qquad j=0,\dots,k.
  27. lim ( x 0 , , x n ) ( z , , z ) f [ x 0 ] + f [ x 0 , x 1 ] ( ξ - x 0 ) + + f [ x 0 , , x n ] ( ξ - x 0 ) ( ξ - x n - 1 ) = \lim_{(x_{0},\dots,x_{n})\to(z,\dots,z)}f[x_{0}]+f[x_{0},x_{1}]\cdot(\xi-x_{0}% )+\dots+f[x_{0},\dots,x_{n}]\cdot(\xi-x_{0})\cdot\dots\cdot(\xi-x_{n-1})=
  28. = f ( z ) + f ( z ) ( ξ - z ) + + f ( n ) ( z ) n ! ( ξ - z ) n =f(z)+f^{\prime}(z)\cdot(\xi-z)+\dots+\frac{f^{(n)}(z)}{n!}\cdot(\xi-z)^{n}
  29. x 0 , , x n x_{0},\ldots,x_{n}
  30. x 0 f ( x 0 ) f ( x 1 ) - f ( x 0 ) x 1 - x 0 x 1 f ( x 1 ) f ( x 2 ) - f ( x 1 ) x 2 - x 1 - f ( x 1 ) - f ( x 0 ) x 1 - x 0 x 2 - x 0 f ( x 2 ) - f ( x 1 ) x 2 - x 1 x 2 f ( x 2 ) x n f ( x n ) \begin{matrix}x_{0}&f(x_{0})&&\\ &&{f(x_{1})-f(x_{0})\over x_{1}-x_{0}}&\\ x_{1}&f(x_{1})&&{{f(x_{2})-f(x_{1})\over x_{2}-x_{1}}-{f(x_{1})-f(x_{0})\over x% _{1}-x_{0}}\over x_{2}-x_{0}}\\ &&{f(x_{2})-f(x_{1})\over x_{2}-x_{1}}&\\ x_{2}&f(x_{2})&&\vdots\\ &&\vdots&\\ \vdots&&&\vdots\\ &&\vdots&\\ x_{n}&f(x_{n})&&\\ \end{matrix}
  31. x 0 = - 3 2 x_{0}=-\tfrac{3}{2}
  32. x 1 = - 3 4 x_{1}=-\tfrac{3}{4}
  33. x 2 = 0 x_{2}=0
  34. x 3 = 3 4 x_{3}=\tfrac{3}{4}
  35. x 4 = 3 2 x_{4}=\tfrac{3}{2}
  36. f ( x 0 ) = - 14.1014 f(x_{0})=-14.1014
  37. f ( x 1 ) = - 0.931596 f(x_{1})=-0.931596
  38. f ( x 2 ) = 0 f(x_{2})=0
  39. f ( x 3 ) = 0.931596 f(x_{3})=0.931596
  40. f ( x 4 ) = 14.1014 f(x_{4})=14.1014
  41. - 3 2 - 14.1014 17.5597 - 3 4 - 0.931596 - 10.8784 1.24213 4.83484 0 0 0 0 1.24213 4.83484 3 4 0.931596 10.8784 17.5597 3 2 14.1014 \begin{matrix}-\tfrac{3}{2}&-14.1014&&&&\\ &&17.5597&&&\\ -\tfrac{3}{4}&-0.931596&&-10.8784&&\\ &&1.24213&&4.83484&\\ 0&0&&0&&0\\ &&1.24213&&4.83484&\\ \tfrac{3}{4}&0.931596&&10.8784&&\\ &&17.5597&&&\\ \tfrac{3}{2}&14.1014&&&&\\ \end{matrix}
  42. - 14.1014 + 17.5597 ( x + 3 2 ) - 10.8784 ( x + 3 2 ) ( x + 3 4 ) + 4.83484 ( x + 3 2 ) ( x + 3 4 ) ( x ) + 0 ( x + 3 2 ) ( x + 3 4 ) ( x ) ( x - 3 4 ) = -14.1014+17.5597(x+\tfrac{3}{2})-10.8784(x+\tfrac{3}{2})(x+\tfrac{3}{4})+4.834% 84(x+\tfrac{3}{2})(x+\tfrac{3}{4})(x)+0(x+\tfrac{3}{2})(x+\tfrac{3}{4})(x)(x-% \tfrac{3}{4})=
  43. = - 0.00005 - 1.4775 x - 0.00001 x 2 + 4.83484 x 3 =-0.00005-1.4775x-0.00001x^{2}+4.83484x^{3}

Newton–Cotes_formulas.html

  1. a b f ( x ) d x i = 0 n w i f ( x i ) \int_{a}^{b}f(x)\,dx\approx\sum_{i=0}^{n}w_{i}\,f(x_{i})
  2. a b f ( x ) d x a b L ( x ) d x = a b ( i = 0 n f ( x i ) l i ( x ) ) d x = i = 0 n f ( x i ) a b l i ( x ) d x w i . \int_{a}^{b}f(x)\,dx\approx\int_{a}^{b}L(x)\,dx=\int_{a}^{b}\bigl(\sum_{i=0}^{% n}f(x_{i})\,l_{i}(x)\bigr)\,dx=\sum_{i=0}^{n}f(x_{i})\underbrace{\int_{a}^{b}l% _{i}(x)\,dx}_{w_{i}}.
  3. a b f ( x ) d x i = 1 n - 1 w i f ( x i ) . \int_{a}^{b}f(x)\,dx\approx\sum_{i=1}^{n-1}w_{i}\,f(x_{i}).
  4. f i f_{i}
  5. f ( x i ) f(x_{i})
  6. b - a 2 ( f 0 + f 1 ) \frac{b-a}{2}(f_{0}+f_{1})
  7. - ( b - a ) 3 12 f ( 2 ) ( ξ ) -\frac{(b-a)^{3}}{12}\,f^{(2)}(\xi)
  8. b - a 6 ( f 0 + 4 f 1 + f 2 ) \frac{b-a}{6}(f_{0}+4f_{1}+f_{2})
  9. - ( b - a ) 5 2880 f ( 4 ) ( ξ ) -\frac{(b-a)^{5}}{2880}\,f^{(4)}(\xi)
  10. b - a 8 ( f 0 + 3 f 1 + 3 f 2 + f 3 ) \frac{b-a}{8}(f_{0}+3f_{1}+3f_{2}+f_{3})
  11. - ( b - a ) 5 6480 f ( 4 ) ( ξ ) -\frac{(b-a)^{5}}{6480}\,f^{(4)}(\xi)
  12. b - a 90 ( 7 f 0 + 32 f 1 + 12 f 2 + 32 f 3 + 7 f 4 ) \frac{b-a}{90}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})
  13. - ( b - a ) 7 1935360 f ( 6 ) ( ξ ) -\frac{(b-a)^{7}}{1935360}\,f^{(6)}(\xi)
  14. ξ \xi
  15. b - a 2 \frac{b-a}{2}
  16. ( b - a ) f 1 (b-a)f_{1}\,
  17. ( b - a ) 3 24 f ( 2 ) ( ξ ) \frac{(b-a)^{3}}{24}\,f^{(2)}(\xi)
  18. b - a 3 \frac{b-a}{3}
  19. b - a 2 ( f 1 + f 2 ) \frac{b-a}{2}(f_{1}+f_{2})
  20. ( b - a ) 3 36 f ( 2 ) ( ξ ) \frac{(b-a)^{3}}{36}\,f^{(2)}(\xi)
  21. b - a 4 \frac{b-a}{4}
  22. b - a 3 ( 2 f 1 - f 2 + 2 f 3 ) \frac{b-a}{3}(2f_{1}-f_{2}+2f_{3})
  23. 7 ( b - a ) 5 23040 f ( 4 ) ( ξ ) \frac{7(b-a)^{5}}{23040}f^{(4)}(\xi)
  24. b - a 5 \frac{b-a}{5}
  25. b - a 24 ( 11 f 1 + f 2 + f 3 + 11 f 4 ) \frac{b-a}{24}(11f_{1}+f_{2}+f_{3}+11f_{4})
  26. 19 ( b - a ) 5 90000 f ( 4 ) ( ξ ) \frac{19(b-a)^{5}}{90000}f^{(4)}(\xi)
  27. [ a , b ] [a,b]
  28. [ a , b ] [a,b]

Nickel–cadmium_battery.html

  1. Cd + 2 O H - Cd ( OH ) 2 + 2 e - \mathrm{Cd+2OH^{-}\rightarrow Cd(OH)_{2}+2e^{-}}
  2. 2 N i O ( OH ) + 2 H 2 O + 2 e - 2 N i ( OH ) 2 + 2 O H - \mathrm{2NiO(OH)+2H_{2}O+2e^{-}\rightarrow 2Ni(OH)_{2}+2OH^{-}}
  3. 2 N i O ( OH ) + Cd + 2 H 2 O 2 N i ( OH ) 2 + Cd ( OH ) 2 . \mathrm{2NiO(OH)+Cd+2H_{2}O\rightarrow 2Ni(OH)_{2}+Cd(OH)_{2}.}

Nicolas_Bourbaki.html

  1. \varnothing

Nilpotent.html

  1. A = ( 0 1 0 0 0 1 0 0 0 ) A=\begin{pmatrix}0&1&0\\ 0&0&1\\ 0&0&0\end{pmatrix}
  2. A = ( 0 1 0 1 ) , B = ( 0 1 0 0 ) . A=\begin{pmatrix}0&1\\ 0&1\end{pmatrix},\;\;B=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}.
  3. ( 1 - x ) ( 1 + x + x 2 + + x n - 1 ) = 1 - x n = 1. (1-x)(1+x+x^{2}+\cdots+x^{n-1})=1-x^{n}=1.
  4. R R
  5. 𝔑 \mathfrak{N}
  6. x x
  7. 𝔭 \mathfrak{p}
  8. x n = 0 𝔭 x^{n}=0\in\mathfrak{p}
  9. 𝔑 \mathfrak{N}
  10. x x
  11. x x
  12. S = { 1 , x , x 2 , } S=\{1,x,x^{2},...\}
  13. S - 1 R S^{-1}R
  14. 𝔭 \mathfrak{p}
  15. 𝔭 S = \mathfrak{p}\cap S=
  16. x x
  17. 𝔑 \mathfrak{N}
  18. 𝔤 \mathfrak{g}
  19. 𝔤 \mathfrak{g}
  20. [ 𝔤 , 𝔤 ] [\mathfrak{g},\mathfrak{g}]
  21. ad x \operatorname{ad}x
  22. \mathbb{C}\otimes\mathbb{H}
  23. 𝕆 \mathbb{C}\otimes\mathbb{O}

Nilpotent_group.html

  1. ad g : G G \operatorname{ad}_{g}\colon G\to G
  2. ad g ( x ) := [ g , x ] \operatorname{ad}_{g}(x):=[g,x]
  3. [ g , x ] = g - 1 x - 1 g x [g,x]=g^{-1}x^{-1}gx
  4. ( ad g ) n ( x ) = e \left(\operatorname{ad}_{g}\right)^{n}(x)=e
  5. x x
  6. G G
  7. ad g \operatorname{ad}_{g}

Nilradical_of_a_ring.html

  1. R red R_{\,\text{red}}

Nimber.html

  1. α β = mex ( { α β : α < α } { α β : β < β } ) , \alpha\oplus\beta=\operatorname{mex}(\{\,\alpha^{\prime}\oplus\beta:\alpha^{% \prime}<\alpha\,\}\cup\{\,\alpha\oplus\beta^{\prime}:\beta^{\prime}<\beta\,\}),
  2. x < s u p > 3 + x + 1 x<sup>3+x+1

Nine-point_circle.html

  1. 1 2 K 2 - 3 R 2 \scriptstyle\frac{1}{2}\sqrt{K^{2}-3R^{2}}

Noether's_theorem.html

  1. d X d t = 0 . \frac{dX}{dt}=0~{}.
  2. I = L ( 𝐪 , 𝐪 ˙ , t ) d t , I=\int L(\mathbf{q},\dot{\mathbf{q}},t)\,dt~{},
  3. 𝐪 ˙ = d 𝐪 d t . \dot{\mathbf{q}}=\frac{d\mathbf{q}}{dt}~{}.
  4. d d t ( L 𝐪 ˙ ) = L 𝐪 . \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\right)=\frac{% \partial L}{\partial\mathbf{q}}~{}.
  5. d d t ( L q ˙ k ) = d p k d t = 0 , \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{k}}\right)=\frac{dp_{k}}{% dt}=0~{},
  6. p k = L q ˙ k p_{k}=\frac{\partial L}{\partial\dot{q}_{k}}
  7. t t = t + δ t t\rightarrow t^{\prime}=t+\delta t
  8. 𝐪 𝐪 = 𝐪 + δ 𝐪 , \mathbf{q}\rightarrow\mathbf{q}^{\prime}=\mathbf{q}+\delta\mathbf{q}~{},
  9. δ t = r ε r T r \delta t=\sum_{r}\varepsilon_{r}T_{r}\!
  10. δ 𝐪 = r ε r 𝐐 r , \delta\mathbf{q}=\sum_{r}\varepsilon_{r}\mathbf{Q}_{r}~{},
  11. ( L 𝐪 ˙ 𝐪 ˙ - L ) T r - L 𝐪 ˙ 𝐐 r \left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\cdot\dot{\mathbf{q}}-L\right% )T_{r}-\frac{\partial L}{\partial\dot{\mathbf{q}}}\cdot\mathbf{Q}_{r}
  12. H = L 𝐪 ˙ 𝐪 ˙ - L . H=\frac{\partial L}{\partial\dot{\mathbf{q}}}\cdot\dot{\mathbf{q}}-L.
  13. p k = L q k ˙ . p_{k}=\frac{\partial L}{\partial\dot{q_{k}}}.
  14. 𝐫 𝐫 + δ θ 𝐧 × 𝐫 . \mathbf{r}\rightarrow\mathbf{r}+\delta\theta\mathbf{n}\times\mathbf{r}.
  15. 𝐐 = 𝐧 × 𝐫 . \mathbf{Q}=\mathbf{n}\times\mathbf{r}.
  16. L 𝐪 ˙ 𝐐 r = 𝐩 ( 𝐧 × 𝐫 ) = 𝐧 ( 𝐫 × 𝐩 ) = 𝐧 𝐋 . \frac{\partial L}{\partial\dot{\mathbf{q}}}\cdot\mathbf{Q}_{r}=\mathbf{p}\cdot% \left(\mathbf{n}\times\mathbf{r}\right)=\mathbf{n}\cdot\left(\mathbf{r}\times% \mathbf{p}\right)=\mathbf{n}\cdot\mathbf{L}.
  17. I = L ( ϕ , μ ϕ , x μ ) d 4 x I=\int L\left(\phi,\partial_{\mu}\phi,x^{\mu}\right)\,d^{4}x
  18. x μ x μ + δ x μ x^{\mu}\rightarrow x^{\mu}+\delta x^{\mu}\!
  19. ϕ ϕ + δ ϕ \phi\rightarrow\phi+\delta\phi
  20. δ x μ = ε r X r μ \delta x^{\mu}=\varepsilon_{r}X^{\mu}_{r}\,
  21. δ ϕ = ε r Ψ r . \delta\phi=\varepsilon_{r}\Psi_{r}~{}.
  22. j r ν = - ( L ϕ , ν ) Ψ r + [ ( L ϕ , ν ) ϕ , σ - L δ σ ν ] X r σ j^{\nu}_{r}=-\left(\frac{\partial L}{\partial\phi_{,\nu}}\right)\cdot\Psi_{r}+% \left[\left(\frac{\partial L}{\partial\phi_{,\nu}}\right)\cdot\phi_{,\sigma}-L% \delta^{\nu}_{\sigma}\right]X_{r}^{\sigma}
  23. μ j μ = 0 \partial_{\mu}j^{\mu}=0
  24. L ( s y m b o l ϕ , μ s y m b o l ϕ , x μ ) L\left(symbol\phi,\partial_{\mu}{symbol\phi},x^{\mu}\right)
  25. T μ = ν [ ( L ϕ , ν ) ϕ , σ - L δ σ ν ] δ μ σ = ( L ϕ , ν ) ϕ , μ - L δ μ ν T_{\mu}{}^{\nu}=\left[\left(\frac{\partial L}{\partial\phi_{,\nu}}\right)\cdot% \phi_{,\sigma}-L\,\delta^{\nu}_{\sigma}\right]\delta_{\mu}^{\sigma}=\left(% \frac{\partial L}{\partial\phi_{,\nu}}\right)\cdot\phi_{,\mu}-L\,\delta_{\mu}^% {\nu}
  26. ψ e i θ ψ , ψ * e - i θ ψ * , \psi\rightarrow e^{i\theta}\psi\ ,\ \psi^{*}\rightarrow e^{-i\theta}\psi^{*}~{},
  27. L = ψ , ν ψ , μ * η ν μ + m 2 ψ ψ * . L=\psi_{,\nu}\psi^{*}_{,\mu}\eta^{\nu\mu}+m^{2}\psi\psi^{*}.
  28. j ν = i ( ψ x μ ψ * - ψ * x μ ψ ) η ν μ , j^{\nu}=i\left(\frac{\partial\psi}{\partial x^{\mu}}\psi^{*}-\frac{\partial% \psi^{*}}{\partial x^{\mu}}\psi\right)\eta^{\nu\mu}~{},
  29. I = t 1 t 2 L [ 𝐪 [ t ] , 𝐪 ˙ [ t ] , t ] d t I=\int_{t_{1}}^{t_{2}}L[\mathbf{q}[t],\dot{\mathbf{q}}[t],t]\,dt
  30. d d t L 𝐪 ˙ [ t ] = L 𝐪 [ t ] . \frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{q}}}[t]=\frac{\partial L}{% \partial\mathbf{q}}[t].
  31. t t = t + ε T t\rightarrow t^{\prime}=t+\varepsilon T\!
  32. 𝐪 [ t ] 𝐪 [ t ] = ϕ [ 𝐪 [ t ] , ε ] = ϕ [ 𝐪 [ t - ε T ] , ε ] \mathbf{q}[t]\rightarrow\mathbf{q}^{\prime}[t^{\prime}]=\phi[\mathbf{q}[t],% \varepsilon]=\phi[\mathbf{q}[t^{\prime}-\varepsilon T],\varepsilon]
  33. 𝐪 ˙ [ t ] 𝐪 ˙ [ t ] = d d t ϕ [ 𝐪 [ t ] , ε ] = ϕ 𝐪 [ 𝐪 [ t - ε T ] , ε ] 𝐪 ˙ [ t - ε T ] . \dot{\mathbf{q}}[t]\rightarrow\dot{\mathbf{q}}^{\prime}[t^{\prime}]=\frac{d}{% dt}\phi[\mathbf{q}[t],\varepsilon]=\frac{\partial\phi}{\partial\mathbf{q}}[% \mathbf{q}[t^{\prime}-\varepsilon T],\varepsilon]\dot{\mathbf{q}}[t^{\prime}-% \varepsilon T].
  34. I [ ε ] = t 1 + ε T t 2 + ε T L [ 𝐪 [ t ] , 𝐪 ˙ [ t ] , t ] d t = t 1 + ε T t 2 + ε T L [ ϕ [ 𝐪 [ t - ε T ] , ε ] , ϕ 𝐪 [ 𝐪 [ t - ε T ] , ε ] 𝐪 ˙ [ t - ε T ] , t ] d t \begin{aligned}\displaystyle I^{\prime}[\varepsilon]&\displaystyle=\int_{t_{1}% +\varepsilon T}^{t_{2}+\varepsilon T}L[\mathbf{q}^{\prime}[t^{\prime}],\dot{% \mathbf{q}}^{\prime}[t^{\prime}],t^{\prime}]\,dt^{\prime}\\ &\displaystyle=\int_{t_{1}+\varepsilon T}^{t_{2}+\varepsilon T}L[\phi[\mathbf{% q}[t^{\prime}-\varepsilon T],\varepsilon],\frac{\partial\phi}{\partial\mathbf{% q}}[\mathbf{q}[t^{\prime}-\varepsilon T],\varepsilon]\dot{\mathbf{q}}[t^{% \prime}-\varepsilon T],t^{\prime}]\,dt^{\prime}\end{aligned}
  35. 0 = d I d ε [ 0 ] = L [ 𝐪 [ t 2 ] , 𝐪 ˙ [ t 2 ] , t 2 ] T - L [ 𝐪 [ t 1 ] , 𝐪 ˙ [ t 1 ] , t 1 ] T + t 1 t 2 L 𝐪 ( - ϕ 𝐪 𝐪 ˙ T + ϕ ε ) + L 𝐪 ˙ ( - 2 ϕ ( 𝐪 ) 2 𝐪 ˙ 2 T + 2 ϕ ε 𝐪 𝐪 ˙ - ϕ 𝐪 𝐪 ¨ T ) d t . \begin{aligned}\displaystyle 0&\displaystyle=\frac{dI^{\prime}}{d\varepsilon}[% 0]=L[\mathbf{q}[t_{2}],\dot{\mathbf{q}}[t_{2}],t_{2}]T-L[\mathbf{q}[t_{1}],% \dot{\mathbf{q}}[t_{1}],t_{1}]T\\ &\displaystyle{}+\int_{t_{1}}^{t_{2}}\frac{\partial L}{\partial\mathbf{q}}% \left(-\frac{\partial\phi}{\partial\mathbf{q}}\dot{\mathbf{q}}T+\frac{\partial% \phi}{\partial\varepsilon}\right)+\frac{\partial L}{\partial\dot{\mathbf{q}}}% \left(-\frac{\partial^{2}\phi}{(\partial\mathbf{q})^{2}}{\dot{\mathbf{q}}}^{2}% T+\frac{\partial^{2}\phi}{\partial\varepsilon\partial\mathbf{q}}\dot{\mathbf{q% }}-\frac{\partial\phi}{\partial\mathbf{q}}\ddot{\mathbf{q}}T\right)\,dt.\end{aligned}
  36. d d t ( L 𝐪 ˙ ϕ 𝐪 𝐪 ˙ T ) = ( d d t L 𝐪 ˙ ) ϕ 𝐪 𝐪 ˙ T + L 𝐪 ˙ ( d d t ϕ 𝐪 ) 𝐪 ˙ T + L 𝐪 ˙ ϕ 𝐪 𝐪 ¨ T = L 𝐪 ϕ 𝐪 𝐪 ˙ T + L 𝐪 ˙ ( 2 ϕ ( 𝐪 ) 2 𝐪 ˙ ) 𝐪 ˙ T + L 𝐪 ˙ ϕ 𝐪 𝐪 ¨ T . \begin{aligned}\displaystyle\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{% \mathbf{q}}}\frac{\partial\phi}{\partial\mathbf{q}}\dot{\mathbf{q}}T\right)&% \displaystyle=\left(\frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{q}}}% \right)\frac{\partial\phi}{\partial\mathbf{q}}\dot{\mathbf{q}}T+\frac{\partial L% }{\partial\dot{\mathbf{q}}}\left(\frac{d}{dt}\frac{\partial\phi}{\partial% \mathbf{q}}\right)\dot{\mathbf{q}}T+\frac{\partial L}{\partial\dot{\mathbf{q}}% }\frac{\partial\phi}{\partial\mathbf{q}}\ddot{\mathbf{q}}\,T\\ &\displaystyle=\frac{\partial L}{\partial\mathbf{q}}\frac{\partial\phi}{% \partial\mathbf{q}}\dot{\mathbf{q}}T+\frac{\partial L}{\partial\dot{\mathbf{q}% }}\left(\frac{\partial^{2}\phi}{(\partial\mathbf{q})^{2}}\dot{\mathbf{q}}% \right)\dot{\mathbf{q}}T+\frac{\partial L}{\partial\dot{\mathbf{q}}}\frac{% \partial\phi}{\partial\mathbf{q}}\ddot{\mathbf{q}}\,T.\end{aligned}
  37. 0 = d I d ε [ 0 ] = L [ 𝐪 [ t 2 ] , 𝐪 ˙ [ t 2 ] , t 2 ] T - L [ 𝐪 [ t 1 ] , 𝐪 ˙ [ t 1 ] , t 1 ] T - L 𝐪 ˙ ϕ 𝐪 𝐪 ˙ [ t 2 ] T + L 𝐪 ˙ ϕ 𝐪 𝐪 ˙ [ t 1 ] T + t 1 t 2 L 𝐪 ϕ ε + L 𝐪 ˙ 2 ϕ ε 𝐪 𝐪 ˙ d t . \begin{aligned}\displaystyle 0&\displaystyle=\frac{dI^{\prime}}{d\varepsilon}[% 0]=L[\mathbf{q}[t_{2}],\dot{\mathbf{q}}[t_{2}],t_{2}]T-L[\mathbf{q}[t_{1}],% \dot{\mathbf{q}}[t_{1}],t_{1}]T-\frac{\partial L}{\partial\dot{\mathbf{q}}}% \frac{\partial\phi}{\partial\mathbf{q}}\dot{\mathbf{q}}[t_{2}]T+\frac{\partial L% }{\partial\dot{\mathbf{q}}}\frac{\partial\phi}{\partial\mathbf{q}}\dot{\mathbf% {q}}[t_{1}]T\\ &\displaystyle{}+\int_{t_{1}}^{t_{2}}\frac{\partial L}{\partial\mathbf{q}}% \frac{\partial\phi}{\partial\varepsilon}+\frac{\partial L}{\partial\dot{% \mathbf{q}}}\frac{\partial^{2}\phi}{\partial\varepsilon\partial\mathbf{q}}\dot% {\mathbf{q}}\,dt.\end{aligned}
  38. d d t ( L 𝐪 ˙ ϕ ε ) = ( d d t L 𝐪 ˙ ) ϕ ε + L 𝐪 ˙ 2 ϕ ε 𝐪 𝐪 ˙ = L 𝐪 ϕ ε + L 𝐪 ˙ 2 ϕ ε 𝐪 𝐪 ˙ . \frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\frac{\partial% \phi}{\partial\varepsilon}\right)=\left(\frac{d}{dt}\frac{\partial L}{\partial% \dot{\mathbf{q}}}\right)\frac{\partial\phi}{\partial\varepsilon}+\frac{% \partial L}{\partial\dot{\mathbf{q}}}\frac{\partial^{2}\phi}{\partial% \varepsilon\partial\mathbf{q}}\dot{\mathbf{q}}=\frac{\partial L}{\partial% \mathbf{q}}\frac{\partial\phi}{\partial\varepsilon}+\frac{\partial L}{\partial% \dot{\mathbf{q}}}\frac{\partial^{2}\phi}{\partial\varepsilon\partial\mathbf{q}% }\dot{\mathbf{q}}.
  39. 0 = L [ 𝐪 [ t 2 ] , 𝐪 ˙ [ t 2 ] , t 2 ] T - L [ 𝐪 [ t 1 ] , 𝐪 ˙ [ t 1 ] , t 1 ] T - L 𝐪 ˙ ϕ 𝐪 𝐪 ˙ [ t 2 ] T + L 𝐪 ˙ ϕ 𝐪 𝐪 ˙ [ t 1 ] T + L 𝐪 ˙ ϕ ε [ t 2 ] - L 𝐪 ˙ ϕ ε [ t 1 ] . \begin{aligned}\displaystyle 0&\displaystyle=L[\mathbf{q}[t_{2}],\dot{\mathbf{% q}}[t_{2}],t_{2}]T-L[\mathbf{q}[t_{1}],\dot{\mathbf{q}}[t_{1}],t_{1}]T-\frac{% \partial L}{\partial\dot{\mathbf{q}}}\frac{\partial\phi}{\partial\mathbf{q}}% \dot{\mathbf{q}}[t_{2}]T+\frac{\partial L}{\partial\dot{\mathbf{q}}}\frac{% \partial\phi}{\partial\mathbf{q}}\dot{\mathbf{q}}[t_{1}]T\\ &\displaystyle{}+\frac{\partial L}{\partial\dot{\mathbf{q}}}\frac{\partial\phi% }{\partial\varepsilon}[t_{2}]-\frac{\partial L}{\partial\dot{\mathbf{q}}}\frac% {\partial\phi}{\partial\varepsilon}[t_{1}].\end{aligned}
  40. ( L 𝐪 ˙ ϕ 𝐪 𝐪 ˙ - L ) T - L 𝐪 ˙ ϕ ε \left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\frac{\partial\phi}{\partial% \mathbf{q}}\dot{\mathbf{q}}-L\right)T-\frac{\partial L}{\partial\dot{\mathbf{q% }}}\frac{\partial\phi}{\partial\varepsilon}
  41. ϕ 𝐪 = 1 \frac{\partial\phi}{\partial\mathbf{q}}=1
  42. ( L 𝐪 ˙ 𝐪 ˙ - L ) T - L 𝐪 ˙ ϕ ε . \left(\frac{\partial L}{\partial\dot{\mathbf{q}}}\dot{\mathbf{q}}-L\right)T-% \frac{\partial L}{\partial\dot{\mathbf{q}}}\frac{\partial\phi}{\partial% \varepsilon}.
  43. x μ ξ μ = x μ + δ x μ x^{\mu}\rightarrow\xi^{\mu}=x^{\mu}+\delta x^{\mu}\!
  44. ϕ A α A ( ξ μ ) = ϕ A ( x μ ) + δ ϕ A ( x μ ) . \phi^{A}\rightarrow\alpha^{A}(\xi^{\mu})=\phi^{A}(x^{\mu})+\delta\phi^{A}(x^{% \mu})\,.
  45. α A ( x μ ) = ϕ A ( x μ ) + δ ¯ ϕ A ( x μ ) . \alpha^{A}(x^{\mu})=\phi^{A}(x^{\mu})+\bar{\delta}\phi^{A}(x^{\mu})\,.
  46. Ω L ( α A , α A , ν , ξ μ ) d 4 ξ - Ω L ( ϕ A , ϕ A , ν , x μ ) d 4 x = 0 \int_{\Omega^{\prime}}L\left(\alpha^{A},{\alpha^{A}}_{,\nu},\xi^{\mu}\right)d^% {4}\xi-\int_{\Omega}L\left(\phi^{A},{\phi^{A}}_{,\nu},x^{\mu}\right)d^{4}x=0
  47. ϕ A , σ = ϕ A x σ . {\phi^{A}}_{,\sigma}=\frac{\partial\phi^{A}}{\partial x^{\sigma}}\,.
  48. Ω { [ L ( α A , α A , ν , x μ ) - L ( ϕ A , ϕ A , ν , x μ ) ] + x σ [ L ( ϕ A , ϕ A , ν , x μ ) δ x σ ] } d 4 x = 0 . \int_{\Omega}\left\{\left[L\left(\alpha^{A},{\alpha^{A}}_{,\nu},x^{\mu}\right)% -L\left(\phi^{A},{\phi^{A}}_{,\nu},x^{\mu}\right)\right]+\frac{\partial}{% \partial x^{\sigma}}\left[L\left(\phi^{A},{\phi^{A}}_{,\nu},x^{\mu}\right)% \delta x^{\sigma}\right]\right\}d^{4}x=0\,.
  49. [ L ( α A , α A , ν , x μ ) - L ( ϕ A , ϕ A , ν , x μ ) ] = L ϕ A δ ¯ ϕ A + L ϕ A , σ δ ¯ ϕ A , σ . \left[L\left(\alpha^{A},{\alpha^{A}}_{,\nu},x^{\mu}\right)-L\left(\phi^{A},{% \phi^{A}}_{,\nu},x^{\mu}\right)\right]=\frac{\partial L}{\partial\phi^{A}}\bar% {\delta}\phi^{A}+\frac{\partial L}{\partial{\phi^{A}}_{,\sigma}}\bar{\delta}{% \phi^{A}}_{,\sigma}\,.
  50. δ ¯ ϕ A , σ = δ ¯ ϕ A x σ = x σ ( δ ¯ ϕ A ) . \bar{\delta}{\phi^{A}}_{,\sigma}=\bar{\delta}\frac{\partial\phi^{A}}{\partial x% ^{\sigma}}=\frac{\partial}{\partial x^{\sigma}}(\bar{\delta}\phi^{A})\,.
  51. x σ ( L ϕ A , σ ) = L ϕ A \frac{\partial}{\partial x^{\sigma}}\left(\frac{\partial L}{\partial{\phi^{A}}% _{,\sigma}}\right)=\frac{\partial L}{\partial\phi^{A}}
  52. [ L ( α A , α A , ν , x μ ) - L ( ϕ A , ϕ A , ν , x μ ) ] = x σ ( L ϕ A , σ ) δ ¯ ϕ A + L ϕ A , σ δ ¯ ϕ A , σ = x σ ( L ϕ A , σ δ ¯ ϕ A ) . \left[L\left(\alpha^{A},{\alpha^{A}}_{,\nu},x^{\mu}\right)-L\left(\phi^{A},{% \phi^{A}}_{,\nu},x^{\mu}\right)\right]=\frac{\partial}{\partial x^{\sigma}}% \left(\frac{\partial L}{\partial{\phi^{A}}_{,\sigma}}\right)\bar{\delta}\phi^{% A}+\frac{\partial L}{\partial{\phi^{A}}_{,\sigma}}\bar{\delta}{\phi^{A}}_{,% \sigma}=\frac{\partial}{\partial x^{\sigma}}\left(\frac{\partial L}{\partial{% \phi^{A}}_{,\sigma}}\bar{\delta}\phi^{A}\right)\,.
  53. Ω x σ { L ϕ A , σ δ ¯ ϕ A + L ( ϕ A , ϕ A , ν , x μ ) δ x σ } d 4 x = 0 . \int_{\Omega}\frac{\partial}{\partial x^{\sigma}}\left\{\frac{\partial L}{% \partial{\phi^{A}}_{,\sigma}}\bar{\delta}\phi^{A}+L\left(\phi^{A},{\phi^{A}}_{% ,\nu},x^{\mu}\right)\delta x^{\sigma}\right\}d^{4}x=0\,.
  54. x σ { L ϕ A , σ δ ¯ ϕ A + L ( ϕ A , ϕ A , ν , x μ ) δ x σ } = 0 . \frac{\partial}{\partial x^{\sigma}}\left\{\frac{\partial L}{\partial{\phi^{A}% }_{,\sigma}}\bar{\delta}\phi^{A}+L\left(\phi^{A},{\phi^{A}}_{,\nu},x^{\mu}% \right)\delta x^{\sigma}\right\}=0\,.
  55. δ x μ = ε X μ \delta x^{\mu}=\varepsilon X^{\mu}\!
  56. δ ϕ A = ε Ψ A = δ ¯ ϕ A + ε X ϕ A \delta\phi^{A}=\varepsilon\Psi^{A}=\bar{\delta}\phi^{A}+\varepsilon\mathcal{L}% _{X}\phi^{A}
  57. X ϕ A \mathcal{L}_{X}\phi^{A}
  58. X μ , ν = 0 {X^{\mu}}_{,\nu}=0\,
  59. X ϕ A = ϕ A x μ X μ . \mathcal{L}_{X}\phi^{A}=\frac{\partial\phi^{A}}{\partial x^{\mu}}X^{\mu}\,.
  60. δ ¯ ϕ A = ε Ψ A - ε X ϕ A . \bar{\delta}\phi^{A}=\varepsilon\Psi^{A}-\varepsilon\mathcal{L}_{X}\phi^{A}\,.
  61. x σ j σ = 0 \frac{\partial}{\partial x^{\sigma}}j^{\sigma}=0
  62. j σ = [ L ϕ A , σ X ϕ A - L X σ ] - ( L ϕ A , σ ) Ψ A . j^{\sigma}=\left[\frac{\partial L}{\partial{\phi^{A}}_{,\sigma}}\mathcal{L}_{X% }\phi^{A}-L\,X^{\sigma}\right]-\left(\frac{\partial L}{\partial{\phi^{A}}_{,% \sigma}}\right)\Psi^{A}\,.
  63. 𝒞 \mathcal{C}
  64. ϕ 1 , , ϕ m \phi_{1},\ldots,\phi_{m}
  65. 𝒮 : 𝒞 𝐑 , \mathcal{S}:\mathcal{C}\rightarrow\mathbf{R},
  66. 𝒮 [ ϕ ] \mathcal{S}[\phi]
  67. ( ϕ , μ ϕ , x ) \mathcal{L}(\phi,\partial_{\mu}\phi,x)
  68. 𝒞 \mathcal{C}
  69. 𝒮 [ ϕ ] = M [ ϕ ( x ) , μ ϕ ( x ) , x ] d n x . \mathcal{S}[\phi]\,=\,\int_{M}\mathcal{L}[\phi(x),\partial_{\mu}\phi(x),x]% \mathrm{d}^{n}x.
  70. 𝒞 \mathcal{C}
  71. 𝒮 \mathcal{S}
  72. δ 𝒮 [ ϕ ] δ ϕ ( x ) 0 \frac{\delta\mathcal{S}[\phi]}{\delta\phi(x)}\approx 0
  73. 𝒞 \mathcal{C}
  74. Q [ N d n x ] N f μ [ ϕ ( x ) , ϕ , ϕ , ] d s μ Q\left[\int_{N}\mathcal{L}\,\mathrm{d}^{n}x\right]\approx\int_{\partial N}f^{% \mu}[\phi(x),\partial\phi,\partial\partial\phi,\ldots]\,\mathrm{d}s_{\mu}
  75. Q [ ( x ) ] μ f μ ( x ) Q[\mathcal{L}(x)]\approx\partial_{\mu}f^{\mu}(x)
  76. ( x ) = [ ϕ ( x ) , μ ϕ ( x ) , x ] . \mathcal{L}(x)=\mathcal{L}[\phi(x),\partial_{\mu}\phi(x),x].
  77. Q [ N d n x ] Q\left[\int_{N}\mathcal{L}\,\mathrm{d}^{n}x\right]
  78. = N [ ϕ - μ ( μ ϕ ) ] Q [ ϕ ] d n x + N ( μ ϕ ) Q [ ϕ ] d s μ =\int_{N}\left[\frac{\partial\mathcal{L}}{\partial\phi}-\partial_{\mu}\frac{% \partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\right]Q[\phi]\,\mathrm{d}^{% n}x+\int_{\partial N}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}Q% [\phi]\,\mathrm{d}s_{\mu}
  79. N f μ d s μ . \approx\int_{\partial N}f^{\mu}\,\mathrm{d}s_{\mu}.
  80. μ [ ( μ ϕ ) Q [ ϕ ] - f μ ] 0. \partial_{\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}Q[% \phi]-f^{\mu}\right]\approx 0.
  81. J μ J^{\mu}\,\!
  82. J μ = ( μ ϕ ) Q [ ϕ ] - f μ , J^{\mu}\,=\,\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}Q[\phi]-f^% {\mu},
  83. N J μ d s μ 0 . \int_{\partial N}J^{\mu}\mathrm{d}s_{\mu}\approx 0~{}.
  84. Q 1 [ ] μ f 1 μ Q_{1}[\mathcal{L}]\approx\partial_{\mu}f_{1}^{\mu}
  85. Q 2 [ ] μ f 2 μ Q_{2}[\mathcal{L}]\approx\partial_{\mu}f_{2}^{\mu}
  86. [ Q 1 , Q 2 ] [ ] = Q 1 [ Q 2 [ ] ] - Q 2 [ Q 1 [ ] ] μ f 12 μ [Q_{1},Q_{2}][\mathcal{L}]=Q_{1}[Q_{2}[\mathcal{L}]]-Q_{2}[Q_{1}[\mathcal{L}]]% \approx\partial_{\mu}f_{12}^{\mu}
  87. j 12 μ = ( ( μ ϕ ) ) ( Q 1 [ Q 2 [ ϕ ] ] - Q 2 [ Q 1 [ ϕ ] ] ) - f 12 μ . j_{12}^{\mu}=\left(\frac{\partial}{\partial(\partial_{\mu}\phi)}\mathcal{L}% \right)(Q_{1}[Q_{2}[\phi]]-Q_{2}[Q_{1}[\phi]])-f_{12}^{\mu}.
  88. Q [ ] μ f μ Q[\mathcal{L}]\approx\partial_{\mu}f^{\mu}
  89. q [ ε ] [ 𝒮 ] = q [ ε ] [ ] d n x = { ( ϕ ) ε Q [ ϕ ] + [ ( μ ϕ ) ] μ ( ε Q [ ϕ ] ) } d n x = { ε Q [ ] + μ ε [ ( μ ϕ ) ] Q [ ϕ ] } d n x ε μ { f μ - [ ( μ ϕ ) ] Q [ ϕ ] } d n x \begin{aligned}\displaystyle q[\varepsilon][\mathcal{S}]&\displaystyle=\int q[% \varepsilon][\mathcal{L}]\,\mathrm{d}^{n}x\\ &\displaystyle=\int\left\{\left(\frac{\partial}{\partial\phi}\mathcal{L}\right% )\varepsilon Q[\phi]+\left[\frac{\partial}{\partial(\partial_{\mu}\phi)}% \mathcal{L}\right]\partial_{\mu}(\varepsilon Q[\phi])\right\}\,\mathrm{d}^{n}x% \\ &\displaystyle=\int\left\{\varepsilon Q[\mathcal{L}]+\partial_{\mu}\varepsilon% \left[\frac{\partial}{\partial\left(\partial_{\mu}\phi\right)}\mathcal{L}% \right]Q[\phi]\right\}\,\mathrm{d}^{n}x\\ &\displaystyle\approx\int\varepsilon\partial_{\mu}\left\{f^{\mu}-\left[\frac{% \partial}{\partial(\partial_{\mu}\phi)}\mathcal{L}\right]Q[\phi]\right\}\,% \mathrm{d}^{n}x\end{aligned}
  90. μ [ f μ - [ ( μ ϕ ) ] Q [ ϕ ] - 2 [ ( μ ν ϕ ) ] ν Q [ ϕ ] + ν [ [ ( μ ν ϕ ) ] Q [ ϕ ] ] - ] 0. \partial_{\mu}\left[f^{\mu}-\left[\frac{\partial}{\partial(\partial_{\mu}\phi)% }\mathcal{L}\right]Q[\phi]-2\left[\frac{\partial}{\partial(\partial_{\mu}% \partial_{\nu}\phi)}\mathcal{L}\right]\partial_{\nu}Q[\phi]+\partial_{\nu}% \left[\left[\frac{\partial}{\partial(\partial_{\mu}\partial_{\nu}\phi)}% \mathcal{L}\right]Q[\phi]\right]-\,\cdots\right]\approx 0.
  91. 𝒮 [ x ] \displaystyle\mathcal{S}[x]
  92. Q [ x ( t ) ] = x ˙ ( t ) Q[x(t)]=\dot{x}(t)
  93. Q [ L ] = m i x ˙ i x ¨ i - i V ( x ) x i x ˙ i = d d t [ m 2 i x ˙ i 2 - V ( x ) ] Q[L]=m\sum_{i}\dot{x}_{i}\ddot{x}_{i}-\sum_{i}\frac{\partial V(x)}{\partial x_% {i}}\dot{x}_{i}=\frac{d}{dt}\left[\frac{m}{2}\sum_{i}\dot{x}_{i}^{2}-V(x)\right]
  94. f = m 2 i x ˙ i 2 - V ( x ) . f=\frac{m}{2}\sum_{i}\dot{x}_{i}^{2}-V(x).
  95. j \displaystyle j
  96. j ˙ = 0 \dot{j}=0
  97. i = 1 3 L x ˙ i x i ˙ - L \sum_{i=1}^{3}\frac{\partial L}{\partial\dot{x}_{i}}\dot{x_{i}}-L
  98. 𝒮 [ x ] \displaystyle\mathcal{S}[\vec{x}]
  99. Q \vec{Q}
  100. Q i [ x α j ( t ) ] = t δ i j . Q_{i}[x^{j}_{\alpha}(t)]=t\delta^{j}_{i}.\,
  101. Q i [ ] = α m α x ˙ α i - α < β i V α β ( x β - x α ) ( t - t ) = α m α x ˙ α i . \begin{aligned}\displaystyle Q_{i}[\mathcal{L}]&\displaystyle=\sum_{\alpha}m_{% \alpha}\dot{x}_{\alpha}^{i}-\sum_{\alpha<\beta}\partial_{i}V_{\alpha\beta}(% \vec{x}_{\beta}-\vec{x}_{\alpha})(t-t)\\ &\displaystyle=\sum_{\alpha}m_{\alpha}\dot{x}_{\alpha}^{i}.\end{aligned}
  102. d d t α m α x α i \frac{\mathrm{d}}{\mathrm{d}t}\sum_{\alpha}m_{\alpha}x^{i}_{\alpha}
  103. f = α m α x α . \vec{f}=\sum_{\alpha}m_{\alpha}\vec{x}_{\alpha}.
  104. j = α ( x ˙ α ) Q [ x α ] - f \vec{j}=\sum_{\alpha}\left(\frac{\partial}{\partial\dot{\vec{x}}_{\alpha}}% \mathcal{L}\right)\cdot\vec{Q}[\vec{x}_{\alpha}]-\vec{f}
  105. = α ( m α x ˙ α t - m α x α ) =\sum_{\alpha}(m_{\alpha}\dot{\vec{x}}_{\alpha}t-m_{\alpha}\vec{x}_{\alpha})
  106. = P t - M x C M =\vec{P}t-M\vec{x}_{CM}
  107. P \vec{P}
  108. x C M \vec{x}_{CM}
  109. j ˙ = 0 P - M x ˙ C M = 0. \dot{\vec{j}}=0\Rightarrow{\vec{P}}-M\dot{\vec{x}}_{CM}=0.
  110. 𝒮 [ ϕ ] \mathcal{S}[\phi]\,
  111. = [ ϕ ( x ) , μ ϕ ( x ) ] d 4 x =\int\mathcal{L}[\phi(x),\partial_{\mu}\phi(x)]\,\mathrm{d}^{4}x
  112. = ( 1 2 μ ϕ μ ϕ - λ ϕ 4 ) d 4 x =\int\left(\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\lambda\phi^{4}% \right)\,\mathrm{d}^{4}x
  113. Q [ ϕ ( x ) ] = x μ μ ϕ ( x ) + ϕ ( x ) . Q[\phi(x)]=x^{\mu}\partial_{\mu}\phi(x)+\phi(x).\!
  114. Q [ ] = μ ϕ ( μ ϕ + x ν μ ν ϕ + μ ϕ ) - 4 λ ϕ 3 ( x μ μ ϕ + ϕ ) . Q[\mathcal{L}]=\partial^{\mu}\phi\left(\partial_{\mu}\phi+x^{\nu}\partial_{\mu% }\partial_{\nu}\phi+\partial_{\mu}\phi\right)-4\lambda\phi^{3}\left(x^{\mu}% \partial_{\mu}\phi+\phi\right).
  115. μ [ 1 2 x μ ν ϕ ν ϕ - λ x μ ϕ 4 ] = μ ( x μ ) \partial_{\mu}\left[\frac{1}{2}x^{\mu}\partial^{\nu}\phi\partial_{\nu}\phi-% \lambda x^{\mu}\phi^{4}\right]=\partial_{\mu}\left(x^{\mu}\mathcal{L}\right)
  116. f μ = x μ . f^{\mu}=x^{\mu}\mathcal{L}.\,
  117. j μ = [ ( μ ϕ ) ] Q [ ϕ ] - f μ j^{\mu}=\left[\frac{\partial}{\partial(\partial_{\mu}\phi)}\mathcal{L}\right]Q% [\phi]-f^{\mu}
  118. = μ ϕ ( x ν ν ϕ + ϕ ) - x μ ( 1 2 ν ϕ ν ϕ - λ ϕ 4 ) . =\partial^{\mu}\phi\left(x^{\nu}\partial_{\nu}\phi+\phi\right)-x^{\mu}\left(% \frac{1}{2}\partial^{\nu}\phi\partial_{\nu}\phi-\lambda\phi^{4}\right).
  119. μ j μ = 0 \partial_{\mu}j^{\mu}=0\!