wpmath0000001_21

Standard_Model.html

  1. M S ¯ \overline{MS}
  2. M S ¯ \overline{MS}
  3. M S ¯ \overline{MS}
  4. M S ¯ \overline{MS}
  5. M S ¯ \overline{MS}
  6. M S ¯ \overline{MS}
  7. M S ¯ \overline{MS}
  8. M S ¯ \overline{MS}
  9. Q C D = i U ¯ ( μ - i g s G μ a T a ) γ μ U + i D ¯ ( μ - i g s G μ a T a ) γ μ D . \mathcal{L}_{QCD}=i\overline{U}(\partial_{\mu}-ig_{s}G_{\mu}^{a}T^{a})\gamma^{% \mu}U+i\overline{D}(\partial_{\mu}-ig_{s}G_{\mu}^{a}T^{a})\gamma^{\mu}D.
  10. G μ a G_{\mu}^{a}
  11. γ μ \gamma^{\mu}
  12. EW = ψ ψ ¯ γ μ ( i μ - g 1 2 Y W B μ - g 1 2 τ L W μ ) ψ \mathcal{L}_{\mathrm{EW}}=\sum_{\psi}\bar{\psi}\gamma^{\mu}\left(i\partial_{% \mu}-g^{\prime}{1\over 2}Y_{\mathrm{W}}B_{\mu}-g{1\over 2}\vec{\tau}_{\mathrm{% L}}\vec{W}_{\mu}\right)\psi
  13. W μ \vec{W}_{\mu}
  14. τ L \vec{\tau}_{\mathrm{L}}
  15. φ = 1 2 ( φ + φ 0 ) , \varphi={1\over\sqrt{2}}\left(\begin{array}[]{c}\varphi^{+}\\ \varphi^{0}\end{array}\right)\;,
  16. H = φ ( μ - i 2 ( g Y W B μ + g τ W μ ) ) ( μ + i 2 ( g Y W B μ + g τ W μ ) ) φ - λ 2 4 ( φ φ - v 2 ) 2 , \mathcal{L}_{\mathrm{H}}=\varphi^{\dagger}\left({\partial^{\mu}}-{i\over 2}% \left(g^{\prime}Y_{\mathrm{W}}B^{\mu}+g\vec{\tau}\vec{W}^{\mu}\right)\right)% \left(\partial_{\mu}+{i\over 2}\left(g^{\prime}Y_{\mathrm{W}}B_{\mu}+g\vec{% \tau}\vec{W}_{\mu}\right)\right)\varphi\ -\ {\lambda^{2}\over 4}\left(\varphi^% {\dagger}\varphi-v^{2}\right)^{2}\;,
  17. H = | ( μ + i 2 ( g Y W B μ + g τ W μ ) ) φ | 2 - λ 2 4 ( φ φ - v 2 ) 2 . \mathcal{L}_{\mathrm{H}}=\left|\left(\partial_{\mu}+{i\over 2}\left(g^{\prime}% Y_{\mathrm{W}}B_{\mu}+g\vec{\tau}\vec{W}_{\mu}\right)\right)\varphi\right|^{2}% \ -\ {\lambda^{2}\over 4}\left(\varphi^{\dagger}\varphi-v^{2}\right)^{2}\;.

Standing_wave.html

  1. y 1 = y 0 sin ( k x - ω t ) y_{1}\;=\;y_{0}\,\sin(kx-\omega t)\,
  2. y 2 = y 0 sin ( k x + ω t ) y_{2}\;=\;y_{0}\,\sin(kx+\omega t)\,
  3. y = y 0 sin ( k x - ω t ) + y 0 sin ( k x + ω t ) . y\;=\;y_{0}\,\sin(kx-\omega t)\;+\;y_{0}\,\sin(kx+\omega t).\,
  4. y = 2 y 0 cos ( ω t ) sin ( k x ) . y\;=\;2\,y_{0}\,\cos(\omega t)\;\sin(kx).\,

Standing_wave_ratio.html

  1. V f V_{f}
  2. V r V_{r}
  3. Γ \Gamma
  4. Γ = V r V f . \Gamma={V_{r}\over V_{f}}.
  5. Γ \Gamma
  6. Γ \Gamma
  7. Γ = - 1 \Gamma=-1
  8. Γ = 0 \Gamma=0
  9. Γ = + 1 \Gamma=+1
  10. Γ \Gamma
  11. V max V_{\max}
  12. | V max | \displaystyle|V_{\max}|
  13. | V min | \displaystyle|V_{\min}|
  14. VSWR = | V max | | V min | = 1 + | Γ | 1 - | Γ | . \,\text{VSWR}={|V_{\max}|\over|V_{\min}|}={{1+|\Gamma|}\over{1-|\Gamma|}}.
  15. Γ \Gamma
  16. Γ \Gamma
  17. Γ \Gamma
  18. SWR = 1 + P r / P f 1 - P r / P f \,\text{SWR}=\frac{1+\sqrt{P_{r}/P_{f}}}{1-\sqrt{P_{r}/P_{f}}}
  19. SWR = ( R L Z 0 ) ± 1 \,\text{SWR}=\left(\frac{R\text{L}}{Z\text{0}}\right)^{\pm 1}
  20. V actual = ( e i 2 π ν t V ) V\text{actual}=\Re(e^{i2\pi\nu t}V)
  21. V f ( x ) \displaystyle V_{f}(x)
  22. e - i 2 π ν t e^{-i2\pi\nu t}
  23. e + i k ( x - x 0 ) e^{+ik(x-x_{0})}
  24. V net ( x ) \displaystyle V\text{net}(x)
  25. | V net ( x ) | 2 \displaystyle|V\text{net}(x)|^{2}
  26. SWR = | V max | | V min | = 1 + | Γ | 1 - | Γ | \,\text{SWR}=\frac{|V\text{max}|}{|V\text{min}|}=\frac{1+|\Gamma|}{1-|\Gamma|}
  27. | V net ( x ) | 2 |V\text{net}(x)|^{2}
  28. | V min | 2 |V\text{min}|^{2}
  29. | V max | 2 |V\text{max}|^{2}

Star.html

  1. z = 6.60 z=6.60
  2. Δ m = m f - m b \Delta{m}=m_{\mathrm{f}}-m_{\mathrm{b}}
  3. 2.512 Δ m = Δ L 2.512^{\Delta{m}}=\Delta{L}

Star_formation.html

  1. z = 6.60 z=6.60

Star_height_problem.html

  1. e 1 \displaystyle e_{1}
  2. e n + 1 e_{n+1}
  3. e n e_{n}
  4. e n e_{n}
  5. { a , b } \{a,b\}
  6. e 1 \displaystyle e_{1}
  7. e n e_{n}
  8. 10 10 10 10^{10^{10}}

Statcoulomb.html

  1. F = q 1 q 2 r 2 F=\frac{q_{1}q_{2}}{r^{2}}
  2. F = q 1 q 2 r 2 F=\frac{q_{1}q_{2}}{r^{2}}
  3. F = q 1 q 2 4 π ϵ 0 r 2 F=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r^{2}}
  4. F = q 1 q 2 4 π ϵ 0 r 2 F=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r^{2}}
  5. 1 C / 4 π ϵ 0 = 2997924580 statC 1\;\mathrm{C}/\sqrt{4\pi\epsilon_{0}}=2997924580\;\mathrm{statC}
  6. Φ D = 4 π Q \Phi_{D}=4\pi Q
  7. Φ D = Q \Phi_{D}=Q
  8. Φ D S 𝐃 d 𝐀 \Phi_{D}\equiv\int_{S}\mathbf{D}\cdot\mathrm{d}\mathbf{A}
  9. 1 C corresponds to 3.7673 × 10 10 statC 1\;\mathrm{C}\,\text{ corresponds to }3.7673\times 10^{10}\;\mathrm{statC}
  10. 1 C 4 π / ϵ 0 = 3.7673 × 10 10 statC 1\;\mathrm{C}\sqrt{4\pi/\epsilon_{0}}=3.7673\times 10^{10}\;\mathrm{statC}

Statics.html

  1. V \overrightarrow{V}
  2. 𝐌 O = 𝐫 × 𝐅 \,\textbf{M}_{O}=\,\textbf{r}\times\,\textbf{F}

Statistical_ensemble_(mathematical_physics).html

  1. ρ̂ ρ̂
  2. X X
  3. X = Tr ( X ^ ρ ) . \langle X\rangle=\operatorname{Tr}(\hat{X}\rho).
  4. Ŷ X̂Ŷ
  5. T r ρ̂ = 1 Trρ̂=1
  6. d ρ̂ / d t = 0 dρ̂/dt=0
  7. Ĥ Ĥ
  8. ρ ^ = i P i | ψ i ψ i | \hat{\rho}=\sum_{i}P_{i}|\psi_{i}\rangle\langle\psi_{i}|
  9. i i
  10. n n
  11. n n
  12. s s
  13. n n
  14. X X
  15. ρ ρ
  16. X = N 1 = 0 N s = 0 ρ X d p 1 d q n . \langle X\rangle=\sum_{N_{1}=0}^{\infty}\ldots\sum_{N_{s}=0}^{\infty}\int% \ldots\int\rho X\,dp_{1}\ldots dq_{n}.
  17. N 1 = 0 N s = 0 ρ d p 1 d q n = 1. \sum_{N_{1}=0}^{\infty}\ldots\sum_{N_{s}=0}^{\infty}\int\ldots\int\rho\,dp_{1}% \ldots dq_{n}=1.
  18. ρ ρ
  19. P P
  20. ρ = 1 h n C P , \rho=\frac{1}{h^{n}C}P,
  21. h h
  22. e n e r g y × t i m e energy×time
  23. ρ ρ
  24. C C
  25. h h
  26. h h
  27. h h
  28. x x
  29. C C
  30. C = 1 C=1
  31. C C
  32. C C
  33. C = N 1 ! N 2 ! N s ! . C=N_{1}!N_{2}!\ldots N_{s}!.
  34. σ ( E ) = lim N 1 N k = 1 N Meas ( E , X k ) \sigma(E)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{k=1}^{N}\operatorname{Meas% }(E,X_{k})
  35. σ ( E ) = Tr ( E S ) . \sigma(E)=\operatorname{Tr}(ES).
  36. h = 1 e n e r g y u n i t t × × t i m e u n i t t h=1energyunitt××timeunitt
  37. h h
  38. k l o g 2 klog2
  39. k T l o g 2 kTlog2

Statistical_hypothesis_testing.html

  1. H 0 H_{0}
  2. H 1 H_{1}
  3. : H 0 : p = 1 4 \,\text{:}\qquad H_{0}:p=\tfrac{1}{4}
  4. : H 1 : p 1 4 \,\text{:}H_{1}:p\neq\tfrac{1}{4}
  5. P ( reject H 0 | H 0 is valid ) = P ( X = 25 | p = 1 4 ) = ( 1 4 ) 25 10 - 15 , P(\,\text{reject }H_{0}|H_{0}\,\text{ is valid})=P(X=25|p=\tfrac{1}{4})=\left(% \tfrac{1}{4}\right)^{25}\approx 10^{-15},
  6. P ( reject H 0 | H 0 is valid ) = P ( X 10 | p = 1 4 ) = k = 10 25 P ( X = k | p = 1 4 ) 0.07. P(\,\text{reject }H_{0}|H_{0}\,\text{ is valid})=P(X\geq 10|p=\tfrac{1}{4})=% \sum_{k=10}^{25}P(X=k|p=\tfrac{1}{4})\approx 0{.}07.
  7. P ( reject H 0 | H 0 is valid ) = P ( X c | p = 1 4 ) 0.01. P(\,\text{reject }H_{0}|H_{0}\,\text{ is valid})=P(X\geq c|p=\tfrac{1}{4})\leq 0% {.}01.
  8. c = 13 c=13
  9. z = x ¯ - μ 0 σ n z=\frac{\overline{x}-\mu_{0}}{\sigma}\sqrt{n}
  10. z = ( x ¯ 1 - x ¯ 2 ) - d 0 σ 1 2 n 1 + σ 2 2 n 2 z=\frac{(\overline{x}_{1}-\overline{x}_{2})-d_{0}}{\sqrt{\frac{\sigma_{1}^{2}}% {n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}
  11. t = x ¯ - μ 0 ( s / n ) , t=\frac{\overline{x}-\mu_{0}}{(s/\sqrt{n})},
  12. d f = n - 1 df=n-1
  13. σ \sigma
  14. t = d ¯ - d 0 ( s d / n ) , t=\frac{\overline{d}-d_{0}}{(s_{d}/\sqrt{n})},
  15. d f = n - 1 df=n-1
  16. σ \sigma
  17. t = ( x ¯ 1 - x ¯ 2 ) - d 0 s p 1 n 1 + 1 n 2 , t=\frac{(\overline{x}_{1}-\overline{x}_{2})-d_{0}}{s_{p}\sqrt{\frac{1}{n_{1}}+% \frac{1}{n_{2}}}},
  18. s p 2 = ( n 1 - 1 ) s 1 2 + ( n 2 - 1 ) s 2 2 n 1 + n 2 - 2 , s_{p}^{2}=\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2},
  19. d f = n 1 + n 2 - 2 df=n_{1}+n_{2}-2
  20. t = ( x ¯ 1 - x ¯ 2 ) - d 0 s 1 2 n 1 + s 2 2 n 2 , t=\frac{(\overline{x}_{1}-\overline{x}_{2})-d_{0}}{\sqrt{\frac{s_{1}^{2}}{n_{1% }}+\frac{s_{2}^{2}}{n_{2}}}},
  21. d f = ( s 1 2 n 1 + s 2 2 n 2 ) 2 ( s 1 2 n 1 ) 2 n 1 - 1 + ( s 2 2 n 2 ) 2 n 2 - 1 df=\frac{\left(\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}\right)^{2}}{% \frac{\left(\frac{s_{1}^{2}}{n_{1}}\right)^{2}}{n_{1}-1}+\frac{\left(\frac{s_{% 2}^{2}}{n_{2}}\right)^{2}}{n_{2}-1}}
  22. z = p ^ - p 0 p 0 ( 1 - p 0 ) n z=\frac{\hat{p}-p_{0}}{\sqrt{p_{0}(1-p_{0})}}\sqrt{n}
  23. H 0 : p 1 = p 2 H_{0}\colon p_{1}=p_{2}
  24. z = ( p ^ 1 - p ^ 2 ) p ^ ( 1 - p ^ ) ( 1 n 1 + 1 n 2 ) z=\frac{(\hat{p}_{1}-\hat{p}_{2})}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}}+% \frac{1}{n_{2}})}}
  25. p ^ = x 1 + x 2 n 1 + n 2 \hat{p}=\frac{x_{1}+x_{2}}{n_{1}+n_{2}}
  26. | d 0 | > 0 |d_{0}|>0
  27. z = ( p ^ 1 - p ^ 2 ) - d 0 p ^ 1 ( 1 - p ^ 1 ) n 1 + p ^ 2 ( 1 - p ^ 2 ) n 2 z=\frac{(\hat{p}_{1}-\hat{p}_{2})-d_{0}}{\sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1}% )}{n_{1}}+\frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_{2}}}}
  28. χ 2 = ( n - 1 ) s 2 σ 0 2 \chi^{2}=(n-1)\frac{s^{2}}{\sigma^{2}_{0}}
  29. χ 2 = k ( observed - expected ) 2 expected \chi^{2}=\sum^{k}\frac{(\,\text{observed}-\,\text{expected})^{2}}{\,\text{% expected}}
  30. F = s 1 2 s 2 2 F=\frac{s_{1}^{2}}{s_{2}^{2}}
  31. s 1 2 s 2 2 s_{1}^{2}\geq s_{2}^{2}
  32. F > F ( α / 2 , n 1 - 1 , n 2 - 1 ) F>F(\alpha/2,n_{1}-1,n_{2}-1)
  33. H 0 : R 2 = 0. H_{0}\colon R^{2}=0.
  34. t = R 2 ( n - k - 1 * ) 1 - R 2 t=\sqrt{\frac{R^{2}(n-k-1^{*})}{1-R^{2}}}
  35. t > t ( α / 2 , n - k - 1 * ) t>t(\alpha/2,n-k-1^{*})
  36. α \alpha
  37. n n
  38. n 1 n_{1}
  39. n 2 n_{2}
  40. x ¯ \overline{x}
  41. μ 0 \mu_{0}
  42. μ 1 \mu_{1}
  43. μ 2 \mu_{2}
  44. σ \sigma
  45. σ 2 \sigma^{2}
  46. s s
  47. k \sum^{k}
  48. s 2 s^{2}
  49. s 1 s_{1}
  50. s 2 s_{2}
  51. t t
  52. d f df
  53. d ¯ \overline{d}
  54. d 0 d_{0}
  55. s d s_{d}
  56. χ 2 \chi^{2}
  57. p ^ \hat{p}
  58. p 0 p_{0}
  59. p 1 p_{1}
  60. p 2 p_{2}
  61. d p d_{p}
  62. min { n 1 , n 2 } \min\{n_{1},n_{2}\}
  63. x 1 = n 1 p 1 x_{1}=n_{1}p_{1}
  64. x 2 = n 2 p 2 x_{2}=n_{2}p_{2}
  65. F F
  66. α \alpha

Statistical_inference.html

  1. n n
  2. n n

Statistical_mechanics.html

  1. N , E , V N,E,V
  2. N , T , V N,T,V
  3. μ , T , V μ,T,V
  4. W W
  5. Z = k e - E k / k B T Z=\sum_{k}e^{-E_{k}/k_{B}T}
  6. 𝒵 = k e - ( E k - μ N k ) / k B T \mathcal{Z}=\sum_{k}e^{-(E_{k}-\mu N_{k})/k_{B}T}
  7. S = k B ln W S=k_{B}\ln W
  8. F = - k B T ln Z F=-k_{B}T\ln Z
  9. Ω = - k B T ln 𝒵 \Omega=-k_{B}T\ln\mathcal{Z}
  10. N = 3.04 × 10 22 \langle N\rangle=3.04\times 10^{22}
  11. σ N = N 2 × 10 11 \sigma_{N}=\sqrt{\langle N\rangle}\approx 2\times 10^{11}

Statistical_model.html

  1. S , 𝒫 S,\mathcal{P}
  2. S S
  3. 𝒫 \mathcal{P}
  4. S S
  5. 𝒫 \mathcal{P}
  6. 𝒫 \mathcal{P}
  7. 𝒫 \mathcal{P}
  8. 𝒫 = { P θ : θ Θ } \mathcal{P}=\{P_{\theta}:\theta\in\Theta\}
  9. Θ \Theta
  10. P θ 1 = P θ 2 θ 1 = θ 2 P_{\theta_{1}}=P_{\theta_{2}}\Rightarrow\theta_{1}=\theta_{2}
  11. S , 𝒫 S,\mathcal{P}
  12. S S
  13. θ \theta
  14. S S
  15. P θ P_{\theta}
  16. Θ \Theta
  17. θ \theta
  18. 𝒫 = { P θ : θ Θ } \mathcal{P}=\{P_{\theta}:\theta\in\Theta\}
  19. S S
  20. 𝒫 \mathcal{P}
  21. 𝒫 \mathcal{P}
  22. S , 𝒫 S,\mathcal{P}
  23. 𝒫 = { P θ : θ Θ } \mathcal{P}=\{P_{\theta}:\theta\in\Theta\}
  24. Θ \Theta
  25. Θ d \Theta\subseteq\mathbb{R}^{d}
  26. \mathbb{R}
  27. 𝒫 = { P μ , σ ( x ) 1 2 π σ exp ( - ( x - μ ) 2 2 σ 2 ) : μ , σ > 0 } \mathcal{P}=\{P_{\mu,\sigma}(x)\equiv\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-% \frac{(x-\mu)^{2}}{2\sigma^{2}}\right):\mu\in\mathbb{R},\sigma>0\}
  28. Θ \Theta
  29. Θ \Theta
  30. d d\rightarrow\infty
  31. n n\rightarrow\infty
  32. d / n 0 d/n\rightarrow 0
  33. n n\rightarrow\infty
  34. 𝒫 \mathcal{P}

Statistical_physics.html

  1. F = m a F=ma
  2. Z Z
  3. q q
  4. Z = q e - E ( q ) k B T Z=\sum_{q}\mathrm{e}^{-\frac{E(q)}{k_{B}T}}
  5. k B k_{B}
  6. T T
  7. E ( q ) E(q)
  8. q q
  9. q q
  10. P ( q ) = e - E ( q ) k B T Z P(q)=\frac{{\mathrm{e}^{-\frac{E(q)}{k_{B}T}}}}{Z}

Steam_turbine.html

  1. r 1 r_{1}
  2. V w 1 V_{w1}
  3. r 2 r_{2}
  4. V w 2 V_{w2}
  5. V 1 V_{1}
  6. V 2 V_{2}
  7. V f 1 V_{f1}
  8. V f 2 V_{f2}
  9. V w 1 + U V_{w1}+U
  10. V w 2 V_{w2}
  11. V r 1 V_{r1}
  12. V r 2 V_{r2}
  13. U 1 U_{1}
  14. U 2 U_{2}
  15. α \alpha
  16. β \beta
  17. T = m ˙ ( r 2 V w 2 - r 1 V w 1 ) T=\dot{m}(r_{2}V_{w2}-r_{1}V_{w1})
  18. r 2 = r 1 = r r_{2}=r_{1}=r
  19. F u = m ˙ ( V w 1 - V w 2 ) F_{u}=\dot{m}(V_{w1}-V_{w2})
  20. W = T * ω {W}={T*\omega}
  21. U = ω * r {U}={\omega*r}
  22. W = m ˙ U ( Δ V w ) W=\dot{m}U({\Delta}V_{w})
  23. η b {\eta_{b}}
  24. η b = W o r k D o n e K i n e t i c E n e r g y S u p p l i e d = 2 U V w V 1 2 {\eta_{b}}=\frac{Work~{}Done}{Kinetic~{}Energy~{}Supplied}=\frac{2UV_{w}}{V_{1% }^{2}}
  25. η s t a g e = W o r k d o n e o n b l a d e E n e r g y s u p p l i e d p e r s t a g e = U Δ V w Δ h {\eta_{stage}}=\frac{Work~{}done~{}on~{}blade}{Energy~{}supplied~{}per~{}stage% }=\frac{U\Delta V_{w}}{\Delta h}
  26. Δ h = h 2 - h 1 {\Delta h}=h_{2}-h_{1}
  27. h 1 + V 1 2 2 = h 2 + V 2 2 2 {h_{1}}+\frac{V_{1}^{2}}{2}={h_{2}}+\frac{V_{2}^{2}}{2}
  28. V 1 V_{1}
  29. V 2 V_{2}
  30. Δ h {\Delta h}
  31. V 2 2 2 \frac{V_{2}^{2}}{2}
  32. η s t a g e = η b * η N {\eta_{stage}}={\eta_{b}}*{\eta_{N}}
  33. η N {\eta_{N}}
  34. V 2 2 2 ( h 1 - h 2 ) \frac{V_{2}^{2}}{2(h_{1}-h_{2})}
  35. h 1 h_{1}
  36. h 2 h_{2}
  37. Δ V w = V w 1 - ( - V w 2 ) {\Delta V_{w}}=V_{w1}-(-V_{w2})
  38. Δ V w = V w 1 + V w 2 {\Delta V_{w}}=V_{w1}+V_{w2}
  39. Δ V w = V r 1 cos β 1 + V r 2 cos β 2 {\Delta V_{w}}={V_{r1}\cos\beta_{1}+V_{r2}\cos\beta_{2}}
  40. Δ V w = V r 1 cos β 1 ( 1 + V r 2 cos β 2 V r 1 cos β 1 ) {\Delta V_{w}}={V_{r1}\cos\beta_{1}}(1+\frac{V_{r2}\cos\beta_{2}}{V_{r1}\cos% \beta_{1}})
  41. c = cos β 2 cos β 1 {c}=\frac{\cos\beta_{2}}{\cos\beta_{1}}
  42. k = V r 2 V r 1 {k}=\frac{V_{r2}}{V_{r1}}
  43. k < 1 k<1
  44. k = 1 k=1
  45. η b = 2 U Δ V w V 1 2 = 2 U ( cos α 1 - U / V 1 ) ( 1 + k c ) V 1 {\eta_{b}}=\frac{2U\Delta V_{w}}{V_{1}^{2}}=\frac{2U(\cos\alpha_{1}-U/V_{1})(1% +kc)}{V_{1}}
  46. ρ {\rho}
  47. U V 1 \frac{U}{V_{1}}
  48. η b {\eta_{b}}
  49. d η b d ρ = 0 {d\eta_{b}\over d\rho}=0
  50. d d ρ ( 2 cos α 1 - ρ 2 ( 1 + k c ) ) = 0 \frac{d}{d\rho}(2{\cos\alpha_{1}-\rho^{2}}(1+kc))=0
  51. ρ = cos α 1 2 {\rho}=\frac{\cos\alpha_{1}}{2}
  52. U V 1 = cos α 1 2 \frac{U}{V_{1}}=\frac{\cos\alpha_{1}}{2}
  53. ρ o p t = U V 1 = cos α 1 2 {\rho_{opt}}=\frac{U}{V_{1}}=\frac{\cos\alpha_{1}}{2}
  54. U V 1 = cos α 1 2 \frac{U}{V_{1}}=\frac{\cos\alpha_{1}}{2}
  55. η b {\eta_{b}}
  56. ( η b ) m a x = 2 ( ρ cos α 1 - ρ 2 ) ( 1 + k c ) = cos 2 α 1 ( 1 + k c ) 2 {(\eta_{b})_{max}}=2(\rho\cos\alpha_{1}-\rho^{2})(1+kc)=\frac{\cos^{2}\alpha_{% 1}(1+kc)}{2}
  57. β 1 = β 2 \beta_{1}=\beta_{2}
  58. c = 1 c=1
  59. ( η b ) m a x = c o s 2 α 1 ( 1 + k ) 2 {(\eta_{b})_{max}}=\frac{cos^{2}\alpha_{1}(1+k)}{2}
  60. ( η b ) m a x = cos 2 α 1 {(\eta_{b})_{max}}={\cos^{2}\alpha_{1}}
  61. ( η b ) m a x = cos 2 α 1 {(\eta_{b})_{max}}={\cos^{2}\alpha_{1}}
  62. cos 2 α 1 = 1 {\cos^{2}\alpha_{1}}=1
  63. α 1 = 0 \alpha_{1}=0
  64. α 1 \alpha_{1}
  65. E = Δ h E={\Delta h}
  66. E {E}
  67. Δ h f {\Delta h_{f}}
  68. Δ h m {\Delta h_{m}}
  69. V r 2 V_{r2}
  70. V r 1 V_{r1}
  71. Δ h m = V r 2 2 - V r 1 2 2 {\Delta h_{m}}=\frac{V_{r2}^{2}-V_{r1}^{2}}{2}
  72. Δ h f {\Delta h_{f}}
  73. V 1 2 - V 0 2 2 \frac{V_{1}^{2}-V_{0}^{2}}{2}
  74. V 0 V_{0}
  75. Δ h f {\Delta h_{f}}
  76. V 1 2 2 \frac{V_{1}^{2}}{2}
  77. E = Δ h f + Δ h m E={\Delta h_{f}+\Delta h_{m}}
  78. E = V 1 2 2 + V r 2 2 - V r 1 2 2 E=\frac{V_{1}^{2}}{2}+\frac{V_{r2}^{2}-V_{r1}^{2}}{2}
  79. α 1 = β 2 \alpha_{1}=\beta_{2}
  80. β 1 = α 2 \beta_{1}=\alpha_{2}
  81. V 1 = V r 2 V_{1}=V_{r2}
  82. V r 1 = V 2 V_{r1}=V_{2}
  83. E = V 1 2 - V r 1 2 2 {E}={V_{1}^{2}}-\frac{V_{r1}^{2}}{2}
  84. V r 1 2 = V 1 2 + U 2 - 2 U V 1 cos α 1 {V_{r1}^{2}}={V_{1}^{2}+U^{2}-2UV_{1}\cos\alpha_{1}}
  85. E = V 1 2 - V 1 2 2 - U 2 2 + 2 U V 1 cos α 1 2 {E}={V_{1}^{2}-\frac{V_{1}^{2}}{2}-\frac{U^{2}}{2}+\frac{2UV_{1}\cos\alpha_{1}% }{2}}
  86. E = V 1 2 - U 2 + 2 U V 1 cos α 1 2 {E}=\frac{V_{1}^{2}-U^{2}+2UV_{1}\cos\alpha_{1}}{2}
  87. W = U * Δ V w = U * ( 2 * V 1 cos α 1 - U ) {W}={U*\Delta V_{w}}={U*(2*V_{1}\cos\alpha_{1}-U)}
  88. η b = 2 U ( 2 V 1 cos α 1 - U ) V 1 2 - U 2 + 2 V 1 U cos α 1 {\eta_{b}}=\frac{2U(2V_{1}\cos\alpha_{1}-U)}{V_{1}^{2}-U^{2}+2V_{1}U\cos\alpha% _{1}}
  89. ρ = U V 1 {\rho}=\frac{U}{V_{1}}
  90. ( η b ) m a x = 2 ρ ( cos α 1 - ρ ) V 1 2 - U 2 + 2 U V 1 cos α 1 {(\eta_{b})_{max}}=\frac{2\rho(\cos\alpha_{1}-\rho)}{V_{1}^{2}-U^{2}+2UV_{1}% \cos\alpha_{1}}
  91. d η b d ρ = 0 {d\eta_{b}\over d\rho}=0
  92. ( 1 - ρ 2 + 2 ρ cos α 1 ) ( 4 cos α 1 - 4 ρ ) - 2 ρ ( 2 cos α 1 - ρ ) ( - 2 ρ + 2 cos α 1 ) = 0 {(1-\rho^{2}+2\rho\cos\alpha_{1})(4\cos\alpha_{1}-4\rho)-2\rho(2\cos\alpha_{1}% -\rho)(-2\rho+2\cos\alpha_{1})=0}
  93. ρ o p t = U V 1 = cos α 1 {\rho_{opt}}=\frac{U}{V_{1}}={\cos\alpha_{1}}
  94. ( η b ) m a x {(\eta_{b})_{max}}
  95. ρ = cos α 1 {\rho}={\cos\alpha_{1}}
  96. ( η b ) r e a c t i o n = 2 cos 2 α 1 1 + cos 2 α 1 {(\eta_{b})_{reaction}}=\frac{2\cos^{2}\alpha_{1}}{1+\cos^{2}\alpha_{1}}
  97. ( η b ) i m p u l s e = cos 2 α 1 {(\eta_{b})_{impulse}}={\cos^{2}\alpha_{1}}
  98. W ˙ m ˙ = h 3 - h 4 \frac{\dot{W}}{\dot{m}}=h_{3}-h_{4}
  99. η t = h 3 - h 4 h 3 - h 4 s \eta_{t}=\frac{h_{3}-h_{4}}{h_{3}-h_{4s}}

Stefan–Boltzmann_constant.html

  1. [ u p h y s c o n s t , u s i g m a , u s y m b o l = y e s , u a f t e r = . ] [u^{\prime}physconst^{\prime},u^{\prime}sigma^{\prime},u^{\prime}symbol=yes^{% \prime},u^{\prime}after=.^{\prime}]
  2. σ 5.6704 × 10 - 5 erg cm - 2 s - 1 K - 4 . \sigma\approx 5.6704\times 10^{-5}\ \textrm{erg}\,\,\textrm{cm}^{-2}\,\textrm{% s}^{-1}\,\textrm{K}^{-4}.
  3. σ 11.7 × 10 - 8 cal cm - 2 day - 1 K - 4 . \sigma\approx 11.7\times 10^{-8}\ \textrm{cal}\,\,\textrm{cm}^{-2}\,\textrm{% day}^{-1}\,\textrm{K}^{-4}.
  4. σ = 0.1714 × 10 - 8 BTU hr - 1 ft - 2 R - 4 . \sigma=0.1714\times 10^{-8}\ \textrm{BTU}\,\textrm{hr}^{-1}\,\textrm{ft}^{-2}% \,\textrm{R}^{-4}.
  5. σ = 2 π 5 k B 4 15 h 3 c 2 = π 2 k B 4 60 3 c 2 = 5.670373 ( 21 ) 10 - 8 J m - 2 s - 1 K - 4 \sigma=\frac{2\pi^{5}k_{\rm B}^{4}}{15h^{3}c^{2}}=\frac{\pi^{2}k_{\rm B}^{4}}{% 60\hbar^{3}c^{2}}=5.670373(21)\,\cdot 10^{-8}\ \textrm{J}\,\textrm{m}^{-2}\,% \textrm{s}^{-1}\,\textrm{K}^{-4}
  6. σ = 2 π 5 R 4 15 h 3 c 2 N A 4 = 32 π 5 h R 4 R 4 15 A r ( e ) 4 M u 4 c 6 α 8 \sigma=\frac{2\pi^{5}R^{4}}{15h^{3}c^{2}N_{\rm A}^{4}}=\frac{32\pi^{5}hR^{4}R_% {\infty}^{4}}{15A_{\rm r}({\rm e})^{4}M_{\rm u}^{4}c^{6}\alpha^{8}}
  7. a = 4 σ c = 7.5657 × 10 - 15 erg cm - 3 K - 4 = 7.5657 × 10 - 16 J m - 3 K - 4 . a=\frac{4\sigma}{c}=7.5657\times 10^{-15}\textrm{erg}\,\textrm{cm}^{-3}\,% \textrm{K}^{-4}=7.5657\times 10^{-16}\textrm{J}\,\textrm{m}^{-3}\,\textrm{K}^{% -4}.

Stefan–Boltzmann_law.html

  1. j j^{\star}
  2. j = σ T 4 . j^{\star}=\sigma T^{4}.
  3. σ = 2 π 5 k 4 15 c 2 h 3 = 5.670373 × 10 - 8 W m - 2 K - 4 , \sigma=\frac{2\pi^{5}k^{4}}{15c^{2}h^{3}}=5.670373\times 10^{-8}\,\mathrm{W\,m% ^{-2}K^{-4}},
  4. L = j π = σ π T 4 . L=\frac{j^{\star}}{\pi}=\frac{\sigma}{\pi}T^{4}.
  5. ε < 1 \varepsilon<1
  6. j = ε σ T 4 . j^{\star}=\varepsilon\sigma T^{4}.
  7. j j^{\star}
  8. ε \varepsilon
  9. ε = 1 \varepsilon=1
  10. ε = ε ( λ ) \varepsilon=\varepsilon(\lambda)
  11. A A
  12. P = A j = A ε σ T 4 . P=Aj^{\star}=A\varepsilon\sigma T^{4}.
  13. L = 4 π R 2 σ T e 4 L=4\pi R^{2}\sigma{T_{e}}^{4}
  14. R R ( T T ) 2 L L \frac{R}{R_{\odot}}\approx\left(\frac{T_{\odot}}{T}\right)^{2}\cdot\sqrt{\frac% {L}{L_{\odot}}}
  15. R R_{\odot}
  16. E S = 4 π r S 2 σ T S 4 E_{S}=4\pi r_{S}^{2}\sigma T_{S}^{4}
  17. E a 0 = E S 4 π a 0 2 E_{a_{0}}=\frac{E_{S}}{4\pi{a_{0}}^{2}}
  18. π r E 2 \pi r_{E}^{2}
  19. E a b s = π r E 2 × E a 0 : E_{abs}=\pi r_{E}^{2}\times E_{a_{0}}:
  20. 4 π r E 2 σ T E 4 \displaystyle 4\pi r_{E}^{2}\sigma T_{E}^{4}
  21. T E 4 \displaystyle T_{E}^{4}
  22. T 4 T^{4}
  23. p p
  24. u u
  25. p = u 3 p=\frac{u}{3}
  26. d U = T d S - p d V dU=TdS-pdV
  27. d V dV
  28. T T
  29. ( U V ) T = T ( S V ) T - p = T ( p T ) V - p \left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial S}{% \partial V}\right)_{T}-p=T\left(\frac{\partial p}{\partial T}\right)_{V}-p
  30. ( S V ) T = ( p T ) V \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial p}{% \partial T}\right)_{V}
  31. U = u V U=uV
  32. ( U V ) T = u ( V V ) T = u \left(\frac{\partial U}{\partial V}\right)_{T}=u\left(\frac{\partial V}{% \partial V}\right)_{T}=u
  33. ( U V ) T = T ( p T ) V - p \left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial p}{% \partial T}\right)_{V}-p
  34. ( U V ) T \left(\frac{\partial U}{\partial V}\right)_{T}
  35. p p
  36. u = T 3 ( u T ) V - u 3 u=\frac{T}{3}\left(\frac{\partial u}{\partial T}\right)_{V}-\frac{u}{3}
  37. ( u T ) V \left(\frac{\partial u}{\partial T}\right)_{V}
  38. u u
  39. T T
  40. d u 4 u = d T T \frac{du}{4u}=\frac{dT}{T}
  41. u = A T 4 u=AT^{4}
  42. A A
  43. n n
  44. P = u n P=\frac{u}{n}
  45. n n
  46. T d S = ( n + 1 ) P d V + n V d P TdS=(n+1)PdV+nVdP\,
  47. 1 P d P d T = ( n + 1 ) T \frac{1}{P}\frac{dP}{dT}=\frac{(n+1)}{T}
  48. P T n + 1 P\propto T^{n+1}
  49. u T n + 1 u\propto T^{n+1}
  50. d Q d t T n + 1 \frac{dQ}{dt}\propto T^{n+1}
  51. n n
  52. σ = 1 p ( n ) π n 2 Γ ( 1 + n 2 ) 1 c n - 1 n ( n - 1 ) h n k ( n + 1 ) Γ ( n + 1 ) ζ ( n + 1 ) \sigma=\frac{1}{p(n)}\frac{\pi^{\frac{n}{2}}}{\Gamma(1+\frac{n}{2})}\frac{1}{c% ^{n-1}}\frac{n(n-1)}{h^{n}}k^{(n+1)}\Gamma(n+1)\zeta(n+1)
  53. ζ ( x ) \zeta(x)
  54. p ( n ) p(n)
  55. n n
  56. p ( 3 ) = 4 p(3)=4
  57. I ( ν , T ) = 2 h ν 3 c 2 1 e h ν k T - 1 . I(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}}-1}.
  58. I ( ν , T ) I(\nu,T)\,
  59. ν \nu\,
  60. h h\,
  61. c c\,
  62. k k\,
  63. I ( ν , T ) A d ν d Ω I(\nu,T)~{}A~{}d\nu~{}d\Omega
  64. P A = 0 I ( ν , T ) d ν d Ω \frac{P}{A}=\int_{0}^{\infty}I(\nu,T)d\nu\int d\Omega\,
  65. P A \displaystyle\frac{P}{A}
  66. P A = 2 π h c 2 0 ν 3 e h ν k T - 1 d ν \frac{P}{A}=\frac{2\pi h}{c^{2}}\int_{0}^{\infty}\frac{\nu^{3}}{e^{\frac{h\nu}% {kT}}-1}d\nu\,
  67. u = h ν k T u=\frac{h\nu}{kT}\,
  68. d u = h k T d ν du=\frac{h}{kT}\,d\nu
  69. P A = 2 π h c 2 ( k T h ) 4 0 u 3 e u - 1 d u . \frac{P}{A}=\frac{2\pi h}{c^{2}}\left(\frac{kT}{h}\right)^{4}\int_{0}^{\infty}% \frac{u^{3}}{e^{u}-1}\,du.
  70. π 4 15 \frac{\pi^{4}}{15}
  71. j = σ T 4 , σ = 2 π 5 k 4 15 c 2 h 3 = π 2 k 4 60 3 c 2 . j^{\star}=\sigma T^{4}~{},~{}~{}\sigma=\frac{2\pi^{5}k^{4}}{15c^{2}h^{3}}=% \frac{\pi^{2}k^{4}}{60\hbar^{3}c^{2}}.
  72. J = 0 x 3 exp ( x ) - 1 d x = Γ ( 4 ) Li 4 ( 1 ) = 6 Li 4 ( 1 ) = 6 ζ ( 4 ) J=\int_{0}^{\infty}\frac{x^{3}}{\exp\left(x\right)-1}\,dx=\Gamma(4)\,\mathrm{% Li}_{4}(1)=6\,\mathrm{Li}_{4}(1)=6\zeta(4)
  73. Li s ( z ) \mathrm{Li}_{s}(z)
  74. ζ ( z ) \zeta(z)
  75. f ( k ) = 0 sin ( k x ) exp ( x ) - 1 d x . f(k)=\int_{0}^{\infty}\frac{\sin\left(kx\right)}{\exp\left(x\right)-1}\,dx.
  76. k k
  77. J J
  78. k 3 k^{3}
  79. f ( k ) f(k)
  80. f ( k ) = lim ε 0 Im ε exp ( i k x ) exp ( x ) - 1 d x . f(k)=\lim_{\varepsilon\rightarrow 0}~{}\,\text{Im}~{}\int_{\varepsilon}^{% \infty}\frac{\exp\left(ikx\right)}{\exp\left(x\right)-1}\,dx.
  81. C ( ε , R ) exp ( i k z ) exp ( z ) - 1 d z \oint_{C(\varepsilon,R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1}\,dz
  82. C ( ε , R ) C(\varepsilon,R)
  83. ε \varepsilon
  84. R R
  85. R + 2 π i R+2\pi i
  86. ε + 2 π i \varepsilon+2\pi i
  87. 2 π i - ε i 2\pi i-\varepsilon i
  88. 2 π i 2\pi i
  89. ε \varepsilon
  90. 2 π i 2\pi i
  91. ε i \varepsilon i
  92. ε \varepsilon
  93. ε \varepsilon
  94. C ( ε , R ) exp ( i k z ) exp ( z ) - 1 d z = 0. \oint_{C(\varepsilon,R)}\frac{\exp\left(ikz\right)}{\exp\left(z\right)-1}\,dz=0.
  95. R R\rightarrow\infty
  96. R R
  97. R + 2 π i R+2\pi i
  98. ε \varepsilon
  99. R R
  100. R + 2 π i R+2\pi i
  101. ε + 2 π i \varepsilon+2\pi i
  102. ε \varepsilon
  103. ε \varepsilon
  104. i π 2 \textstyle\frac{i\pi}{2}
  105. [ 1 - exp ( - 2 π k ) ] ε exp ( i k x ) exp ( x ) - 1 d x = i ε 2 π - ε exp ( - k y ) exp ( i y ) - 1 d y + i π 2 [ 1 + exp ( - 2 π k ) ] + 𝒪 ( ε ) (1) \left[1-\exp\left(-2\pi k\right)\right]\int_{\varepsilon}^{\infty}\frac{\exp% \left(ikx\right)}{\exp\left(x\right)-1}\,dx=i\int_{\varepsilon}^{2\pi-% \varepsilon}\frac{\exp\left(-ky\right)}{\exp\left(iy\right)-1}\,dy+i\frac{\pi}% {2}\left[1+\exp\left(-2\pi k\right)\right]+\mathcal{O}\left(\varepsilon\right)% \qquad\,\text{ (1)}
  106. ε \varepsilon
  107. R R
  108. R + 2 π i R+2\pi i
  109. 2 π i + ε 2\pi i+\varepsilon
  110. 1 exp ( i y ) - 1 = exp ( - i y 2 ) exp ( i y 2 ) - exp ( - i y 2 ) = 1 2 i exp ( - i y 2 ) sin ( y 2 ) \frac{1}{\exp\left(iy\right)-1}=\frac{\exp\left(-i\frac{y}{2}\right)}{\exp% \left(i\frac{y}{2}\right)-\exp\left(-i\frac{y}{2}\right)}=\frac{1}{2i}\frac{% \exp\left(-i\frac{y}{2}\right)}{\sin\left(\frac{y}{2}\right)}
  111. ε 0 \varepsilon\rightarrow 0
  112. f ( k ) = - 1 2 k + π 2 coth ( π k ) f(k)=-\frac{1}{2k}+\frac{\pi}{2}\coth\left(\pi k\right)
  113. coth ( x ) = 1 + exp ( - 2 x ) 1 - exp ( - 2 x ) . \coth\left(x\right)=\frac{1+\exp\left(-2x\right)}{1-\exp\left(-2x\right)}.
  114. coth ( x ) \coth(x)
  115. coth ( x ) = 1 x + 1 3 x - 1 45 x 3 + \coth(x)=\frac{1}{x}+\frac{1}{3}x-\frac{1}{45}x^{3}+\cdots
  116. k 3 k^{3}
  117. f ( k ) f(k)
  118. - π 4 90 \textstyle-\frac{\pi^{4}}{90}
  119. J = π 4 15 \textstyle J=\frac{\pi^{4}}{15}
  120. j = 2 π 5 k 4 15 h 3 c 2 T 4 j^{\star}=\frac{2\pi^{5}k^{4}}{15h^{3}c^{2}}T^{4}

Steiner_system.html

  1. S ( 2 , q + 1 , q 2 + q + 1 ) S(2,q+1,q^{2}+q+1)
  2. q 2 + q + 1 q^{2}+q+1
  3. q + 1 q+1
  4. \equiv
  5. \equiv
  6. \equiv
  7. \not\equiv
  8. 1 < t < k < n 1<t<k<n
  9. ( n t ) {\textstyle\left({{n}\atop{t}}\right)}
  10. ( k t ) {\textstyle\left({{k}\atop{t}}\right)}
  11. ( n t ) = b ( k t ) {\textstyle\left({{n}\atop{t}}\right)}=b{\textstyle\left({{k}\atop{t}}\right)}
  12. b = ( n t ) ( k t ) b=\frac{{\textstyle\left({{n}\atop{t}}\right)}}{{\textstyle\left({{k}\atop{t}}% \right)}}
  13. ( n - 1 t - 1 ) = r ( k - 1 t - 1 ) {\textstyle\left({{n-1}\atop{t-1}}\right)}=r{\textstyle\left({{k-1}\atop{t-1}}% \right)}
  14. r = ( n - 1 t - 1 ) ( k - 1 t - 1 ) r=\frac{{\textstyle\left({{n-1}\atop{t-1}}\right)}}{{\textstyle\left({{k-1}% \atop{t-1}}\right)}}
  15. b k = r n bk=rn
  16. b n b\geq n
  17. t t t^{\prime}\leq t
  18. \equiv
  19. 𝐅 \mathbf{F}
  20. S S
  21. 𝐅 \mathbf{F}
  22. { , 1 , 3 , 4 , 5 , 9 } , \{\infty,1,3,4,5,9\},
  23. z = f ( z ) = a z + b c z + d , where a , b , c , d are in F 11 and a d - b c is a non-zero square in F 11 . z^{\prime}=f(z)=\frac{az+b}{cz+d}\,\text{, where }a,b,c,d\,\text{ are in }F_{1% 1}\,\text{ and }ad-bc\,\text{ is a non-zero square in }F_{11}.
  24. f ( d / c ) = f(−d/c)=∞
  25. f ( ) = a / c f(∞)=a/c
  26. S S
  27. S S

Step-index_profile.html

  1. = n 1 - n 2 n 1 1 \triangle\,=\frac{n_{1}-n_{2}}{n_{1}}\ll\ 1
  2. \triangle
  3. pulse dispersion = n 1 c \,\text{pulse dispersion}=\frac{\triangle\ n_{1}\ \ell}{c}\,\!
  4. \triangle\,\!
  5. n 1 n_{1}\,\!
  6. \ell\,\!
  7. c = 3 × 10 8 ms - 1 c=3\times 10^{8}\,\mathrm{ms}^{-1}\,\!

Steradian.html

  1. Ω = A r 2 sr \Omega=\frac{A}{r^{2}}\,\mathrm{sr}\,
  2. A A
  3. π \pi
  4. r r
  5. h h
  6. r r
  7. θ \displaystyle\theta
  8. Ω = 2 π ( 1 - cos θ ) sr \Omega=2\pi\left(1-\cos\theta\right)\,\mathrm{sr}
  9. θ = l r rad \theta=\frac{l}{r}\,\mathrm{rad}
  10. Ω = A r 2 sr \Omega=\frac{A}{r^{2}}\,\mathrm{sr}

Stimulated_emission.html

  1. E 2 - E 1 = h ν 0 E_{2}-E_{1}=h\,\nu_{0}
  2. ν 0 \nu_{0}
  3. N 2 t = - N 1 t = - B 21 ρ ( ν ) N 2 \frac{\partial N_{2}}{\partial t}=-\frac{\partial N_{1}}{\partial t}=-B_{21}\ % \rho(\nu)N_{2}
  4. N 2 t = - N 1 t = B 12 ρ ( ν ) N 1 \frac{\partial N_{2}}{\partial t}=-\frac{\partial N_{1}}{\partial t}=B_{12}\ % \rho(\nu)N_{1}
  5. B 12 = B 21 B_{12}=B_{21}
  6. N 1 ( n e t ) t = - N 2 ( n e t ) t = B 21 ρ ( ν ) ( N 2 - N 1 ) = B 21 ρ ( ν ) Δ N \frac{\partial N_{1}\ (net)}{\partial t}=-\frac{\partial N_{2}\ (net)}{% \partial t}=B_{21}\ \rho(\nu)(N_{2}-N_{1})=B_{21}\ \rho(\nu)\ \Delta N
  7. Δ N > 0 \Delta N>0
  8. N 2 > N 1 N_{2}>N_{1}
  9. Δ N > 0 \Delta N>0
  10. ν 0 \nu_{0}
  11. ν 0 \nu_{0}
  12. g ( ν ) = 1 π ( Γ / 2 ) ( ν - ν 0 ) 2 + ( Γ / 2 ) 2 g^{\prime}(\nu)={1\over\pi}{(\Gamma/2)\over(\nu-\nu_{0})^{2}+(\Gamma/2)^{2}}
  13. Γ \Gamma\,
  14. ν = ν 0 \nu=\nu_{0}
  15. ν 0 \nu_{0}
  16. g ( ν ) = g ( ν ) g ( ν 0 ) = ( Γ / 2 ) 2 ( ν - ν 0 ) 2 + ( Γ / 2 ) 2 g(\nu)={g^{\prime}(\nu)\over g^{\prime}(\nu_{0})}={(\Gamma/2)^{2}\over(\nu-\nu% _{0})^{2}+(\Gamma/2)^{2}}
  17. ν 0 \nu_{0}
  18. ν \nu
  19. P = h ν g ( ν ) B 21 ρ ( ν ) Δ N P=h\nu\ g(\nu)B_{21}\ \rho(\nu)\ \Delta N
  20. σ 21 ( ν ) = A 21 λ 2 8 π n 2 g ( ν ) \sigma_{21}(\nu)=A_{21}{\lambda^{2}\over 8\pi n^{2}}g(\nu)
  21. Δ N 21 = ( N 2 - g 2 g 1 N 1 ) \Delta N_{21}=\left(N_{2}-{g_{2}\over g_{1}}N_{1}\right)
  22. d I d z = σ 21 ( ν ) Δ N 21 I ( z ) {dI\over dz}=\sigma_{21}(\nu)\cdot\Delta N_{21}\cdot I(z)
  23. d I d z = γ 0 ( ν ) I ( z ) {dI\over dz}=\gamma_{0}(\nu)\cdot I(z)
  24. γ 0 ( ν ) = σ 21 ( ν ) Δ N 21 \gamma_{0}(\nu)=\sigma_{21}(\nu)\cdot\Delta N_{21}
  25. d I I ( z ) = γ 0 ( ν ) d z {dI\over I(z)}=\gamma_{0}(\nu)\cdot dz
  26. ln ( I ( z ) I i n ) = γ 0 ( ν ) z \ln\left({I(z)\over I_{in}}\right)=\gamma_{0}(\nu)\cdot z
  27. I ( z ) = I i n e γ 0 ( ν ) z I(z)=I_{in}e^{\gamma_{0}(\nu)z}
  28. I i n = I ( z = 0 ) I_{in}=I(z=0)\,
  29. I S = h ν σ ( ν ) τ S I_{S}={h\nu\over\sigma(\nu)\cdot\tau_{S}}
  30. ν \nu
  31. d I d z = γ 0 ( ν ) 1 + g ¯ ( ν ) I ( z ) I S I ( z ) {dI\over dz}={\gamma_{0}(\nu)\over 1+\bar{g}(\nu){I(z)\over I_{S}}}\cdot I(z)
  32. I S I_{S}
  33. d I I ( z ) [ 1 + g ¯ ( ν ) I ( z ) I S ] = γ 0 ( ν ) d z {dI\over I(z)}\left[1+\bar{g}(\nu){I(z)\over I_{S}}\right]=\gamma_{0}(\nu)% \cdot dz
  34. ln ( I ( z ) I i n ) + g ¯ ( ν ) I ( z ) - I i n I S = γ 0 ( ν ) z \ln\left({I(z)\over I_{in}}\right)+\bar{g}(\nu){I(z)-I_{in}\over I_{S}}=\gamma% _{0}(\nu)\cdot z
  35. ln ( I ( z ) I i n ) + g ¯ ( ν ) I i n I S ( I ( z ) I i n - 1 ) = γ 0 ( ν ) z \ln\left({I(z)\over I_{in}}\right)+\bar{g}(\nu){I_{in}\over I_{S}}\left({I(z)% \over I_{in}}-1\right)=\gamma_{0}(\nu)\cdot z
  36. G = G ( z ) = I ( z ) I i n G=G(z)={I(z)\over I_{in}}
  37. ln ( G ) + g ¯ ( ν ) I i n I S ( G - 1 ) = γ 0 ( ν ) z \ln\left(G\right)+\bar{g}(\nu){I_{in}\over I_{S}}\left(G-1\right)=\gamma_{0}(% \nu)\cdot z
  38. I i n I S I_{in}\ll I_{S}\,
  39. ln ( G ) = ln ( G 0 ) = γ 0 ( ν ) z \ln(G)=\ln(G_{0})=\gamma_{0}(\nu)\cdot z
  40. G = G 0 = e γ 0 ( ν ) z G=G_{0}=e^{\gamma_{0}(\nu)z}
  41. I i n I S I_{in}\gg I_{S}\,
  42. G 1 G\rightarrow 1
  43. I ( z ) = I i n + γ 0 ( ν ) z g ¯ ( ν ) I S I(z)=I_{in}+{\gamma_{0}(\nu)\cdot z\over\bar{g}(\nu)}I_{S}

Stimulus–response_model.html

  1. E ( Y ) = f ( x ) E(Y)=f(x)
  2. E ( Y ) = α + β x . E(Y)=\alpha+\beta x.

Stochastic_process.html

  1. ( Ω , , P ) (\Omega,\mathcal{F},P)
  2. ( S , Σ ) (S,\Sigma)
  3. Ω \Omega
  4. { X t : t T } \{X_{t}:t\in T\}
  5. X t X_{t}
  6. Ω \Omega
  7. T = ( t 1 , , t k ) T k T^{\prime}=(t_{1},\ldots,t_{k})\in T^{k}
  8. X T = ( X t 1 , X t 2 , , X t k ) X_{T^{\prime}}=(X_{t_{1}},X_{t_{2}},\ldots,X_{t_{k}})
  9. S k S^{k}
  10. T ( ) = ( X T - 1 ( ) ) \mathbb{P}_{T^{\prime}}(\cdot)=\mathbb{P}(X_{T^{\prime}}^{-1}(\cdot))
  11. S k S^{k}
  12. τ \tau
  13. τ \tau
  14. f : X Y f:X\to Y
  15. f ( x 1 ) , , f ( x n ) f(x_{1}),\dots,f(x_{n})
  16. f ( x 1 ) , , f ( x n - 1 ) f(x_{1}),\dots,f(x_{n-1})
  17. [ f ( x 1 ) , , f ( x n ) ] [f(x_{1}),\dots,f(x_{n})]
  18. Y n Y_{n}
  19. { f ( x i ) } \{f(x_{i})\}
  20. ( Ω , , P ) (\Omega,\mathcal{F},P)
  21. Ω \Omega
  22. { t , t T } \{\mathcal{F}_{t},t\in T\}
  23. T T
  24. \mathcal{F}
  25. T \in T
  26. X X
  27. T T
  28. T \in T
  29. X t X_{t}
  30. t \mathcal{F}_{t}
  31. X = { X t : t T } X=\{X_{t}:t\in T\}
  32. t \mathcal{F}_{t}
  33. X s X_{s}
  34. t = σ ( { X s - 1 ( A ) : s t , A Σ } ) \mathcal{F}_{t}=\sigma(\{X_{s}^{-1}(A):s\leq t,A\in\Sigma\})
  35. t t
  36. X t X_{t}
  37. \mathbb{N}
  38. X t X_{t}
  39. t t
  40. X t X_{t}
  41. X t X_{t}
  42. t t
  43. t t
  44. f ( i ) f(i)

Stoichiometry.html

  1. 2.00 g NaCl 58.44 g NaCl mol - 1 = 0.034 mol \frac{2.00\mbox{ g NaCl}~{}}{58.44\mbox{ g NaCl mol}~{}^{-1}}=0.034\ \,\text{mol}
  2. ( 2.00 g NaCl 1 ) ( 1 mol NaCl 58.44 g NaCl ) = 0.034 mol \left(\frac{2.00\mbox{ g NaCl}~{}}{1}\right)\left(\frac{1\mbox{ mol NaCl}~{}}{% 58.44\mbox{ g NaCl}~{}}\right)=0.034\ \,\text{mol}
  3. ( 0.27 mol CH 3 OH 1 ) ( 4 mol H 2 O 2 mol CH 3 OH ) = 0.54 mol H 2 O \left(\frac{0.27\mbox{ mol }~{}\mathrm{CH_{3}OH}}{1}\right)\left(\frac{4\mbox{% mol }~{}\mathrm{H_{2}O}}{2\mbox{ mol }~{}\mathrm{CH_{3}OH}}\right)=0.54\ \,% \text{mol }\mathrm{H_{2}O}
  4. ( 16.00 g Cu 1 ) ( 1 mol Cu 63.55 g Cu ) = 0.2518 mol Cu \left(\frac{16.00\mbox{ g Cu}~{}}{1}\right)\left(\frac{1\mbox{ mol Cu}~{}}{63.% 55\mbox{ g Cu}~{}}\right)=0.2518\ \,\text{mol Cu}
  5. ( 0.2518 mol Cu 1 ) ( 2 mol Ag 1 mol Cu ) = 0.5036 mol Ag \left(\frac{0.2518\mbox{ mol Cu}~{}}{1}\right)\left(\frac{2\mbox{ mol Ag}~{}}{% 1\mbox{ mol Cu}~{}}\right)=0.5036\ \,\text{mol Ag}
  6. ( 0.5036 mol Ag 1 ) ( 107.87 g Ag 1 mol Ag ) = 54.32 g Ag \left(\frac{0.5036\mbox{ mol Ag}~{}}{1}\right)\left(\frac{107.87\mbox{ g Ag}~{% }}{1\mbox{ mol Ag}~{}}\right)=54.32\ \,\text{g Ag}
  7. m Ag = ( 16.00 g Cu 1 ) ( 1 mol Cu 63.55 g Cu ) ( 2 mol Ag 1 mol Cu ) ( 107.87 g Ag 1 mol Ag ) = 54.32 g m_{\mathrm{Ag}}=\left(\frac{16.00\mbox{ g }~{}\mathrm{Cu}}{1}\right)\left(% \frac{1\mbox{ mol }~{}\mathrm{Cu}}{63.55\mbox{ g }~{}\mathrm{Cu}}\right)\left(% \frac{2\mbox{ mol }~{}\mathrm{Ag}}{1\mbox{ mol }~{}\mathrm{Cu}}\right)\left(% \frac{107.87\mbox{ g }~{}\mathrm{Ag}}{1\mbox{ mol Ag}~{}}\right)=54.32\mbox{ g% }~{}
  8. m H 2 O = ( 120. g C 3 H 8 1 ) ( 1 mol C 3 H 8 44.09 g C 3 H 8 ) ( 4 mol H 2 O 1 mol C 3 H 8 ) ( 18.02 g H 2 O 1 mol H 2 O ) = 196 g m_{\mathrm{H_{2}O}}=\left(\frac{120.\mbox{ g }~{}\mathrm{C_{3}H_{8}}}{1}\right% )\left(\frac{1\mbox{ mol }~{}\mathrm{C_{3}H_{8}}}{44.09\mbox{ g }~{}\mathrm{C_% {3}H_{8}}}\right)\left(\frac{4\mbox{ mol }~{}\mathrm{H_{2}O}}{1\mbox{ mol }~{}% \mathrm{C_{3}H_{8}}}\right)\left(\frac{18.02\mbox{ g }~{}\mathrm{H_{2}O}}{1% \mbox{ mol }~{}\mathrm{H_{2}O}}\right)=196\mbox{ g}~{}
  9. m Al = ( 85.0 g Fe 2 O 3 1 ) ( 1 mol Fe 2 O 3 159.7 g Fe 2 O 3 ) ( 2 mol Al 1 mol Fe 2 O 3 ) ( 26.98 g Al 1 mol Al ) = 28.7 g m_{\mathrm{Al}}=\left(\frac{85.0\mbox{ g }~{}\mathrm{Fe_{2}O_{3}}}{1}\right)% \left(\frac{1\mbox{ mol }~{}\mathrm{Fe_{2}O_{3}}}{159.7\mbox{ g }~{}\mathrm{Fe% _{2}O_{3}}}\right)\left(\frac{2\mbox{ mol Al}~{}}{1\mbox{ mol }~{}\mathrm{Fe_{% 2}O_{3}}}\right)\left(\frac{26.98\mbox{ g Al}~{}}{1\mbox{ mol Al}~{}}\right)=2% 8.7\mbox{ g}~{}
  10. m PbO = ( 200.0 g PbS 1 ) ( 1 mol PbS 239.27 g PbS ) ( 2 mol PbO 2 mol PbS ) ( 223.2 g PbO 1 mol PbO ) = 186.6 g m_{\mathrm{PbO}}=\left(\frac{200.0\mbox{ g }~{}\mathrm{PbS}}{1}\right)\left(% \frac{1\mbox{ mol }~{}\mathrm{PbS}}{239.27\mbox{ g }~{}\mathrm{PbS}}\right)% \left(\frac{2\mbox{ mol }~{}\mathrm{PbO}}{2\mbox{ mol }~{}\mathrm{PbS}}\right)% \left(\frac{223.2\mbox{ g }~{}\mathrm{PbO}}{1\mbox{ mol }~{}\mathrm{PbO}}% \right)=186.6\mbox{ g}~{}
  11. m PbO = ( 200.0 g O 2 1 ) ( 1 mol O 2 32.00 g O 2 ) ( 2 mol PbO 3 mol O 2 ) ( 223.2 g PbO 1 mol PbO ) = 930.0 g m_{\mathrm{PbO}}=\left(\frac{200.0\mbox{ g }~{}\mathrm{O_{2}}}{1}\right)\left(% \frac{1\mbox{ mol }~{}\mathrm{O_{2}}}{32.00\mbox{ g }~{}\mathrm{O_{2}}}\right)% \left(\frac{2\mbox{ mol }~{}\mathrm{PbO}}{3\mbox{ mol }~{}\mathrm{O_{2}}}% \right)\left(\frac{223.2\mbox{ g }~{}\mathrm{PbO}}{1\mbox{ mol }~{}\mathrm{PbO% }}\right)=930.0\mbox{ g}~{}
  12. percent yield = actual yield theoretical yield × 100 \mbox{percent yield}~{}=\frac{\mbox{actual yield}~{}}{\mbox{theoretical yield}% ~{}}\times\!\,100
  13. percent yield = 170.0 g PbO 186.6 g PbO × 100 = 91.12 % \mbox{percent yield}~{}=\frac{\mbox{170.0 g PbO}~{}}{\mbox{186.6 g PbO}~{}}% \times\!\,100=91.12\%
  14. m HCl = ( 90.0 g FeCl 3 1 ) ( 1 mol FeCl 3 162 g FeCl 3 ) ( 6 mol HCl 2 mol FeCl 3 ) ( 36.5 g HCl 1 mol HCl ) = 60.8 g m_{\mathrm{HCl}}=\left(\frac{90.0\mbox{ g }~{}\mathrm{FeCl_{3}}}{1}\right)% \left(\frac{1\mbox{ mol }~{}\mathrm{FeCl_{3}}}{162\mbox{ g }~{}\mathrm{FeCl_{3% }}}\right)\left(\frac{6\mbox{ mol }~{}\mathrm{HCl}}{2\mbox{ mol }~{}\mathrm{% FeCl_{3}}}\right)\left(\frac{36.5\mbox{ g }~{}\mathrm{HCl}}{1\mbox{ mol }~{}% \mathrm{HCl}}\right)=60.8\mbox{ g}~{}
  15. m HCl = ( 52.0 g H 2 S 1 ) ( 1 mol H 2 S 34.1 g H 2 S ) ( 6 mol HCl 3 mol H 2 S ) ( 36.5 g HCl 1 mol HCl ) = 111 g m_{\mathrm{HCl}}=\left(\frac{52.0\mbox{ g }~{}\mathrm{H_{2}S}}{1}\right)\left(% \frac{1\mbox{ mol }~{}\mathrm{H_{2}S}}{34.1\mbox{ g }~{}\mathrm{H_{2}S}}\right% )\left(\frac{6\mbox{ mol }~{}\mathrm{HCl}}{3\mbox{ mol }~{}\mathrm{H_{2}S}}% \right)\left(\frac{36.5\mbox{ g }~{}\mathrm{HCl}}{1\mbox{ mol }~{}\mathrm{HCl}% }\right)=111\mbox{ g}~{}
  16. m H 2 S = ( 90.0 g FeCl 3 1 ) ( 1 mol FeCl 3 162 g FeCl 3 ) ( 3 mol H 2 S 2 mol FeCl 3 ) ( 34.1 g H 2 S 1 mol H 2 S ) = 28.4 g reacted m_{\mathrm{H_{2}S}}=\left(\frac{90.0\mbox{ g }~{}\mathrm{FeCl_{3}}}{1}\right)% \left(\frac{1\mbox{ mol }~{}\mathrm{FeCl_{3}}}{162\mbox{ g }~{}\mathrm{FeCl_{3% }}}\right)\left(\frac{3\mbox{ mol }~{}\mathrm{H_{2}S}}{2\mbox{ mol }~{}\mathrm% {FeCl_{3}}}\right)\left(\frac{34.1\mbox{ g }~{}\mathrm{H_{2}S}}{1\mbox{ mol }~% {}\mathrm{H_{2}S}}\right)={28.4\mbox{ g }~{}\mathrm{reacted}}
  17. 52.0 g H 2 S - 28.4 g H 2 S = 23.6 g H 2 S {52.0\mbox{ g }~{}\mathrm{H_{2}S}}-{28.4\mbox{ g }~{}\mathrm{H_{2}S}}={23.6% \mbox{ g }~{}\mathrm{H_{2}S}}
  18. excess {\mathrm{excess}}
  19. ν i = Δ N i Δ ξ \nu_{i}=\frac{\Delta N_{i}}{\Delta\xi}\,
  20. Δ N i = ν i Δ ξ \Delta N_{i}=\nu_{i}\Delta\xi\,
  21. A B A\rightarrow B
  22. ν i k = N i ξ k \nu_{ik}=\frac{\partial N_{i}}{\partial\xi_{k}}\,
  23. d N i = k ν i k d ξ k . dN_{i}=\sum_{k}\nu_{ik}d\xi_{k}.\,
  24. 𝐍 \mathbf{N}
  25. n \mathit{n}
  26. m \mathit{m}
  27. m \mathit{m}
  28. n \mathit{n}
  29. 𝐍 = [ - 1 0 0 0 1 1 0 0 0 - 1 - 1 0 0 0 1 - 1 0 0 0 1 ] \mathbf{N}=\begin{bmatrix}-1&0&0&0\\ 1&1&0&0\\ 0&-1&-1&0\\ 0&0&1&-1\\ 0&0&0&1\\ \end{bmatrix}
  30. 𝐯 \mathbf{v}
  31. 𝐒 \mathbf{S}
  32. d 𝐒 d t = 𝐍 𝐯 . \frac{d\mathbf{S}}{dt}=\mathbf{N}\cdot\mathbf{v}.
  33. 100 g NH 3 1 mol NH 3 17.034 g NH 3 = 5.871 mol NH 3 100\,\mathrm{g\,NH_{3}}\cdot\frac{1\,\mathrm{mol\,NH_{3}}}{17.034\,\mathrm{g\,% NH_{3}}}=5.871\,\mathrm{mol\,NH_{3}}
  34. P V \displaystyle PV
  35. ρ = m V \rho=\frac{m}{V}
  36. n = m M n=\frac{m}{M}
  37. ρ = M P R T \rho=\frac{MP}{R\,T}

Stokes'_theorem.html

  1. Ω ω = Ω d ω . \int_{\partial\Omega}\omega=\int_{\Omega}\mathrm{d}\omega.
  2. Σ × 𝐅 d 𝚺 = Σ 𝐅 d 𝐫 . \iint_{\Sigma}\nabla\times\mathbf{F}\cdot\mathrm{d}\mathbf{\Sigma}=\oint_{% \partial\Sigma}\mathbf{F}\cdot\mathrm{d}\mathbf{r}.
  3. a b f ( x ) d x = F ( b ) - F ( a ) . \int_{a}^{b}f(x)\,\mathrm{d}x=F(b)-F(a).
  4. d F d x = f ( x ) \tfrac{\mathrm{d}F}{\mathrm{d}x}=f(x)
  5. [ a , b ] f ( x ) d x = [ a , b ] d F = { a } - { b } + F = F ( b ) - F ( a ) . \int_{[a,b]}f(x)\,\mathrm{d}x=\int_{[a,b]}\mathrm{d}F=\int_{\{a\}^{-}\cup\{b\}% ^{+}}F=F(b)-F(a).
  6. Ω α = ϕ ( U ) ( ϕ - 1 ) * α , \int_{\Omega}\alpha=\int_{\phi(U)}\left(\phi^{-1}\right)^{*}\alpha,
  7. Ω α i U i ψ i α , \int_{\Omega}\alpha\equiv\sum_{i}\int_{U_{i}}\psi_{i}\alpha,
  8. Ω d ω = Ω ω ( = Ω ω ) . \int_{\Omega}\mathrm{d}\omega=\int_{\partial\Omega}\omega\ \ \left(=\oint_{% \partial\Omega}\omega\right).
  9. I ( ω ) ( c ) = c ω I(\omega)(c)=\oint_{c}\omega
  10. c M c \partial\,\sum\nolimits_{c}M_{c}
  11. c M c = \sum\nolimits_{c}\partial M_{c}=\varnothing
  12. c i ω = a i , \oint_{c_{i}}\omega=a_{i},
  13. Σ × 𝐅 d 𝚺 = Σ 𝐅 d 𝐫 , \int_{\Sigma}\nabla\times\mathbf{F}\cdot\mathrm{d}\mathbf{\Sigma}=\oint_{% \partial\Sigma}\mathbf{F}\cdot\mathrm{d}\mathbf{r},
  14. Σ { ( R y - Q z ) d y d z + ( P z - R x ) d z d x + ( Q x - P y ) d x d y } \iint_{\Sigma}\left\{\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{% \partial z}\right)\mathrm{d}y\mathrm{d}z+\left(\frac{\partial P}{\partial z}-% \frac{\partial R}{\partial x}\right)\mathrm{d}z\mathrm{d}x+\left(\frac{% \partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\mathrm{d}x\mathrm% {d}y\right\}
  15. = Σ { P d x + Q d y + R d z } , =\oint_{\partial\Sigma}\left\{P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z\right\},
  16. Σ [ g ( × 𝐅 ) + ( g ) × 𝐅 ] d 𝚺 = Σ g 𝐅 d 𝐫 , \int_{\Sigma}\left[g\left(\nabla\times\mathbf{F}\right)+\left(\nabla g\right)% \times\mathbf{F}\right]\cdot\mathrm{d}\mathbf{\Sigma}=\oint_{\partial\Sigma}g% \mathbf{F}\cdot\mathrm{d}\mathbf{r},
  17. Σ [ 𝐅 ( 𝐆 ) - 𝐆 ( 𝐅 ) + ( 𝐆 ) 𝐅 - ( 𝐅 ) 𝐆 ] d 𝚺 = Σ ( 𝐅 × 𝐆 ) d 𝐫 . \int_{\Sigma}\left[\mathbf{F}\left(\nabla\cdot\mathbf{G}\right)-\mathbf{G}% \left(\nabla\cdot\mathbf{F}\right)+\left(\mathbf{G}\cdot\nabla\right)\mathbf{F% }-\left(\mathbf{F}\cdot\nabla\right)\mathbf{G}\right]\cdot\mathrm{d}\mathbf{% \Sigma}=\oint_{\partial\Sigma}\left(\mathbf{F}\times\mathbf{G}\right)\cdot% \mathrm{d}\mathbf{r}.
  18. Σ ( 𝐅 d 𝚺 ) - ( 𝐅 ) d 𝚺 = Σ d 𝐫 × 𝐅 . \int_{\Sigma}\nabla(\mathbf{F}\cdot\mathrm{d}\mathbf{\Sigma})-(\nabla\cdot% \mathbf{F})\mathrm{d}\mathbf{\Sigma}=\oint_{\partial\Sigma}\mathrm{d}\mathbf{r% }\times\mathbf{F}.
  19. t d d t \scriptstyle\int\frac{\partial}{\partial t}...\to\frac{\mathrm{d}}{\mathrm{d}t% }\int...
  20. × 𝐄 = - 𝐁 t \ \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  21. C 𝐄 d 𝐥 = S × 𝐄 d 𝐀 = - S 𝐁 t d 𝐀 \ \oint_{C}\mathbf{E}\cdot\mathrm{d}\mathbf{l}=\iint_{S}\nabla\times\mathbf{E}% \cdot\mathrm{d}\mathbf{A}=-\,\iint_{S}\frac{\partial\mathbf{B}}{\partial t}% \cdot\mathrm{d}\mathbf{A}
  22. × 𝐇 = 𝐉 + 𝐃 t \ \ \nabla\times\mathbf{H}=\mathbf{J}+\frac{\partial\mathbf{D}}{\partial t}\
  23. C 𝐇 d 𝐥 = S × 𝐇 d 𝐀 = S 𝐉 d 𝐀 + S 𝐃 t d 𝐀 \,\ \begin{aligned}\displaystyle\oint_{C}\mathbf{H}\cdot\mathrm{d}\mathbf{l}&% \displaystyle=\iint_{S}\nabla\times\mathbf{H}\cdot\mathrm{d}\mathbf{A}\\ &\displaystyle=\iint_{S}\mathbf{J}\cdot\mathrm{d}\mathbf{A}+\iint_{S}\frac{% \partial\mathbf{D}}{\partial t}\cdot\mathrm{d}\mathbf{A}\end{aligned}\,
  24. × 𝐄 = - 1 c 𝐁 t , \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t},
  25. × 𝐇 = 1 c 𝐃 t + 4 π c 𝐉 , \nabla\times\mathbf{H}=\frac{1}{c}\frac{\partial\mathbf{D}}{\partial t}+\frac{% 4\pi}{c}\mathbf{J},
  26. Vol 𝐅 d Vol = Vol 𝐅 d 𝚺 \int_{\mathrm{Vol}}\nabla\cdot\mathbf{F}\ \mathrm{d}_{\mathrm{Vol}}=\oint_{% \partial\mathrm{Vol}}\mathbf{F}\cdot\mathrm{d}\mathbf{\Sigma}
  27. 𝐅 = f 𝐜 \mathbf{F}=f\mathbf{c}
  28. 𝐜 \mathbf{c}
  29. 𝐜 Vol f d Vol = 𝐜 Vol f d 𝚺 \mathbf{c}\cdot\int_{\mathrm{Vol}}\nabla f\ \mathrm{d}_{\mathrm{Vol}}=\mathbf{% c}\cdot\oint_{\partial\mathrm{Vol}}f\mathrm{d}\mathbf{\Sigma}
  30. c , \vec{c},
  31. Vol f d Vol = Vol f d 𝚺 \int_{\mathrm{Vol}}\nabla f\ \mathrm{d}_{\mathrm{Vol}}=\oint_{\partial\mathrm{% Vol}}f\mathrm{d}\mathbf{\Sigma}
  32. W { d t o t a l U } \oint_{W}\,\{d_{\,total\,}U\}
  33. α 1 := T \alpha_{1}:=T
  34. α 2 := V , \alpha_{2}:=V,
  35. α 3 := P \alpha_{3}:=P
  36. { d t o t a l U } = i = 1 3 U α i d α i , \{d_{\,total\,}U\}=\sum_{i=1}^{3}\,\frac{\partial U}{\partial\alpha_{i}}% \mathrm{d}\alpha_{i},
  37. W { d t o t a l U } = ! 0. \oint_{W}\,\{d_{\,total\,}U\}\,\stackrel{!}{=}\,0.

Stone–Čech_compactification.html

  1. C \prod C
  2. X [ 0 , 1 ] C X\to[0,1]^{C}
  3. x ( f ( x ) ) f C x\mapsto(f(x))_{f\in C}
  4. { F : U F } \{F:U\in F\}
  5. 1 \aleph_{1}
  6. ( 𝐍 ) C ( β 𝐍 ) \ell^{\infty}(\mathbf{N})\to C(\beta\mathbf{N})
  7. A - n = { k 𝐍 k + n A } . A-n=\{k\in\mathbf{N}\mid k+n\in A\}.
  8. F + G = { A 𝐍 { n 𝐍 A - n F } G } ; F+G=\Big\{A\subset\mathbf{N}\mid\{n\in\mathbf{N}\mid A-n\in F\}\in G\Big\};
  9. β 𝐍 β 𝐍 \beta\mathbf{N}\to\beta\mathbf{N}
  10. G F + G G\mapsto F+G

Stone–Weierstrass_theorem.html

  1. a a , b aa,b
  2. a a , b aa,b
  3. X X
  4. C ( X ) C(X)
  5. f f
  6. a a , b aa,b
  7. ε > 0 ε>0
  8. p ( x ) p(x)
  9. x x
  10. a a , b aa,b
  11. C a a , b Caa,b
  12. C a a , b Caa,b
  13. C a a , b Caa,b
  14. C a a , b Caa,b
  15. a a , b aa,b
  16. [ u ! ! ] f g [ u ! ! ] [ u ! ! ] f [ u ! ! ] · [ u ! ! ] g [ u ! ! ] [u^{\prime}!!^{\prime}]fg[u^{\prime}!!^{\prime}]≤[u^{\prime}!!^{\prime}]f[u^{% \prime}!!^{\prime}]·[u^{\prime}!!^{\prime}]g[u^{\prime}!!^{\prime}]
  17. f , g f,g
  18. C a a , b Caa,b
  19. C a a , b Caa,b
  20. C a a , b Caa,b
  21. X X
  22. C ( X , 𝐑 ) C(X,\mathbf{R})
  23. X X
  24. C ( X , 𝐑 ) C(X,\mathbf{R})
  25. A A
  26. X X
  27. x x
  28. y y
  29. X X
  30. p p
  31. A A
  32. p ( x ) p ( y ) p(x)≠p(y)
  33. X X
  34. A A
  35. C ( X , 𝐑 ) C(X,\mathbf{R})
  36. A A
  37. C ( X , 𝐑 ) C(X,\mathbf{R})
  38. a a , b aa,b
  39. C a a , b Caa,b
  40. X X
  41. X X
  42. f f
  43. ε > 0 ε>0
  44. K X K⊂X
  45. X K X\ K
  46. A A
  47. A A
  48. x x
  49. X X
  50. f f
  51. A A
  52. f ( x ) 0 f(x)≠0
  53. X X
  54. A A
  55. A A
  56. X X
  57. f f
  58. a a , b × c c , d aa,b×cc,d
  59. ε > 0 ε>0
  60. p p
  61. x x
  62. a a , b aa,b
  63. y y
  64. c c , d cc,d
  65. X X
  66. Y Y
  67. f : X × Y 𝐑 f:X×Y→\mathbf{R}
  68. ε > 0 ε>0
  69. n > 0 n>0
  70. X X
  71. Y Y
  72. 0 , 11 0,11
  73. 0 , 11 0,11
  74. C ( X , 𝐂 ) C(X,\mathbf{C})
  75. X X
  76. X X
  77. S S
  78. C ( X , 𝐂 ) C(X,\mathbf{C})
  79. S S
  80. C ( X , 𝐂 ) C(X,\mathbf{C})
  81. S S
  82. S S
  83. 1 1
  84. f f
  85. f f
  86. C ( X , 𝐇 ) C(X,\mathbf{H})
  87. X X
  88. X X
  89. A A
  90. C ( X , 𝐇 ) C(X,\mathbf{H})
  91. A A
  92. C ( X , 𝐇 ) C(X,\mathbf{H})
  93. 𝔄 \mathfrak{A}
  94. 𝔄 \mathfrak{A}
  95. 𝔄 \mathfrak{A}
  96. 𝔅 \mathfrak{B}
  97. 𝔄 \mathfrak{A}
  98. 𝔄 = 𝔅 \mathfrak{A}=\mathfrak{B}
  99. 𝔄 \mathfrak{A}
  100. 𝔅 \mathfrak{B}
  101. 𝔄 \mathfrak{A}
  102. 𝔄 = 𝔅 \mathfrak{A}=\mathfrak{B}
  103. X X
  104. C ( X , 𝐑 ) C(X,\mathbf{R})
  105. L L
  106. C ( X , 𝐑 ) C(X,\mathbf{R})
  107. f , g L f,g∈L
  108. L L
  109. X X
  110. L L
  111. C ( X , 𝐑 ) C(X,\mathbf{R})
  112. x x
  113. y y
  114. X X
  115. a a
  116. b b
  117. f L f∈L
  118. f ( x ) = a f(x)=a
  119. f ( y ) = b f(y)=b
  120. L L
  121. C ( X , 𝐑 ) C(X,\mathbf{R})
  122. | f | |f|
  123. f f
  124. C ( X , 𝐑 ) C(X,\mathbf{R})
  125. X X
  126. B B
  127. C ( X , 𝐑 ) C(X,\mathbf{R})
  128. B B
  129. B B
  130. f B f∈B
  131. a f B af∈B
  132. a 𝐑 a∈\mathbf{R}
  133. f , g B f,g∈B
  134. B B
  135. C ( X , 𝐑 ) C(X,\mathbf{R})
  136. X X
  137. L L
  138. C ( X , 𝐑 ) C(X,\mathbf{R})
  139. φ C ( X , 𝐑 ) φ∈C(X,\mathbf{R})
  140. L L
  141. X X
  142. ε > 0 ε>0
  143. f L f∈L
  144. | f ( x ) φ ( x ) | < ε |f(x)−φ(x)|<ε

Stratified_sampling.html

  1. μ s = 1 N h = 1 L N J μ h \mu_{s}=\frac{1}{N}\sum_{h=1}^{L}N_{J}\mu_{h}
  2. σ s = h = 1 L ( N h N ) 2 ( N h - n h N h ) σ h 2 n h \sigma_{s}=\sum_{h=1}^{L}\left(\frac{N_{h}}{N}\right)^{2}\left(\frac{N_{h}-n_{% h}}{N_{h}}\right)\frac{\sigma_{h}^{2}}{n_{h}}
  3. N = N=
  4. N h = N_{h}=
  5. n h = n_{h}=
  6. L = L=
  7. σ h = \sigma_{h}=
  8. h h
  9. μ h = \mu_{h}=
  10. h h

String_(computer_science).html

  1. Σ * = n { 0 } Σ n \Sigma^{*}=\bigcup_{n\in\mathbb{N}\cup\{0\}}\Sigma^{n}
  2. L : Σ * { 0 } L:\Sigma^{*}\mapsto\mathbb{N}\cup\{0\}
  3. L ( s t ) = L ( s ) + L ( t ) s , t Σ * L(st)=L(s)+L(t)\quad\forall s,t\in\Sigma^{*}

String_instrument.html

  1. f 1 l f\propto\frac{1}{l}
  2. f T f\propto\sqrt{T}
  3. f 1 μ f\propto{1\over\sqrt{\mu}}

String_searching_algorithm.html

  1. Θ ( n ) \Theta(n)
  2. z z
  3. O ( m ) O(m)

String_theory.html

  1. S O ( 32 ) SO(32)
  2. S O ( 32 ) SO(32)
  3. R R
  4. 1 / R 1/R
  5. p p
  6. n n
  7. n n
  8. p p
  9. S S
  10. S = c 3 k A 4 G S=\frac{c^{3}kA}{4\hbar G}
  11. c c
  12. k k
  13. ħ ħ
  14. G G
  15. A A
  16. 1 / 4 1/4
  17. 10 < s u p > - 11 10<sup>-11

Strouhal_number.html

  1. St = f L U , \mathrm{St}={fL\over U},
  2. St = k a π c , \mathrm{St}={ka\over\pi c},
  3. St = f U C 3 \mathrm{St}={f\over U}{C^{3}}

Structural_engineering.html

  1. K * l K*l
  2. l l

Subgroup.html

  1. / 8 \mathbb{Z}/8\mathbb{Z}
  2. / 2 \mathbb{Z}/2\mathbb{Z}
  3. [ G : H ] = | G | | H | [G:H]={|G|\over|H|}
  4. G = { 0 , 2 , 4 , 6 , 1 , 3 , 5 , 7 } G=\left\{0,2,4,6,1,3,5,7\right\}

Subset.html

  1. A B A\subseteq B
  2. B A . B\supseteq A.
  3. A B . A\subsetneq B.
  4. B A . B\supsetneq A.
  5. 𝒫 ( S ) \mathcal{P}(S)
  6. A B A⊆B
  7. \preceq
  8. 𝒫 ( S ) \mathcal{P}(S)

Subset_sum_problem.html

  1. O ( s N ) O(sN)
  2. s s
  3. N N
  4. O ~ ( s N ) \tilde{O}(s\sqrt{N})
  5. s s

Substance_theory.html

  1. a a
  2. b b
  3. b b
  4. a a
  5. b b
  6. a a
  7. b b
  8. c c
  9. c c
  10. a a
  11. c c
  12. b b
  13. a a
  14. b b
  15. a a
  16. b b
  17. a a
  18. b b
  19. a a
  20. b b

Substitution_cipher.html

  1. 26 2 = 676 26^{2}=676
  2. 26 n \mathbb{Z}_{26}^{n}

Subwoofer.html

  1. η 0 = ( 4 π 2 F s 3 V a s c 3 Q e s ) × 100 % \eta_{0}=\left(\frac{4\pi^{2}F_{s}^{3}V_{as}}{c^{3}Q_{es}}\right)\times 100\%
  2. V b = V a s α V_{b}=\frac{V_{as}}{\alpha}
  3. α = Q t c 2 Q t s 2 - 1 \alpha=\frac{Q_{tc}^{2}}{Q_{ts}^{2}}-1
  4. F c = ( Q t c F s ) Q t s F_{c}=\frac{(Q_{tc}F_{s})}{Q_{ts}}
  5. V d = x max × S d V_{\mathrm{d}}=x_{\mathrm{max}}\times S_{\mathrm{d}}

Sunlight.html

  1. E ext = E sc ( 1 + 0.033412 cos ( 2 π dn - 3 365 ) ) , E_{\rm ext}=E_{\rm sc}\cdot\left(1+0.033412\cdot\cos\left(2\pi\frac{{\rm dn}-3% }{365}\right)\right),
  2. E dn = E ext e - c m , E_{\rm dn}=E_{\rm ext}\,e^{-cm},
  3. c c
  4. m m
  5. 2 A r 2 d t = d θ , \tfrac{2A}{r^{2}}dt=d\theta,
  6. A A
  7. 0 T 2 A r 2 d t = 0 2 π d θ = constant . \int_{0}^{T}\tfrac{2A}{r^{2}}dt=\int_{0}^{2\pi}d\theta=\mathrm{constant}.
  8. P P

Super-Kamiokande.html

  1. R mean {R\text{mean}}
  2. R mean = i = 1 N multi - 1 j = i + 1 N multi | r i - r j | N nulti C 2 {R\text{mean}}=\frac{\sum_{i=1}^{{N\text{multi}}-1}\sum_{j=i+1}^{{N\text{multi% }}}|{r\text{i}}-{r\text{j}}|}{{N\text{nulti}}{C\text{2}}}
  3. R mean {R\text{mean}}
  4. R mean {R\text{mean}}
  5. R mean {R\text{mean}}
  6. N multi {N\text{multi}}
  7. R mean {R\text{mean}}
  8. N multi {N\text{multi}}
  9. v x + e - v x + e - {v\text{x}}+{e\text{-}}\to{v\text{x}}+{e\text{-}}
  10. δ θ 30 N \delta\theta\sim{30^{\circ}\over\sqrt{N}}
  11. 2 e - + 4 p H 2 e + 2 υ e + 26.73 M e V 2e^{-}+4p\to{}^{2}\!He+2\upsilon_{e}+26.73MeV
  12. v x + e - v x + e - v_{x}+e^{-}\to v_{x}+e^{-}
  13. v e v_{e}
  14. v e v_{e}
  15. c o s θ S u n cos\theta_{Sun}
  16. θ S u n \theta_{Sun}
  17. B 8 {}^{8}\!B
  18. 2.40 ± 0.03 ( s t a t . ) ( s y s . ) - 0.07 + 0.08 × 10 6 c m - 2 s - 1 2.40\pm 0.03(stat.){}_{-0.07}^{+0.08}\!(sys.)\times 10^{6}cm^{-2}s^{-1}
  19. D a t a S S M B P 98 = 0.465 ± 0.005 ( s t a t . ) ( s y s . ) - 0.013 + 0.015 {Data\over SSM_{BP98}}=0.465\pm 0.005(stat.){}_{-0.013}^{+0.015}\!(sys.)

Superconducting_magnetic_energy_storage.html

  1. E = 1 2 L I 2 E=\frac{1}{2}LI^{2}
  2. E = 1 2 R N 2 I 2 f ( ξ , δ ) E=\frac{1}{2}RN^{2}I^{2}f\left(\xi,\delta\right)

Superconductivity.html

  1. 2 𝐇 = λ - 2 𝐇 \nabla^{2}\mathbf{H}=\lambda^{-2}\mathbf{H}\,
  2. 𝐣 s t = n s e 2 m 𝐄 , × 𝐣 s = - n s e 2 m 𝐁 . \frac{\partial\mathbf{j}_{s}}{\partial t}=\frac{n_{s}e^{2}}{m}\mathbf{E},% \qquad\mathbf{\nabla}\times\mathbf{j}_{s}=-\frac{n_{s}e^{2}}{m}\mathbf{B}.

Superellipse.html

  1. | x a | n + | y b | n = 1 , \left|\frac{x}{a}\right|^{n}\!+\left|\frac{y}{b}\right|^{n}\!=1,
  2. s = 2 - 1 / n s=2^{-1/n}
  3. x ( θ ) = \plusmn a cos 2 n θ y ( θ ) = \plusmn b sin 2 n θ } 0 θ < π 2 \left.\begin{aligned}\displaystyle x\left(\theta\right)&\displaystyle=\plusmn a% \cos^{\frac{2}{n}}\theta\\ \displaystyle y\left(\theta\right)&\displaystyle=\plusmn b\sin^{\frac{2}{n}}% \theta\end{aligned}\right\}\qquad 0\leq\theta<\frac{\pi}{2}
  4. x ( θ ) = | cos θ | 2 n a sgn ( cos θ ) y ( θ ) = | sin θ | 2 n b sgn ( sin θ ) . \begin{aligned}\displaystyle x\left(\theta\right)&\displaystyle={|\cos\theta|}% ^{\frac{2}{n}}\cdot a\operatorname{sgn}(\cos\theta)\\ \displaystyle y\left(\theta\right)&\displaystyle={|\sin\theta|}^{\frac{2}{n}}% \cdot b\operatorname{sgn}(\sin\theta).\end{aligned}
  5. Area = 4 a b ( Γ ( 1 + 1 n ) ) 2 Γ ( 1 + 2 n ) . \mathrm{Area}=4ab\frac{\left(\Gamma\left(1+\tfrac{1}{n}\right)\right)^{2}}{% \Gamma\left(1+\tfrac{2}{n}\right)}.
  6. | x a | n + | y b | n = 1 , \left|\frac{x}{a}\right|^{n}\!+\left|\frac{y}{b}\right|^{n}\!=1,
  7. ( a cos θ ) n n - 1 + ( b sin θ ) n n - 1 = r n n - 1 . (a\cos\theta)^{\tfrac{n}{n-1}}+(b\sin\theta)^{\tfrac{n}{n-1}}=r^{\tfrac{n}{n-1% }}.
  8. | x a | m + | y b | n = 1 ; m , n > 0. \left|\frac{x}{a}\right|^{m}+\left|\frac{y}{b}\right|^{n}=1;\qquad m,n>0.
  9. x ( θ ) = | cos θ | 2 m a sgn ( cos θ ) y ( θ ) = | sin θ | 2 n b sgn ( sin θ ) . \begin{aligned}\displaystyle x\left(\theta\right)&\displaystyle={|\cos\theta|}% ^{\frac{2}{m}}\cdot a\operatorname{sgn}(\cos\theta)\\ \displaystyle y\left(\theta\right)&\displaystyle={|\sin\theta|}^{\frac{2}{n}}% \cdot b\operatorname{sgn}(\sin\theta).\end{aligned}
  10. θ \theta

Superfluid_helium-4.html

  1. F = M 4 d v s d t . \vec{F}=M_{4}\frac{\mathrm{d}\vec{v}_{s}}{\mathrm{d}t}.
  2. v s \vec{v}_{s}
  3. F = - ( μ + M 4 g z ) . \vec{F}=-\vec{\nabla}(\mu+M_{4}gz).
  4. M 4 d v s d t = - ( μ + M 4 g z ) . M_{4}\frac{\mathrm{d}\vec{v}_{s}}{\mathrm{d}t}=-\vec{\nabla}(\mu+M_{4}gz).
  5. d μ = V m d p - S m d T . \mathrm{d}\mu=V_{m}\mathrm{d}p-S_{m}\mathrm{d}T.
  6. μ ( p , T ) = μ ( 0 , 0 ) + 0 p V m ( p , 0 ) d p - 0 T S m ( p , T ) d T . \mu(p,T)=\mu(0,0)+\int_{0}^{p}V_{m}(p^{\prime},0)\mathrm{d}p^{\prime}-\int_{0}% ^{T}S_{m}(p,T^{\prime})\mathrm{d}T^{\prime}.
  7. 0 p V m ( p , 0 ) d p = V m 0 p \int_{0}^{p}V_{m}(p^{\prime},0)\mathrm{d}p^{\prime}=V_{m0}p
  8. 0 T S m ( p , T ) d T = V m 0 p f . \int_{0}^{T}S_{m}(p,T^{\prime})\mathrm{d}T^{\prime}=V_{m0}p_{f}.
  9. μ ( p , T ) = μ 0 + V m 0 ( p - p f ) . \mu(p,T)=\mu_{0}+V_{m0}(p-p_{f}).
  10. ρ 0 d v s d t = - ( p + ρ 0 g z - p f ) . \rho_{0}\frac{\mathrm{d}\vec{v}_{s}}{\mathrm{d}t}=-\vec{\nabla}(p+\rho_{0}gz-p% _{f}).
  11. p l + ρ 0 g z l - p f l = p r + ρ 0 g z r - p f r p_{l}+\rho_{0}gz_{l}-p_{fl}=p_{r}+\rho_{0}gz_{r}-p_{fr}
  12. 0 = p r - p f r . 0=p_{r}-p_{fr}.
  13. V ˙ n \dot{V}_{n}

Superheterodyne_receiver.html

  1. f img = { f + 2 f IF , if f LO > f (high side injection) f - 2 f IF , if f LO < f (low side injection) f_{\mathrm{img}}=\begin{cases}f+2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}% >f\mbox{ (high side injection)}\\ f-2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}<f\mbox{ (low side injection)}% \end{cases}
  2. f o f_{o}\!
  3. 2 f IF 2f_{\mathrm{IF}}\!

Superparamagnetism.html

  1. τ N \tau_{N}
  2. τ N = τ 0 exp ( K V k B T ) \tau_{N}=\tau_{0}\exp\left(\frac{KV}{k_{B}T}\right)
  3. τ N \tau_{N}
  4. τ 0 \tau_{0}
  5. τ m \tau_{m}
  6. τ m τ N \tau_{m}\gg\tau_{N}
  7. τ m τ N \tau_{m}\ll\tau_{N}
  8. τ m = τ N \tau_{m}=\tau_{N}
  9. τ m = τ N \tau_{m}=\tau_{N}
  10. T B = K V k B ln ( τ m τ 0 ) T_{B}=\frac{KV}{k_{B}\ln\left(\frac{\tau_{m}}{\tau_{0}}\right)}
  11. tanh ( x / 3 ) \tanh(x/3)
  12. M ( H ) n μ tanh ( μ 0 H μ k B T ) M(H)\approx n\mu\tanh\left(\frac{\mu_{0}H\mu}{k_{B}T}\right)
  13. M ( H ) n μ L ( μ 0 H μ k B T ) M(H)\approx n\mu L\left(\frac{\mu_{0}H\mu}{k_{B}T}\right)
  14. μ 0 \mu_{0}
  15. μ \mu
  16. L ( x ) = 1 / tanh ( x ) - 1 / x L(x)=1/\tanh(x)-1/x
  17. M ( H ) M(H)
  18. χ \chi
  19. χ = n μ 0 μ 2 k B T \chi=\frac{n\mu_{0}\mu^{2}}{k_{B}T}
  20. χ = n μ 0 μ 2 3 k B T \chi=\frac{n\mu_{0}\mu^{2}}{3k_{B}T}
  21. T > T B T>T_{B}
  22. T T B T\ll T_{B}
  23. T T B T\gg T_{B}
  24. T B T_{B}
  25. χ ( ω ) = χ s p + i ω τ χ b 1 + i ω τ \chi(\omega)=\frac{\chi_{sp}+i\omega\tau\chi_{b}}{1+i\omega\tau}
  26. ω 2 π \tfrac{\omega}{2\pi}
  27. χ s p = n μ 0 μ 2 3 k B T \chi_{sp}=\tfrac{n\mu_{0}\mu^{2}}{3k_{B}T}
  28. χ b = n μ 0 μ 2 3 K V \chi_{b}=\tfrac{n\mu_{0}\mu^{2}}{3KV}
  29. τ = τ N 2 \tau=\tfrac{\tau_{N}}{2}
  30. τ d M d t + M = τ χ b d H d t + χ s p H \tau\frac{\mathrm{d}M}{\mathrm{d}t}+M=\tau\chi_{b}\frac{\mathrm{d}H}{\mathrm{d% }t}+\chi_{sp}H

Supersaturation.html

  1. P V = n R T PV=nRT

Superscalar.html

  1. n k n^{k}
  2. k 2 log n k^{2}\log n
  3. n n
  4. k k

Supervised_learning.html

  1. x x
  2. x x
  3. x x
  4. N N
  5. { ( x 1 , y 1 ) , , ( x N , y N ) } \{(x_{1},y_{1}),...,(x_{N},\;y_{N})\}
  6. x i x_{i}
  7. y i y_{i}
  8. g : X Y g:X\to Y
  9. X X
  10. Y Y
  11. g g
  12. G G
  13. g g
  14. f : X × Y f:X\times Y\to\mathbb{R}
  15. g g
  16. y y
  17. g ( x ) = arg max y f ( x , y ) g(x)=\arg\max_{y}\;f(x,y)
  18. F F
  19. G G
  20. F F
  21. g g
  22. g ( x ) = P ( y | x ) g(x)=P(y|x)
  23. f f
  24. f ( x , y ) = P ( x , y ) f(x,y)=P(x,y)
  25. f f
  26. g g
  27. ( x i , y i ) (x_{i},\;y_{i})
  28. L : Y × Y 0 L:Y\times Y\to\mathbb{R}^{\geq 0}
  29. ( x i , y i ) (x_{i},\;y_{i})
  30. y ^ \hat{y}
  31. L ( y i , y ^ ) L(y_{i},\hat{y})
  32. R ( g ) R(g)
  33. g g
  34. g g
  35. R e m p ( g ) = 1 N i L ( y i , g ( x i ) ) R_{emp}(g)=\frac{1}{N}\sum_{i}L(y_{i},g(x_{i}))
  36. g g
  37. R ( g ) R(g)
  38. g g
  39. g g
  40. P ( y | x ) P(y|x)
  41. L ( y , y ^ ) = - log P ( y | x ) L(y,\hat{y})=-\log P(y|x)
  42. G G
  43. g g
  44. g ( x ) = j = 1 d β j x j g(x)=\sum_{j=1}^{d}\beta_{j}x_{j}
  45. j β j 2 \sum_{j}\beta_{j}^{2}
  46. L 2 L_{2}
  47. L 1 L_{1}
  48. j | β j | \sum_{j}|\beta_{j}|
  49. L 0 L_{0}
  50. β j \beta_{j}
  51. C ( g ) C(g)
  52. g g
  53. J ( g ) = R e m p ( g ) + λ C ( g ) . J(g)=R_{emp}(g)+\lambda C(g).
  54. λ \lambda
  55. λ = 0 \lambda=0
  56. λ \lambda
  57. λ \lambda
  58. g g
  59. - log P ( g ) -\log P(g)
  60. J ( g ) J(g)
  61. g g
  62. g g
  63. f ( x , y ) = P ( x , y ) f(x,y)=P(x,y)
  64. - i log P ( x i , y i ) , -\sum_{i}\log P(x_{i},y_{i}),
  65. f f

Support_vector_machine.html

  1. k ( x , y ) k(x,y)
  2. α i \alpha_{i}
  3. x i x_{i}
  4. x x
  5. i α i k ( x i , x ) = constant . \textstyle\sum_{i}\alpha_{i}k(x_{i},x)=\mathrm{constant}.
  6. k ( x , y ) k(x,y)
  7. y y
  8. x x
  9. x x
  10. x i x_{i}
  11. x x
  12. p p
  13. p p
  14. ( p - 1 ) (p-1)
  15. 𝒟 \mathcal{D}
  16. n n
  17. 𝒟 = { ( 𝐱 i , y i ) 𝐱 i p , y i { - 1 , 1 } } i = 1 n \mathcal{D}=\left\{(\mathbf{x}_{i},y_{i})\mid\mathbf{x}_{i}\in\mathbb{R}^{p},% \,y_{i}\in\{-1,1\}\right\}_{i=1}^{n}
  18. y i y_{i}
  19. 𝐱 i \mathbf{x}_{i}
  20. 𝐱 i \mathbf{x}_{i}
  21. p p
  22. y i = 1 y_{i}=1
  23. y i = - 1 y_{i}=-1
  24. 𝐱 \mathbf{x}
  25. 𝐰 𝐱 - b = 0 , \mathbf{w}\cdot\mathbf{x}-b=0,\,
  26. \cdot
  27. 𝐰 {\mathbf{w}}
  28. b 𝐰 \tfrac{b}{\|\mathbf{w}\|}
  29. 𝐰 {\mathbf{w}}
  30. 𝐰 𝐱 - b = 1 \mathbf{w}\cdot\mathbf{x}-b=1\,
  31. 𝐰 𝐱 - b = - 1. \mathbf{w}\cdot\mathbf{x}-b=-1.\,
  32. 2 𝐰 \tfrac{2}{\|\mathbf{w}\|}
  33. 𝐰 \|\mathbf{w}\|
  34. i i
  35. 𝐰 𝐱 i - b 1 for 𝐱 i \mathbf{w}\cdot\mathbf{x}_{i}-b\geq 1\qquad\,\text{ for }\mathbf{x}_{i}
  36. 𝐰 𝐱 i - b - 1 for 𝐱 i \mathbf{w}\cdot\mathbf{x}_{i}-b\leq-1\qquad\,\text{ for }\mathbf{x}_{i}
  37. y i ( 𝐰 𝐱 i - b ) 1 , for all 1 i n . ( 1 ) y_{i}(\mathbf{w}\cdot\mathbf{x}_{i}-b)\geq 1,\quad\,\text{ for all }1\leq i% \leq n.\qquad\qquad(1)
  38. 𝐰 , b {\mathbf{w},b}
  39. 𝐰 \|\mathbf{w}\|
  40. i = 1 , , n i=1,\dots,n
  41. y i ( 𝐰 𝐱 𝐢 - b ) 1. y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)\geq 1.\,
  42. 𝐰 \|\mathbf{w}\|
  43. 𝐰 \mathbf{w}
  44. 𝐰 \|\mathbf{w}\|
  45. 1 2 𝐰 2 \tfrac{1}{2}\|\mathbf{w}\|^{2}
  46. 1 2 \frac{1}{2}
  47. 𝐰 \mathbf{w}
  48. b b
  49. arg min ( 𝐰 , b ) 1 2 𝐰 2 \arg\min_{(\mathbf{w},b)}\frac{1}{2}\|\mathbf{w}\|^{2}
  50. i = 1 , , n i=1,\dots,n
  51. y i ( 𝐰 𝐱 𝐢 - b ) 1. y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)\geq 1.
  52. s y m b o l α symbol{\alpha}
  53. arg min 𝐰 , b max s y m b o l α 0 { 1 2 𝐰 2 - i = 1 n α i [ y i ( 𝐰 𝐱 𝐢 - b ) - 1 ] } \arg\min_{\mathbf{w},b}\max_{symbol{\alpha}\geq 0}\left\{\frac{1}{2}\|\mathbf{% w}\|^{2}-\sum_{i=1}^{n}{\alpha_{i}[y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)-1]}\right\}
  54. y i ( 𝐰 𝐱 𝐢 - b ) - 1 > 0 y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)-1>0
  55. α i \alpha_{i}
  56. 𝐰 = i = 1 n α i y i 𝐱 𝐢 . \mathbf{w}=\sum_{i=1}^{n}{\alpha_{i}y_{i}\mathbf{x_{i}}}.
  57. α i \alpha_{i}
  58. 𝐱 𝐢 \mathbf{x_{i}}
  59. y i ( 𝐰 𝐱 𝐢 - b ) = 1 y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)=1
  60. 𝐰 𝐱 𝐢 - b = 1 y i = y i b = 𝐰 𝐱 𝐢 - y i \mathbf{w}\cdot\mathbf{x_{i}}-b=\frac{1}{y_{i}}=y_{i}\iff b=\mathbf{w}\cdot% \mathbf{x_{i}}-y_{i}
  61. b b
  62. b b
  63. y i y_{i}
  64. x i x_{i}
  65. N S V N_{SV}
  66. b = 1 N S V i = 1 N S V ( 𝐰 𝐱 𝐢 - y i ) b=\frac{1}{N_{SV}}\sum_{i=1}^{N_{SV}}{(\mathbf{w}\cdot\mathbf{x_{i}}-y_{i})}
  67. 𝐰 2 = 𝐰 T 𝐰 \|\mathbf{w}\|^{2}=\mathbf{w}^{T}\cdot\mathbf{w}
  68. 𝐰 = i = 1 n α i y i 𝐱 𝐢 \mathbf{w}=\sum_{i=1}^{n}{\alpha_{i}y_{i}\mathbf{x_{i}}}
  69. α i \alpha_{i}
  70. L ~ ( α ) = i = 1 n α i - 1 2 i , j α i α j y i y j 𝐱 i T 𝐱 j = i = 1 n α i - 1 2 i , j α i α j y i y j k ( 𝐱 i , 𝐱 j ) \tilde{L}(\mathbf{\alpha})=\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i,j}% \alpha_{i}\alpha_{j}y_{i}y_{j}\mathbf{x}_{i}^{T}\mathbf{x}_{j}=\sum_{i=1}^{n}% \alpha_{i}-\frac{1}{2}\sum_{i,j}\alpha_{i}\alpha_{j}y_{i}y_{j}k(\mathbf{x}_{i}% ,\mathbf{x}_{j})
  71. i = 1 , , n i=1,\dots,n
  72. α i 0 , \alpha_{i}\geq 0,\,
  73. b b
  74. i = 1 n α i y i = 0. \sum_{i=1}^{n}\alpha_{i}y_{i}=0.
  75. k ( 𝐱 i , 𝐱 j ) = 𝐱 i 𝐱 j k(\mathbf{x}_{i},\mathbf{x}_{j})=\mathbf{x}_{i}\cdot\mathbf{x}_{j}
  76. W W
  77. α \alpha
  78. 𝐰 = i α i y i 𝐱 i . \mathbf{w}=\sum_{i}\alpha_{i}y_{i}\mathbf{x}_{i}.
  79. b = 0 b=0
  80. i = 1 n α i y i = 0 \sum_{i=1}^{n}\alpha_{i}y_{i}=0
  81. ξ i \xi_{i}
  82. x i x_{i}
  83. y i ( 𝐰 𝐱 𝐢 - b ) 1 - ξ i 1 i n . ( 2 ) y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)\geq 1-\xi_{i}\quad 1\leq i\leq n.\quad% \quad(2)
  84. ξ i \xi_{i}
  85. arg min 𝐰 , ξ , b { 1 2 𝐰 2 + C i = 1 n ξ i } \arg\min_{\mathbf{w},\mathbf{\xi},b}\left\{\frac{1}{2}\|\mathbf{w}\|^{2}+C\sum% _{i=1}^{n}\xi_{i}\right\}
  86. i = 1 , n i=1,\dots n
  87. y i ( 𝐰 𝐱 𝐢 - b ) 1 - ξ i , ξ i 0 y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)\geq 1-\xi_{i},~{}~{}~{}~{}\xi_{i}\geq 0
  88. i [ 1 - y i ( w x i + b ) ] + + λ w 2 \sum_{i}[1-y_{i}(w\cdot x_{i}+b)]_{+}+\lambda\|w\|^{2}
  89. [ 1 - y i ( w x i + b ) ] + = [ ξ i ] + = ξ i , λ = 1 / 2 C [1-y_{i}(w\cdot x_{i}+b)]_{+}=[\xi_{i}]_{+}=\xi_{i},\quad\lambda=1/2C
  90. 𝐰 \|\mathbf{w}\|
  91. arg min 𝐰 , ξ , b max s y m b o l α , s y m b o l β { 1 2 𝐰 2 + C i = 1 n ξ i - i = 1 n α i [ y i ( 𝐰 𝐱 𝐢 - b ) - 1 + ξ i ] - i = 1 n β i ξ i } \arg\min_{\mathbf{w},\mathbf{\xi},b}\max_{symbol{\alpha},symbol{\beta}}\left\{% \frac{1}{2}\|\mathbf{w}\|^{2}+C\sum_{i=1}^{n}\xi_{i}-\sum_{i=1}^{n}{\alpha_{i}% [y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)-1+\xi_{i}]}-\sum_{i=1}^{n}\beta_{i}\xi% _{i}\right\}
  92. α i , β i 0 \alpha_{i},\beta_{i}\geq 0
  93. α i \alpha_{i}
  94. L ~ ( α ) = i = 1 n α i - 1 2 i , j α i α j y i y j k ( 𝐱 i , 𝐱 j ) \tilde{L}(\mathbf{\alpha})=\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i,j}% \alpha_{i}\alpha_{j}y_{i}y_{j}k(\mathbf{x}_{i},\mathbf{x}_{j})
  95. i = 1 , , n i=1,\dots,n
  96. 0 α i C , 0\leq\alpha_{i}\leq C,\,
  97. i = 1 n α i y i = 0. \sum_{i=1}^{n}\alpha_{i}y_{i}=0.
  98. k ( 𝐱 𝐢 , 𝐱 𝐣 ) = ( 𝐱 𝐢 𝐱 𝐣 ) d k(\mathbf{x_{i}},\mathbf{x_{j}})=(\mathbf{x_{i}}\cdot\mathbf{x_{j}})^{d}
  99. k ( 𝐱 𝐢 , 𝐱 𝐣 ) = ( 𝐱 𝐢 𝐱 𝐣 + 1 ) d k(\mathbf{x_{i}},\mathbf{x_{j}})=(\mathbf{x_{i}}\cdot\mathbf{x_{j}}+1)^{d}
  100. k ( 𝐱 𝐢 , 𝐱 𝐣 ) = exp ( - γ 𝐱 𝐢 - 𝐱 𝐣 2 ) k(\mathbf{x_{i}},\mathbf{x_{j}})=\exp(-\gamma\|\mathbf{x_{i}}-\mathbf{x_{j}}\|% ^{2})
  101. γ > 0 \gamma>0
  102. γ = 1 / 2 σ 2 \gamma=1/{2\sigma^{2}}
  103. k ( 𝐱 𝐢 , 𝐱 𝐣 ) = tanh ( κ 𝐱 𝐢 𝐱 𝐣 + c ) k(\mathbf{x_{i}},\mathbf{x_{j}})=\tanh(\kappa\mathbf{x_{i}}\cdot\mathbf{x_{j}}% +c)
  104. κ > 0 \kappa>0
  105. c < 0 c<0
  106. φ ( 𝐱 𝐢 ) \varphi(\mathbf{x_{i}})
  107. k ( 𝐱 𝐢 , 𝐱 𝐣 ) = φ ( 𝐱 𝐢 ) φ ( 𝐱 𝐣 ) k(\mathbf{x_{i}},\mathbf{x_{j}})=\varphi(\mathbf{x_{i}})\cdot\varphi(\mathbf{x% _{j}})
  108. 𝐰 = i α i y i φ ( 𝐱 i ) \textstyle\mathbf{w}=\sum_{i}\alpha_{i}y_{i}\varphi(\mathbf{x}_{i})
  109. 𝐰 φ ( 𝐱 ) = i α i y i k ( 𝐱 i , 𝐱 ) \textstyle\mathbf{w}\cdot\varphi(\mathbf{x})=\sum_{i}\alpha_{i}y_{i}k(\mathbf{% x}_{i},\mathbf{x})
  110. 𝐰 φ ( 𝐱 ) = k ( 𝐰 , 𝐱 ) \mathbf{w}\cdot\varphi(\mathbf{x})=k(\mathbf{w^{\prime}},\mathbf{x})
  111. γ \gamma
  112. γ \gamma
  113. γ \gamma
  114. C { 2 - 5 , 2 - 3 , , 2 13 , 2 15 } C\in\{2^{-5},2^{-3},\dots,2^{13},2^{15}\}
  115. γ { 2 - 15 , 2 - 13 , , 2 1 , 2 3 } \gamma\in\{2^{-15},2^{-13},\dots,2^{1},2^{3}\}
  116. γ \gamma
  117. 𝒟 \mathcal{D}
  118. 𝒟 = { 𝐱 i | 𝐱 i p } i = 1 k \mathcal{D}^{\star}=\{\mathbf{x}^{\star}_{i}|\mathbf{x}^{\star}_{i}\in\mathbb{% R}^{p}\}_{i=1}^{k}\,
  119. 𝐰 , b , 𝐲 {\mathbf{w},b,\mathbf{y^{\star}}}
  120. 1 2 𝐰 2 \frac{1}{2}\|\mathbf{w}\|^{2}
  121. i = 1 , , n i=1,\dots,n
  122. j = 1 , , k j=1,\dots,k
  123. y i ( 𝐰 𝐱 𝐢 - b ) 1 , y_{i}(\mathbf{w}\cdot\mathbf{x_{i}}-b)\geq 1,\,
  124. y j ( 𝐰 𝐱 𝐣 - b ) 1 , y^{\star}_{j}(\mathbf{w}\cdot\mathbf{x^{\star}_{j}}-b)\geq 1,
  125. y j { - 1 , 1 } . y^{\star}_{j}\in\{-1,1\}.\,
  126. 1 2 w 2 \frac{1}{2}\|w\|^{2}
  127. { y i - w , x i - b ϵ w , x i + b - y i ϵ \begin{cases}y_{i}-\langle w,x_{i}\rangle-b\leq\epsilon\\ \langle w,x_{i}\rangle+b-y_{i}\leq\epsilon\end{cases}
  128. x i x_{i}
  129. y i y_{i}
  130. w , x i + b \langle w,x_{i}\rangle+b
  131. ϵ \epsilon
  132. ϵ \epsilon

Surface.html

  1. A B B - 1 A - 1 ABB^{-1}A^{-1}
  2. A B A B ABAB
  3. A B A - 1 B - 1 ABA^{-1}B^{-1}
  4. A B A B - 1 ABAB^{-1}
  5. χ \chi
  6. χ ( M # N ) = χ ( M ) + χ ( N ) - 2. \chi(M\#N)=\chi(M)+\chi(N)-2.\,
  7. g 1 g\geq 1
  8. k 1 k\geq 1
  9. Σ g , k , \Sigma_{g,k},
  10. S K d A = 2 π χ ( S ) . \int_{S}K\;dA=2\pi\chi(S).

Surface_area.html

  1. r r
  2. 4 π r 2 4\pi r^{2}
  3. S A ( S ) S\mapsto A(S)
  4. A ( S ) = A ( S 1 ) + + A ( S r ) . A(S)=A(S_{1})+\cdots+A(S_{r}).
  5. S D : r = r ( u , v ) , ( u , v ) D S_{D}:\vec{r}=\vec{r}(u,v),\quad(u,v)\in D
  6. r . \vec{r}.
  7. A ( S D ) = D | r u × r v | d u d v . A(S_{D})=\iint_{D}\left|\vec{r}_{u}\times\vec{r}_{v}\right|\,du\,dv.
  8. r u × r v \vec{r}_{u}\times\vec{r}_{v}
  9. 6 s 2 6s^{2}\,
  10. 2 ( w + h + w h ) 2(\ell w+\ell h+wh)\,
  11. b h + l ( a + b + c ) bh+l(a+b+c)
  12. 2 B + P h 2B+Ph\,
  13. 4 π r 2 = π d 2 4\pi r^{2}=\pi d^{2}\,
  14. 2 r 2 θ 2r^{2}\theta\,
  15. ( 2 π r ) ( 2 π R ) = 4 π 2 R r (2\pi r)(2\pi R)=4\pi^{2}Rr
  16. 2 π r 2 + 2 π r h = 2 π r ( r + h ) 2\pi r^{2}+2\pi rh=2\pi r(r+h)\,
  17. π r ( r 2 + h 2 ) = π r s \pi r\left(\sqrt{r^{2}+h^{2}}\right)=\pi rs\,
  18. s = r 2 + h 2 s=\sqrt{r^{2}+h^{2}}
  19. π r ( r + r 2 + h 2 ) = π r ( r + s ) \pi r\left(r+\sqrt{r^{2}+h^{2}}\right)=\pi r(r+s)\,
  20. B + P L 2 B+\frac{PL}{2}
  21. b 2 + 2 b s = b 2 + 2 b ( b 2 ) 2 + h 2 b^{2}+2bs=b^{2}+2b\sqrt{\left(\frac{b}{2}\right)^{2}+h^{2}}
  22. l w + l ( w 2 ) 2 + h 2 + w ( l 2 ) 2 + h 2 lw+l\sqrt{\left(\frac{w}{2}\right)^{2}+h^{2}}+w\sqrt{\left(\frac{l}{2}\right)^% {2}+h^{2}}
  23. 3 a 2 \sqrt{3}a^{2}
  24. Sphere surface area = 4 π r 2 = ( 2 π r 2 ) × 2 Cylinder surface area = 2 π r ( h + r ) = 2 π r ( 2 r + r ) = ( 2 π r 2 ) × 3 \begin{array}[]{rlll}\,\text{Sphere surface area}&=4\pi r^{2}&&=(2\pi r^{2})% \times 2\\ \,\text{Cylinder surface area}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3% \end{array}

Surjective_function.html

  1. f ( x ) = y f(x)=y
  2. f : X Y f\colon X\rightarrow Y
  3. f f
  4. y Y , x X , f ( x ) = y \forall y\in Y,\,\exists x\in X,\;\;f(x)=y

Surreal_number.html

  1. x L X L x_{L}\in X_{L}
  2. x L x_{L}
  3. y R Y R y_{R}\in Y_{R}
  4. y R y_{R}
  5. i < n S i \cup_{i<n}S_{i}
  6. - x = - { X L | X R } = { - X R | - X L } -x=-\{X_{L}|X_{R}\}=\{-X_{R}|-X_{L}\}
  7. - S = { - s : s S } -S=\{-s:s\in S\}
  8. x + y = { X L | X R } + { Y L | Y R } = { X L + y , x + Y L | X R + y , x + Y R } x+y=\{X_{L}|X_{R}\}+\{Y_{L}|Y_{R}\}=\{X_{L}+y,x+Y_{L}|X_{R}+y,x+Y_{R}\}
  9. X + y = { x + y : x X } , x + Y = { x + y : y Y } X+y=\{x+y:x\in X\},x+Y=\{x+y:y\in Y\}
  10. X R y + x Y R - X R Y R X_{R}y+xY_{R}-X_{R}Y_{R}
  11. x y \displaystyle xy
  12. a 2 b \frac{a}{2^{b}}
  13. n N S n \cup_{n\in N}S_{n}
  14. x S * : x > 0 {x\in S_{*}:x>0}
  15. x S * : x < 0 {x\in S_{*}:x<0}
  16. 𝐍𝐨 \mathbf{No}
  17. ω = { S * | } = { 1 , 2 , 3 , 4 , | } . \omega=\{S_{*}|\}=\{1,2,3,4,...|\}.
  18. 1 3 = { y S * : 3 y < 1 | y S * : 3 y > 1 } \tfrac{1}{3}=\{y\in S_{*}:3y<1|y\in S_{*}:3y>1\}
  19. π = { 3 , 25 8 , 201 64 , | 4 , 7 2 , 13 4 , 51 16 , } \pi=\{3,\frac{25}{8},\frac{201}{64},...|4,\frac{7}{2},\frac{13}{4},\frac{51}{1% 6},...\}
  20. ϵ = { S - S 0 | S + } = { 0 | 1 , 1 2 , 1 4 , 1 8 , } = { 0 | y S * : y > 0 } \epsilon=\{S_{-}\cup S_{0}|S_{+}\}=\{0|1,\tfrac{1}{2},\tfrac{1}{4},\tfrac{1}{8% },...\}=\{0|y\in S_{*}:y>0\}
  21. S ω 2 S_{\omega^{2}}
  22. y L y\in L
  23. y R y\in R
  24. { 1 , 2 , 3 , 4 , | } + { 0 | } = { 1 , 2 , 3 , 4 , , ω | } \{1,2,3,4,...|\}+\{0|\}=\{1,2,3,4,...,\omega|\}
  25. k < ω S ω + k \bigcup_{k<\omega}S_{\omega+k}
  26. a + b i a+bi
  27. 𝐍𝐨 , < , b \langle\mathbf{No},\mathrm{<},b\rangle

Svante_Arrhenius.html

  1. Δ F = α ln ( C / C 0 ) \Delta F=\alpha\ln(C/C_{0})

Syllogism.html

  1. s ( A ) s ( B ) = s(A)\cap s(B)=
  2. s ( A ) s ( B ) s(A)\cap s(B)\neq

Sylow_theorems.html

  1. ( p - 1 ) ! - 1 ( mod p ) (p-1)!\ \equiv\ -1\;\;(\mathop{{\rm mod}}p)
  2. \nmid
  3. | Ω | = ( p k m p k ) = j = 0 p k - 1 p k m - j p k - j = m j = 1 p k - 1 p k - ν p ( j ) m - j / p ν p ( j ) p k - ν p ( j ) - j / p ν p ( j ) |\Omega|={p^{k}m\choose p^{k}}=\prod_{j=0}^{p^{k}-1}\frac{p^{k}m-j}{p^{k}-j}=m% \prod_{j=1}^{p^{k}-1}\frac{p^{k-\nu_{p}(j)}m-j/p^{\nu_{p}(j)}}{p^{k-\nu_{p}(j)% }-j/p^{\nu_{p}(j)}}
  4. | Ω | = ω R | G ω | . |\Omega|=\sum_{\omega\in R}|G\omega|\mathrm{.}
  5. \nmid
  6. \nmid

Symmetric_group.html

  1. f = ( 1 3 ) ( 4 5 ) = ( 1 2 3 4 5 3 2 1 5 4 ) f=(1\ 3)(4\ 5)=\begin{pmatrix}1&2&3&4&5\\ 3&2&1&5&4\end{pmatrix}
  2. g = ( 1 2 5 ) ( 3 4 ) = ( 1 2 3 4 5 2 5 4 3 1 ) . g=(1\ 2\ 5)(3\ 4)=\begin{pmatrix}1&2&3&4&5\\ 2&5&4&3&1\end{pmatrix}.
  3. f g = f g = ( 1 2 4 ) ( 3 5 ) = ( 1 2 3 4 5 2 4 5 1 3 ) . fg=f\circ g=(1\ 2\ 4)(3\ 5)=\begin{pmatrix}1&2&3&4&5\\ 2&4&5&1&3\end{pmatrix}.
  4. ( 1 2 3 4 5 6 ) 2 = ( 1 3 5 ) ( 2 4 6 ) . (1~{}2~{}3~{}4~{}5~{}6)^{2}=(1~{}3~{}5)(2~{}4~{}6).
  5. sgn f = { + 1 , if f is even - 1 , if f is odd . \operatorname{sgn}f=\begin{cases}+1,&\,\text{if }f\mbox{ is even}\\ -1,&\,\text{if }f\,\text{ is odd}.\end{cases}
  6. sgn : S n { + 1 , - 1 } \operatorname{sgn}\colon\mathrm{S}_{n}\rightarrow\{+1,-1\}
  7. h = ( 1 2 3 4 5 4 2 1 3 5 ) h=\begin{pmatrix}1&2&3&4&5\\ 4&2&1&3&5\end{pmatrix}
  8. ( 1 2 n n n - 1 1 ) . \begin{pmatrix}1&2&\cdots&n\\ n&n-1&\cdots&1\end{pmatrix}.
  9. n / 2 \lfloor n/2\rfloor
  10. ( 1 n ) ( 2 n - 1 ) , or k = 1 n - 1 k = n ( n - 1 ) 2 adjacent transpositions: (1\,n)(2\,n-1)\cdots,\,\text{ or }\sum_{k=1}^{n-1}k=\frac{n(n-1)}{2}\,\text{ % adjacent transpositions: }
  11. ( n n - 1 ) ( n - 1 n - 2 ) ( 2 1 ) ( n - 1 n - 2 ) ( n - 2 n - 3 ) , (n\,n-1)(n-1\,n-2)\cdots(2\,1)(n-1\,n-2)(n-2\,n-3)\cdots,
  12. sgn ( ρ n ) = ( - 1 ) n / 2 = ( - 1 ) n ( n - 1 ) / 2 = { + 1 n 0 , 1 ( mod 4 ) - 1 n 2 , 3 ( mod 4 ) \mathrm{sgn}(\rho_{n})=(-1)^{\lfloor n/2\rfloor}=(-1)^{n(n-1)/2}=\begin{cases}% +1&n\equiv 0,1\;\;(\mathop{{\rm mod}}4)\\ -1&n\equiv 2,3\;\;(\mathop{{\rm mod}}4)\end{cases}
  13. ( - 1 ) n / 2 . (-1)^{\lfloor n/2\rfloor}.
  14. k = ( 1 2 3 4 5 1 4 3 2 5 ) k=\begin{pmatrix}1&2&3&4&5\\ 1&4&3&2&5\end{pmatrix}
  15. ( 2 4 ) ( 1 2 3 ) ( 4 5 ) ( 2 4 ) = ( 1 4 3 ) ( 2 5 ) . (2~{}4)\circ(1~{}2~{}3)(4~{}5)\circ(2~{}4)=(1~{}4~{}3)(2~{}5).
  16. σ 1 , , σ n - 1 \sigma_{1},\ldots,\sigma_{n-1}
  17. σ i 2 = 1 , {\sigma_{i}}^{2}=1,
  18. σ i σ j = σ j σ i if j i ± 1 , \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\,\text{ if }j\neq i\pm 1,
  19. σ i σ i + 1 σ i = σ i + 1 σ i σ i + 1 . \sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}.
  20. σ i \sigma_{i}
  21. Aut ( S n ) \mathrm{Aut}(\mathrm{S}_{n})
  22. Out ( S n ) \mathrm{Out}(\mathrm{S}_{n})
  23. Z ( S n ) \mathrm{Z}(\mathrm{S}_{n})
  24. n 2 , 6 n\neq 2,6
  25. S n \mathrm{S}_{n}
  26. n = 2 n=2
  27. S 2 \mathrm{S}_{2}
  28. n = 6 n=6
  29. S 6 C 2 \mathrm{S}_{6}\rtimes\mathrm{C}_{2}
  30. C 2 \mathrm{C}_{2}
  31. n = 6 n=6
  32. Aut ( S 6 ) = S 6 C 2 . \mathrm{Aut}(\mathrm{S}_{6})=\mathrm{S}_{6}\rtimes\mathrm{C}_{2}.
  33. H 1 ( S n , 𝐙 ) = { 0 n < 2 𝐙 / 2 n 2. H_{1}(\mathrm{S}_{n},\mathbf{Z})=\begin{cases}0&n<2\\ \mathbf{Z}/2&n\geq 2.\end{cases}
  34. H 2 ( S n , 𝐙 ) = { 0 n < 4 𝐙 / 2 n 4. H_{2}(\mathrm{S}_{n},\mathbf{Z})=\begin{cases}0&n<4\\ \mathbf{Z}/2&n\geq 4.\end{cases}
  35. H 1 ( A 3 ) H 1 ( A 4 ) C 3 , H_{1}(\mathrm{A}_{3})\cong H_{1}(\mathrm{A}_{4})\cong\mathrm{C}_{3},
  36. H 2 ( A 6 ) H 2 ( A 7 ) C 6 , H_{2}(\mathrm{A}_{6})\cong H_{2}(\mathrm{A}_{7})\cong\mathrm{C}_{6},
  37. A 4 C 3 \mathrm{A}_{4}\twoheadrightarrow\mathrm{C}_{3}
  38. S 4 S 3 , \mathrm{S}_{4}\twoheadrightarrow\mathrm{S}_{3},
  39. S 4 S 3 \mathrm{S}_{4}\twoheadrightarrow\mathrm{S}_{3}

Symmetry_group.html

  1. 𝐀 = A \rhosymbol ρ ^ + A \phisymbol ϕ ^ + A z s y m b o l z ^ \mathbf{A}=A_{\rhosymbol}{\hat{\rho}}+A_{\phisymbol}{\hat{\phi}}+A_{z}symbol{% \hat{z}}
  2. A ρ , A ϕ , A_{\rho},A_{\phi},
  3. A z A_{z}
  4. A ϕ = 0 A_{\phi}=0

Symplectic_manifold.html

  1. V H ( ω ) = d ω ( V H ) = 0 \mathcal{L}_{V_{H}}(\omega)=d\omega(V_{H})=0
  2. Ω = ( 0 I n - I n 0 ) . \Omega=\left(\begin{array}[]{c|c}0&I_{n}\\ \hline-I_{n}&0\end{array}\right).
  3. ( M , ω ) (M,\omega)
  4. ω \omega
  5. L M L\subset M
  6. ω | L = 0 \omega|_{L}=0
  7. dim L = 1 / 2 dim M \,\text{dim }L=1/2\cdot\,\text{dim }M
  8. Ω = Ω 1 + i Ω 2 \Omega=\Omega_{1}+i\Omega_{2}
  9. M M
  10. Ω 1 \Omega_{1}
  11. Ω 2 \Omega_{2}
  12. L L
  13. Ω 2 \Omega_{2}
  14. L L
  15. Ω 1 \Omega_{1}
  16. L L
  17. L L
  18. τ i 1 = i 2 σ , ν π 1 = π 2 τ , τ * ω 2 = ω 1 , \tau\circ i_{1}=i_{2}\circ\sigma,\ \nu\circ\pi_{1}=\pi_{2}\circ\tau,\ \tau^{*}% \omega_{2}=\omega_{1}\,,
  19. ( n + 2 ) (n+2)

T1.html

  1. 𝕋 1 \mathbb{T}^{1}

Tachyon.html

  1. E 2 = p 2 c 2 + m 2 c 4 E^{2}=p^{2}c^{2}+m^{2}c^{4}\;
  2. E = m c 2 1 - v 2 c 2 . E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.
  3. E E
  4. v v

Tactical_voting.html

  1. R i = j i p i j ( u i - u j ) R_{i}=\sum_{j\neq i}\;p_{ij}\cdot(u_{i}-u_{j})\,
  2. G ( p , v , u ) = i = 1 k v i R i G(p,v,u)=\sum_{i=1}^{k}\;v_{i}\cdot R_{i}\,

Tacticity.html

  1. M S L = m m m m + 3 2 m r r r + 2 r m m r + 1 2 r m r m + 1 2 r m r r 1 2 m m m r + r m m r + 1 2 r m r m + 1 2 r m r r MSL=\frac{mmmm+\tfrac{3}{2}mrrr+2rmmr+\tfrac{1}{2}rmrm+\tfrac{1}{2}rmrr}{% \tfrac{1}{2}mmmr+rmmr+\tfrac{1}{2}rmrm+\tfrac{1}{2}rmrr}

Tangent.html

  1. f ( x + h ) f(x+h)
  2. f ( x ) f(x)
  3. h h
  4. f ( a + h ) - f ( a ) h . \frac{f(a+h)-f(a)}{h}.
  5. y - f ( a ) = k ( x - a ) . y-f(a)=k(x-a).\,
  6. y = f ( a ) + f ( a ) ( x - a ) . y=f(a)+f^{\prime}(a)(x-a).\,
  7. d y d x , \frac{dy}{dx},
  8. y - Y = d y d x ( X ) ( x - X ) y-Y=\frac{dy}{dx}(X)\cdot(x-X)
  9. x = X x=X
  10. f ( x ) f\,(x)
  11. ( x - X ) 2 (x-X)^{2}
  12. g ( x ) g(x)
  13. y = g ( x ) . y=g(x).
  14. d y d x = - f x f y . \frac{dy}{dx}=-\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y% }}.
  15. f x ( X , Y ) ( x - X ) + f y ( X , Y ) ( y - Y ) = 0. \frac{\partial f}{\partial x}(X,Y)\cdot(x-X)+\frac{\partial f}{\partial y}(X,Y% )\cdot(y-Y)=0.
  16. f y ( X , Y ) = 0 \frac{\partial f}{\partial y}(X,Y)=0
  17. f x ( X , Y ) 0 \frac{\partial f}{\partial x}(X,Y)\neq 0
  18. f y ( X , Y ) = f x ( X , Y ) = 0 , \frac{\partial f}{\partial y}(X,Y)=\frac{\partial f}{\partial x}(X,Y)=0,
  19. g x X + g y Y + g z Z = n g ( X , Y , Z ) = 0. \frac{\partial g}{\partial x}\cdot X+\frac{\partial g}{\partial y}\cdot Y+% \frac{\partial g}{\partial z}\cdot Z=ng(X,Y,Z)=0.
  20. g x ( X , Y , Z ) x + g y ( X , Y , Z ) y + g z ( X , Y , Z ) z = 0. \frac{\partial g}{\partial x}(X,Y,Z)\cdot x+\frac{\partial g}{\partial y}(X,Y,% Z)\cdot y+\frac{\partial g}{\partial z}(X,Y,Z)\cdot z=0.
  21. f = u n + u n - 1 + + u 1 + u 0 f=u_{n}+u_{n-1}+\dots+u_{1}+u_{0}\,
  22. g = u n + u n - 1 z + + u 1 z n - 1 + u 0 z n = 0. g=u_{n}+u_{n-1}z+\dots+u_{1}z^{n-1}+u_{0}z^{n}=0.\,
  23. f x ( X , Y ) x + f y ( X , Y ) y + g z ( X , Y , 1 ) = 0 \frac{\partial f}{\partial x}(X,Y)\cdot x+\frac{\partial f}{\partial y}(X,Y)% \cdot y+\frac{\partial g}{\partial z}(X,Y,1)=0
  24. x = x ( t ) , y = y ( t ) x=x(t),\quad y=y(t)
  25. d y d x = d y d t d x d t \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}
  26. t = T , X = x ( T ) , Y = y ( T ) \,t=T,\,X=x(T),\,Y=y(T)
  27. d x d t ( T ) ( y - Y ) = d y d t ( T ) ( x - X ) . \frac{dx}{dt}(T)\cdot(y-Y)=\frac{dy}{dt}(T)\cdot(x-X).
  28. d x d t ( T ) = d y d t ( T ) = 0 , \frac{dx}{dt}(T)=\frac{dy}{dt}(T)=0,
  29. - 1 d y d x -\frac{1}{\frac{dy}{dx}}
  30. ( x - X ) + d y d x ( y - Y ) = 0. (x-X)+\frac{dy}{dx}(y-Y)=0.
  31. f y ( x - X ) - f x ( y - Y ) = 0. \frac{\partial f}{\partial y}(x-X)-\frac{\partial f}{\partial x}(y-Y)=0.
  32. x = x ( t ) , y = y ( t ) x=x(t),\quad y=y(t)
  33. d x d t ( x - X ) + d y d t ( y - Y ) = 0. \frac{dx}{dt}(x-X)+\frac{dy}{dt}(y-Y)=0.
  34. ( x 2 + y 2 - 2 a x ) 2 = a 2 ( x 2 + y 2 ) . (x^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,
  35. a 2 ( 3 x 2 - y 2 ) = 0 a^{2}(3x^{2}-y^{2})=0\,
  36. y = ± 3 x . y=\pm\sqrt{3}x.
  37. ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( r 1 ± r 2 ) 2 . \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}\pm r_{2}% \right)^{2}.\,
  38. ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( r 1 + r 2 ) 2 . \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}+r_{2}% \right)^{2}.\,
  39. ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( r 1 - r 2 ) 2 . \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}% \right)^{2}.\,

Tangent_space.html

  1. T x M \scriptstyle T_{x}M
  2. v T x M \scriptstyle v\in T_{x}M
  3. x M \scriptstyle x\in M
  4. d d t \scriptstyle\frac{d}{dt}
  5. D ( f g ) = D ( f ) g ( x ) + f ( x ) D ( g ) D(fg)=D(f)\cdot g(x)+f(x)\cdot D(g)
  6. ( D 1 + D 2 ) ( f ) = D 1 ( f ) + D 2 ( f ) (D_{1}+D_{2})(f)=D_{1}(f)+D_{2}(f)
  7. ( λ D ) ( f ) = λ D ( f ) (\lambda D)(f)=\lambda D(f)
  8. γ ( 0 ) D γ \gamma^{\prime}(0)\longmapsto D_{\gamma}
  9. D γ ( f ) = d d t ( f γ ) | t = 0 = ( f γ ) ( 0 ) D_{\gamma}(f)=\left.\frac{d}{dt}(f\circ\gamma)\right|_{t=0}=(f\circ\gamma)^{% \prime}(0)
  10. D v f ( x ) = d d t f ( x + t v ) | t = 0 = i = 1 n v i f x i ( x ) . D_{v}f(x)=\left.\frac{d}{dt}f(x+tv)\right|_{t=0}=\sum_{i=1}^{n}v^{i}\frac{% \partial f}{\partial x^{i}}(x).
  11. D v ( f ) = v ( f ) D_{v}(f)=v(f)\,
  12. D v ( f ) = ( f γ ) ( 0 ) . D_{v}(f)=(f\circ\gamma)^{\prime}(0).
  13. d φ x : T x M T φ ( x ) N . \mathrm{d}\varphi_{x}\colon T_{x}M\to T_{\varphi(x)}N.
  14. d φ x ( γ ( 0 ) ) = ( φ γ ) ( 0 ) . \mathrm{d}\varphi_{x}(\gamma^{\prime}(0))=(\varphi\circ\gamma)^{\prime}(0).
  15. d φ x ( X ) ( f ) = X ( f φ ) . \mathrm{d}\varphi_{x}(X)(f)=X(f\circ\varphi).
  16. D φ x , ( φ * ) x , φ ( x ) . D\varphi_{x},\quad(\varphi_{*})_{x},\quad\varphi^{\prime}(x).

Tangram.html

  1. 1 \scriptstyle{1}
  2. 2 / 2 \scriptstyle{\sqrt{2}/2}
  3. 1 / 4 \scriptstyle{1/4}
  4. 2 / 2 \scriptstyle{\sqrt{2}/2}
  5. 1 / 2 \scriptstyle{1/2}
  6. 1 / 8 \scriptstyle{1/8}
  7. 1 / 2 \scriptstyle{1/2}
  8. 2 / 4 \scriptstyle{\sqrt{2}/4}
  9. 1 / 16 \scriptstyle{1/16}
  10. 2 / 4 \scriptstyle{\sqrt{2}/4}
  11. 1 / 8 \scriptstyle{1/8}
  12. 1 / 2 \scriptstyle{1/2}
  13. 2 / 4 \scriptstyle{\sqrt{2}/4}
  14. 1 / 8 \scriptstyle{1/8}

Tariff.html

  1. P t a r i f f P_{tariff}
  2. P w P_{w}
  3. ( A + B + C + D ) - ( C + E ) - A = B + D - E (A+B+C+D)-(C+E)-A=B+D-E

Tau_Ceti.html

  1. v eq sin i 1 km / s . v_{\mathrm{eq}}\cdot\sin i\approx 1\ \,\text{km}/\,\text{s}.
  2. [ Fe H ] = - 0.50 \left[\frac{\mathrm{Fe}}{\mathrm{H}}\right]=-0.50
  3. 1 / 20 {1}/{20}
  4. m = M v + 5 ( ( log 10 3.64 ) - 1 ) = 2.6 \begin{smallmatrix}m=M_{v}+5\cdot((\log_{10}3.64)-1)=2.6\end{smallmatrix}
  5. μ = μ δ 2 + μ α 2 cos 2 δ = 1907.79 mas/y \begin{smallmatrix}\mu=\sqrt{{\mu_{\delta}}^{2}+{\mu_{\alpha}}^{2}\cdot\cos^{2% }\delta}=1907.79\,\,\text{mas/y}\end{smallmatrix}
  6. 18 2 + 29 2 + 13 2 = 36.5 km/s. \begin{smallmatrix}\sqrt{18^{2}+29^{2}+13^{2}}=36.5\,\,\text{km/s.}\end{smallmatrix}

Taylor's_theorem.html

  1. f ( x ) = f ( a ) + f ( a ) ( x - a ) + h 1 ( x ) ( x - a ) , lim x a h 1 ( x ) = 0. f(x)=f(a)+f^{\prime}(a)(x-a)+h_{1}(x)(x-a),\qquad\lim_{x\to a}h_{1}(x)=0.
  2. P 1 ( x ) = f ( a ) + f ( a ) ( x - a ) P_{1}(x)=f(a)+f^{\prime}(a)(x-a)
  3. R 1 ( x ) = f ( x ) - P 1 ( x ) = h 1 ( x ) ( x - a ) . R_{1}(x)=f(x)-P_{1}(x)=h_{1}(x)(x-a).
  4. P 2 ( x ) = f ( a ) + f ( a ) ( x - a ) + f ′′ ( a ) 2 ( x - a ) 2 . P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2}(x-a)^{2}.\,
  5. f ( x ) = P 2 ( x ) + h 2 ( x ) ( x - a ) 2 , lim x a h 2 ( x ) = 0. f(x)=P_{2}(x)+h_{2}(x)(x-a)^{2},\qquad\lim_{x\to a}h_{2}(x)=0.
  6. R 2 ( x ) = f ( x ) - P 2 ( x ) = h 2 ( x ) ( x - a ) 2 R_{2}(x)=f(x)-P_{2}(x)=h_{2}(x)(x-a)^{2}
  7. f ( x ) = f ( a ) + f ( a ) ( x - a ) + f ′′ ( a ) 2 ! ( x - a ) 2 + + f ( k ) ( a ) k ! ( x - a ) k + h k ( x ) ( x - a ) k , f(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+\cdots+% \frac{f^{(k)}(a)}{k!}(x-a)^{k}+h_{k}(x)(x-a)^{k},
  8. and lim x a h k ( x ) = 0. \mbox{and}~{}\quad\lim_{x\to a}h_{k}(x)=0.
  9. P k ( x ) = f ( a ) + f ( a ) ( x - a ) + f ′′ ( a ) 2 ! ( x - a ) 2 + + f ( k ) ( a ) k ! ( x - a ) k P_{k}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+% \cdots+\frac{f^{(k)}(a)}{k!}(x-a)^{k}
  10. f ( x ) = p ( x ) + h k ( x ) ( x - a ) k , lim x a h k ( x ) = 0 , f(x)=p(x)+h_{k}(x)(x-a)^{k},\quad\lim_{x\to a}h_{k}(x)=0,
  11. R k ( x ) = f ( x ) - P k ( x ) , \ R_{k}(x)=f(x)-P_{k}(x),
  12. R k ( x ) = o ( | x - a | k ) , x a . R_{k}(x)=o(|x-a|^{k}),\quad x\to a.
  13. R k ( x ) = f ( k + 1 ) ( ξ ) k ! ( x - ξ ) k G ( x ) - G ( a ) G ( ξ ) R_{k}(x)=\frac{f^{(k+1)}(\xi)}{k!}(x-\xi)^{k}\frac{G(x)-G(a)}{G^{\prime}(\xi)}
  14. q f ( k + 1 ) ( x ) Q q\leq f^{(k+1)}(x)\leq Q
  15. q ( x - a ) k + 1 ( k + 1 ) ! R k ( x ) Q ( x - a ) k + 1 ( k + 1 ) ! , q\frac{(x-a)^{k+1}}{(k+1)!}\leq R_{k}(x)\leq Q\frac{(x-a)^{k+1}}{(k+1)!},
  16. | f ( k + 1 ) ( x ) | M |f^{(k+1)}(x)|\leq M
  17. | R k ( x ) | M | x - a | k + 1 ( k + 1 ) ! M r k + 1 ( k + 1 ) ! |R_{k}(x)|\leq M\frac{|x-a|^{k+1}}{(k+1)!}\leq M\frac{r^{k+1}}{(k+1)!}
  18. ( * ) e 0 = 1 , d d x e x = e x , e x > 0 , x . (*)\qquad e^{0}=1,\qquad\frac{d}{dx}e^{x}=e^{x},\qquad e^{x}>0,\qquad x\in% \mathbb{R}.
  19. P k ( x ) = 1 + x + x 2 2 ! + + x k k ! , R k ( x ) = e ξ ( k + 1 ) ! x k + 1 , P_{k}(x)=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{k}}{k!},\qquad R_{k}(x)=\frac{e^% {\xi}}{(k+1)!}x^{k+1},
  20. e x 1 + x 1 - x 2 2 = 2 1 + x 2 - x 2 4 , 0 x 1 e^{x}\leq\frac{1+x}{1-\frac{x^{2}}{2}}=2\frac{1+x}{2-x^{2}}\leq 4,\qquad 0\leq x\leq 1
  21. | R k ( x ) | 4 | x | k + 1 ( k + 1 ) ! 4 ( k + 1 ) ! , - 1 x 1 , |R_{k}(x)|\leq\frac{4|x|^{k+1}}{(k+1)!}\leq\frac{4}{(k+1)!},\qquad-1\leq x\leq 1,
  22. 4 ( k + 1 ) ! < 10 - 5 4 10 5 < ( k + 1 ) ! k 9. \frac{4}{(k+1)!}<10^{-5}\quad\Leftrightarrow\quad 4\cdot 10^{5}<(k+1)!\quad% \Leftrightarrow\quad k\geq 9.
  23. e x = 1 + x + x 2 2 ! + + x 9 9 ! + R 9 ( x ) , | R 9 ( x ) | < 10 - 5 , - 1 x 1. e^{x}=1+x+\frac{x^{2}}{2!}+\ldots+\frac{x^{9}}{9!}+R_{9}(x),\qquad|R_{9}(x)|<1% 0^{-5},\qquad-1\leq x\leq 1.
  24. f ( x ) = k = 0 c k ( x - a ) k = c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + , | x - a | < r . f(x)=\sum_{k=0}^{\infty}c_{k}(x-a)^{k}=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}+\cdots,% \qquad|x-a|<r.
  25. 1 R = lim sup k | c k | 1 k . \frac{1}{R}=\limsup_{k\to\infty}|c_{k}|^{\frac{1}{k}}.
  26. P k ( x ) = j = 0 k c j ( x - a ) j , c j = f ( j ) ( a ) j ! P_{k}(x)=\sum_{j=0}^{k}c_{j}(x-a)^{j},\qquad c_{j}=\frac{f^{(j)}(a)}{j!}
  27. R k ( x ) = j = k + 1 c j ( x - a ) j = ( x - a ) k h k ( x ) , | x - a | < r . R_{k}(x)=\sum_{j=k+1}^{\infty}c_{j}(x-a)^{j}=(x-a)^{k}h_{k}(x),\qquad|x-a|<r.
  28. h k : ( a - r , a + r ) \R ; h k ( x ) = ( x - a ) j = 0 c k + 1 + j ( x - a ) j h_{k}:(a-r,a+r)\to\R;\qquad h_{k}(x)=(x-a)\sum_{j=0}^{\infty}c_{k+1+j}(x-a)^{j}
  29. f ( x ) k = 0 c k ( x - a ) k = c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + f(x)\approx\sum_{k=0}^{\infty}c_{k}(x-a)^{k}=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}+\ldots
  30. ( * ) | R k ( x ) | M k , r | x - a | k + 1 ( k + 1 ) ! (*)\quad|R_{k}(x)|\leq M_{k,r}\frac{|x-a|^{k+1}}{(k+1)!}
  31. T f : ( a - r , a + r ) ; T f ( x ) = k = 0 f ( k ) ( a ) k ! ( x - a ) k . T_{f}:(a-r,a+r)\to\mathbb{R};\qquad T_{f}(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(% a)}{k!}(x-a)^{k}.
  32. f : ; f ( x ) = { e - 1 x 2 x > 0 , 0 x 0. f:\mathbb{R}\to\mathbb{R};\qquad f(x)=\begin{cases}e^{-\frac{1}{x^{2}}}&x>0,\\ 0&x\leq 0.\end{cases}
  33. f ( k ) ( x ) = { p k ( x ) x 3 k e - 1 x 2 x > 0 0 x 0 f^{(k)}(x)=\begin{cases}\frac{p_{k}(x)}{x^{3k}}e^{-\frac{1}{x^{2}}}&x>0\\ 0&x\leq 0\end{cases}
  34. e - 1 x 2 e^{-\frac{1}{x^{2}}}
  35. f : f:\mathbb{C}\to\mathbb{C}
  36. f ( z ) = 1 2 π i γ f ( w ) w - z d w , f ( z ) = 1 2 π i γ f ( w ) ( w - z ) 2 d w , , f ( k ) ( z ) = k ! 2 π i γ f ( w ) ( w - z ) k + 1 d w . \begin{aligned}&\displaystyle f(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-% z}dw,\quad f^{\prime}(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{(w-z)^{2}}dw% ,\\ &\displaystyle\ldots,\quad f^{(k)}(z)=\frac{k!}{2\pi i}\int_{\gamma}\frac{f(w)% }{(w-z)^{k+1}}dw.\end{aligned}
  37. | f ( k ) ( z ) | k ! 2 π γ M r | w - z | k + 1 d w = k ! M r r k , M r = max | w - c | = r | f ( w ) | |f^{(k)}(z)|\leq\frac{k!}{2\pi}\int_{\gamma}\frac{M_{r}}{|w-z|^{k+1}}dw=\frac{% k!M_{r}}{r^{k}},\quad M_{r}=\max_{|w-c|=r}|f(w)|
  38. T f ( z ) = k = 0 f ( k ) ( c ) k ! ( z - c ) k T_{f}(z)=\sum_{k=0}^{\infty}\frac{f^{(k)}(c)}{k!}(z-c)^{k}
  39. T f ( z ) = k = 0 ( z - c ) k 2 π i γ f ( w ) ( w - c ) k + 1 d w = 1 2 π i γ f ( w ) w - c k = 0 ( z - c w - c ) k d w = 1 2 π i γ f ( w ) w - c ( 1 1 - z - c w - c ) d w = 1 2 π i γ f ( w ) w - z d w = f ( z ) , \begin{aligned}\displaystyle T_{f}(z)=&\displaystyle\sum_{k=0}^{\infty}\frac{(% z-c)^{k}}{2\pi i}\int_{\gamma}\frac{f(w)}{(w-c)^{k+1}}dw=\frac{1}{2\pi i}\int_% {\gamma}\frac{f(w)}{w-c}\sum_{k=0}^{\infty}\left(\frac{z-c}{w-c}\right)^{k}dw% \\ \displaystyle=&\displaystyle\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-c}\left% (\frac{1}{1-\frac{z-c}{w-c}}\right)dw=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}% {w-z}dw=f(z),\end{aligned}
  40. f ( z ) = P k ( z ) + R k ( z ) , P k ( z ) = j = 0 k f ( j ) ( c ) j ! ( z - c ) j , f(z)=P_{k}(z)+R_{k}(z),\quad P_{k}(z)=\sum_{j=0}^{k}\frac{f^{(j)}(c)}{j!}(z-c)% ^{j},
  41. R k ( z ) = j = k + 1 ( z - c ) j 2 π i γ f ( w ) ( w - c ) j + 1 d w = ( z - c ) k + 1 2 π i γ f ( w ) d w ( w - c ) k + 1 ( w - z ) , z W . R_{k}(z)=\sum_{j=k+1}^{\infty}\frac{(z-c)^{j}}{2\pi i}\int_{\gamma}\frac{f(w)}% {(w-c)^{j+1}}dw=\frac{(z-c)^{k+1}}{2\pi i}\int_{\gamma}\frac{f(w)dw}{(w-c)^{k+% 1}(w-z)},\qquad z\in W.
  42. | R k ( z ) | j = k + 1 M r | z - c | j r j = M r r k + 1 | z - c | k + 1 1 - | z - c | r M r β k + 1 1 - β , | z - c | r β < 1. |R_{k}(z)|\leq\sum_{j=k+1}^{\infty}\frac{M_{r}|z-c|^{j}}{r^{j}}=\frac{M_{r}}{r% ^{k+1}}\frac{|z-c|^{k+1}}{1-\frac{|z-c|}{r}}\leq\frac{M_{r}\beta^{k+1}}{1-% \beta},\qquad\frac{|z-c|}{r}\leq\beta<1.
  43. f ( x ) = 1 1 + x 2 f(x)=\frac{1}{1+x^{2}}
  44. f : { } { } ; f ( z ) = 1 1 + z 2 f:\mathbb{C}\cup\{\infty\}\to\mathbb{C}\cup\{\infty\};\quad f(z)=\frac{1}{1+z^% {2}}
  45. f ( s y m b o l x ) = f ( s y m b o l a ) + L ( s y m b o l x - s y m b o l a ) + h ( s y m b o l x ) | 𝐱 - 𝐚 | , lim s y m b o l x \tosymbol a h ( s y m b o l x ) = 0. f(symbol{x})=f(symbol{a})+L(symbol{x}-symbol{a})+h(symbol{x})|\mathbf{x}-% \mathbf{a}|,\qquad\lim_{symbol{x}\tosymbol{a}}h(symbol{x})=0.
  46. d f ( s y m b o l a ) ( s y m b o l v ) = f x 1 ( s y m b o l a ) v 1 + + f x n ( s y m b o l a ) v n . df(symbol{a})(symbol{v})=\frac{\partial f}{\partial x_{1}}(symbol{a})v_{1}+% \cdots+\frac{\partial f}{\partial x_{n}}(symbol{a})v_{n}.
  47. | α | = α 1 + + α n , α ! = α 1 ! α n ! , s y m b o l x α = x 1 α 1 x n α n |\alpha|=\alpha_{1}+\cdots+\alpha_{n},\quad\alpha!=\alpha_{1}!\cdots\alpha_{n}% !,\quad symbol{x}^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}
  48. D α f = | α | f x 1 α 1 x n α n , | α | k D^{\alpha}f=\frac{\partial^{|\alpha|}f}{\partial x_{1}^{\alpha_{1}}\cdots% \partial x_{n}^{\alpha_{n}}},\qquad|\alpha|\leq k
  49. f ( s y m b o l x ) = | α | k D α f ( s y m b o l a ) α ! ( s y m b o l x - s y m b o l a ) α + | β | = k + 1 R β ( s y m b o l x ) ( s y m b o l x - s y m b o l a ) β , R β ( s y m b o l x ) = | β | β ! 0 1 ( 1 - t ) | β | - 1 D β f ( s y m b o l a + t ( s y m b o l x - s y m b o l a ) ) d t . \begin{aligned}&\displaystyle f(symbol{x})=\sum_{|\alpha|\leq k}\frac{D^{% \alpha}f(symbol{a})}{\alpha!}(symbol{x}-symbol{a})^{\alpha}+\sum_{|\beta|=k+1}% R_{\beta}(symbol{x})(symbol{x}-symbol{a})^{\beta},\\ &\displaystyle R_{\beta}(symbol{x})=\frac{|\beta|}{\beta!}\int_{0}^{1}(1-t)^{|% \beta|-1}D^{\beta}f\big(symbol{a}+t(symbol{x}-symbol{a})\big)\,dt.\end{aligned}
  50. | R β ( s y m b o l x ) | 1 β ! max | α | = | β | max s y m b o l y B | D α f ( s y m b o l y ) | , s y m b o l x B . \left|R_{\beta}(symbol{x})\right|\leq\frac{1}{\beta!}\max_{|\alpha|=|\beta|}% \max_{symbol{y}\in B}|D^{\alpha}f(symbol{y})|,\qquad symbol{x}\in B.
  51. P 3 ( s y m b o l x ) = f ( s y m b o l a ) + f x 1 ( s y m b o l a ) v 1 + f x 2 ( s y m b o l a ) v 2 + 2 f 2 x 1 ( s y m b o l a ) v 1 2 2 ! + 2 f x 1 x 2 ( s y m b o l a ) v 1 v 2 + 2 f 2 x 2 ( s y m b o l a ) v 2 2 2 ! + 3 f x 1 3 ( s y m b o l a ) v 1 3 3 ! + 3 f 2 x 1 x 2 ( s y m b o l a ) v 1 2 v 2 2 ! + 3 f x 1 2 x 2 ( s y m b o l a ) v 1 v 2 2 2 ! + 3 f 3 x 2 ( s y m b o l a ) v 2 3 3 ! \begin{aligned}\displaystyle P_{3}(symbol{x})=f(symbol{a})+&\displaystyle\frac% {\partial f}{\partial x_{1}}(symbol{a})v_{1}+\frac{\partial f}{\partial x_{2}}% (symbol{a})v_{2}+\frac{\partial^{2}f}{\partial^{2}x_{1}}(symbol{a})\frac{v_{1}% ^{2}}{2!}+\frac{\partial^{2}f}{\partial x_{1}\partial x_{2}}(symbol{a})v_{1}v_% {2}+\frac{\partial^{2}f}{\partial^{2}x_{2}}(symbol{a})\frac{v_{2}^{2}}{2!}\\ &\displaystyle+\frac{\partial^{3}f}{\partial x_{1}^{3}}(symbol{a})\frac{v_{1}^% {3}}{3!}+\frac{\partial^{3}f}{\partial^{2}x_{1}\partial x_{2}}(symbol{a})\frac% {v_{1}^{2}v_{2}}{2!}+\frac{\partial^{3}f}{\partial x_{1}\partial^{2}x_{2}}(% symbol{a})\frac{v_{1}v_{2}^{2}}{2!}+\frac{\partial^{3}f}{\partial^{3}x_{2}}(% symbol{a})\frac{v_{2}^{3}}{3!}\end{aligned}
  52. h k ( x ) = { f ( x ) - P ( x ) ( x - a ) k x a 0 x = a h_{k}(x)=\begin{cases}\frac{f(x)-P(x)}{(x-a)^{k}}&x\not=a\\ 0&x=a\end{cases}
  53. P ( x ) = f ( a ) + f ( a ) ( x - a ) + f ′′ ( a ) 2 ! ( x - a ) 2 + + f ( k ) ( a ) k ! ( x - a ) k . P(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+\cdots+% \frac{f^{(k)}(a)}{k!}(x-a)^{k}.
  54. lim x a h k ( x ) = 0. \lim_{x\to a}h_{k}(x)=0.\,
  55. f ( j ) ( a ) = P ( j ) ( a ) f^{(j)}(a)=P^{(j)}(a)
  56. h k ( x ) h_{k}(x)
  57. x = a x=a
  58. lim x a f ( x ) - P ( x ) ( x - a ) k = lim x a d d x ( f ( x ) - P ( x ) ) d d x ( x - a ) k = = lim x a d k - 1 d x k - 1 ( f ( x ) - P ( x ) ) d k - 1 d x k - 1 ( x - a ) k = 1 k ! lim x a f ( k - 1 ) ( x ) - P ( k - 1 ) ( x ) x - a = 1 k ! ( f ( k ) ( a ) - f ( k ) ( a ) ) = 0 \begin{aligned}\displaystyle\lim_{x\to a}\frac{f(x)-P(x)}{(x-a)^{k}}&% \displaystyle=\lim_{x\to a}\frac{\frac{d}{dx}(f(x)-P(x))}{\frac{d}{dx}(x-a)^{k% }}=\cdots=\lim_{x\to a}\frac{\frac{d^{k-1}}{dx^{k-1}}(f(x)-P(x))}{\frac{d^{k-1% }}{dx^{k-1}}(x-a)^{k}}\\ &\displaystyle=\frac{1}{k!}\lim_{x\to a}\frac{f^{(k-1)}(x)-P^{(k-1)}(x)}{x-a}% \\ &\displaystyle=\frac{1}{k!}(f^{(k)}(a)-f^{(k)}(a))=0\end{aligned}
  59. F ( t ) = f ( t ) + f ( t ) ( x - t ) + f ′′ ( t ) 2 ! ( x - t ) 2 + + f ( k ) ( t ) k ! ( x - t ) k . F(t)=f(t)+f^{\prime}(t)(x-t)+\frac{f^{\prime\prime}(t)}{2!}(x-t)^{2}+\cdots+% \frac{f^{(k)}(t)}{k!}(x-t)^{k}.
  60. ( * ) F ( ξ ) G ( ξ ) = F ( x ) - F ( a ) G ( x ) - G ( a ) (*)\quad\frac{F^{\prime}(\xi)}{G^{\prime}(\xi)}=\frac{F(x)-F(a)}{G(x)-G(a)}
  61. F ( t ) = f ( t ) + ( f ′′ ( t ) ( x - t ) - f ( t ) ) + ( f ( 3 ) ( t ) 2 ! ( x - t ) 2 - f ( 2 ) ( t ) 1 ! ( x - t ) ) + + ( f ( k + 1 ) ( t ) k ! ( x - t ) k - f ( k ) ( t ) ( k - 1 ) ! ( x - t ) k - 1 ) = f ( k + 1 ) ( t ) k ! ( x - t ) k , \begin{aligned}\displaystyle F^{\prime}(t)=&\displaystyle f^{\prime}(t)+\big(f% ^{\prime\prime}(t)(x-t)-f^{\prime}(t)\big)+\left(\frac{f^{(3)}(t)}{2!}(x-t)^{2% }-\frac{f^{(2)}(t)}{1!}(x-t)\right)+\cdots\\ &\displaystyle\cdots+\left(\frac{f^{(k+1)}(t)}{k!}(x-t)^{k}-\frac{f^{(k)}(t)}{% (k-1)!}(x-t)^{k-1}\right)=\frac{f^{(k+1)}(t)}{k!}(x-t)^{k},\end{aligned}
  62. R k ( x ) = f ( k + 1 ) ( ξ ) k ! ( x - ξ ) k G ( x ) - G ( a ) G ( ξ ) . R_{k}(x)=\frac{f^{(k+1)}(\xi)}{k!}(x-\xi)^{k}\frac{G(x)-G(a)}{G^{\prime}(\xi)}.
  63. G ( t ) = ( x - t ) k + 1 \ G(t)=(x-t)^{k+1}
  64. G ( t ) = t - a \ G(t)=t-a
  65. G ( t ) = a t f ( k + 1 ) ( s ) k ! ( x - s ) k d s , G(t)=\int_{a}^{t}\frac{f^{(k+1)}(s)}{k!}(x-s)^{k}\,ds,
  66. f ( x ) = f ( a ) + a x f ( t ) d t . f(x)=f(a)+\int_{a}^{x}\,f^{\prime}(t)\,dt.
  67. f ( x ) = f ( a ) + ( x f ( x ) - a f ( a ) ) - a x t f ′′ ( t ) d t = f ( a ) + x ( f ( a ) + a x f ′′ ( t ) d t ) - a f ( a ) - a x t f ′′ ( t ) d t = f ( a ) + ( x - a ) f ( a ) + a x ( x - t ) f ′′ ( t ) d t , \begin{aligned}\displaystyle f(x)&\displaystyle=f(a)+\Big(xf^{\prime}(x)-af^{% \prime}(a)\Big)-\int_{a}^{x}tf^{\prime\prime}(t)\,dt\\ &\displaystyle=f(a)+x\left(f^{\prime}(a)+\int_{a}^{x}f^{\prime\prime}(t)\,dt% \right)-af^{\prime}(a)-\int_{a}^{x}tf^{\prime\prime}(t)\,dt\\ &\displaystyle=f(a)+(x-a)f^{\prime}(a)+\int_{a}^{x}\,(x-t)f^{\prime\prime}(t)% \,dt,\end{aligned}
  68. ( * ) f ( x ) = f ( a ) + f ( a ) 1 ! ( x - a ) + + f ( k ) ( a ) k ! ( x - a ) k + a x f ( k + 1 ) ( t ) k ! ( x - t ) k d t . (*)\quad f(x)=f(a)+\frac{f^{\prime}(a)}{1!}(x-a)+\cdots+\frac{f^{(k)}(a)}{k!}(% x-a)^{k}+\int_{a}^{x}\frac{f^{(k+1)}(t)}{k!}(x-t)^{k}\,dt.
  69. a x f ( k + 1 ) ( t ) k ! ( x - t ) k d t = - [ f ( k + 1 ) ( t ) ( k + 1 ) k ! ( x - t ) k + 1 ] a x + a x f ( k + 2 ) ( t ) ( k + 1 ) k ! ( x - t ) k + 1 d t = f ( k + 1 ) ( a ) ( k + 1 ) ! ( x - a ) k + 1 + a x f ( k + 2 ) ( t ) ( k + 1 ) ! ( x - t ) k + 1 d t . \begin{aligned}\displaystyle\int_{a}^{x}\frac{f^{(k+1)}(t)}{k!}(x-t)^{k}\,dt=&% \displaystyle-\left[\frac{f^{(k+1)}(t)}{(k+1)k!}(x-t)^{k+1}\right]_{a}^{x}+% \int_{a}^{x}\frac{f^{(k+2)}(t)}{(k+1)k!}(x-t)^{k+1}\,dt\\ \displaystyle=&\displaystyle\ \frac{f^{(k+1)}(a)}{(k+1)!}(x-a)^{k+1}+\int_{a}^% {x}\frac{f^{(k+2)}(t)}{(k+1)!}(x-t)^{k+1}\,dt.\\ \end{aligned}
  70. f ( 𝐱 ) = g ( 1 ) = g ( 0 ) + j = 1 k 1 j ! g ( j ) ( 0 ) + 0 1 ( 1 - t ) k k ! g ( k + 1 ) ( t ) d t . f(\mathbf{x})=g(1)=g(0)+\sum_{j=1}^{k}\frac{1}{j!}g^{(j)}(0)\ +\ \int_{0}^{1}% \frac{(1-t)^{k}}{k!}g^{(k+1)}(t)\,dt.
  71. g ( j ) ( t ) = d j d t j f ( u ( t ) ) = d j d t j f ( 𝐚 + t ( 𝐱 - 𝐚 ) ) = | α | = j ( j α ) ( D α f ) ( 𝐚 + t ( 𝐱 - 𝐚 ) ) ( 𝐱 - 𝐚 ) α \begin{aligned}\displaystyle g^{(j)}(t)&\displaystyle=\frac{d^{j}}{dt^{j}}f(u(% t))=\frac{d^{j}}{dt^{j}}f(\mathbf{a}+t(\mathbf{x}-\mathbf{a}))\\ &\displaystyle=\sum_{|\alpha|=j}\left(\begin{matrix}j\\ \alpha\end{matrix}\right)(D^{\alpha}f)(\mathbf{a}+t(\mathbf{x}-\mathbf{a}))(% \mathbf{x}-\mathbf{a})^{\alpha}\end{aligned}
  72. ( j α ) \left(\begin{matrix}j\\ \alpha\end{matrix}\right)
  73. 1 j ! ( j α ) = 1 α ! \frac{1}{j!}\left(\begin{matrix}j\\ \alpha\end{matrix}\right)=\frac{1}{\alpha!}
  74. f ( 𝐱 ) = f ( 𝐚 ) + | α | k 1 α ! ( D α f ) ( 𝐚 ) ( 𝐱 - 𝐚 ) α + | α | = k + 1 k + 1 α ! ( 𝐱 - 𝐚 ) α 0 1 ( 1 - t ) k ( D α f ) ( 𝐚 + t ( 𝐱 - 𝐚 ) ) d t . f(\mathbf{x})=f(\mathbf{a})+\sum_{|\alpha|\leq k}\frac{1}{\alpha!}(D^{\alpha}f% )(\mathbf{a})(\mathbf{x}-\mathbf{a})^{\alpha}+\sum_{|\alpha|=k+1}\frac{k+1}{% \alpha!}(\mathbf{x}-\mathbf{a})^{\alpha}\int_{0}^{1}(1-t)^{k}(D^{\alpha}f)(% \mathbf{a}+t(\mathbf{x}-\mathbf{a}))\,dt.

Taylor_series.html

  1. a a
  2. f ( a ) + f ( a ) 1 ! ( x - a ) + f ′′ ( a ) 2 ! ( x - a ) 2 + f ′′′ ( a ) 3 ! ( x - a ) 3 + . f(a)+\frac{f^{\prime}(a)}{1!}(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^{2}+% \frac{f^{\prime\prime\prime}(a)}{3!}(x-a)^{3}+\cdots.
  3. n = 0 f ( n ) ( a ) n ! ( x - a ) n \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}\,(x-a)^{n}
  4. 1 + x + x 2 + x 3 + 1+x+x^{2}+x^{3}+\cdots\!
  5. 1 - ( x - 1 ) + ( x - 1 ) 2 - ( x - 1 ) 3 + . 1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots.\!
  6. - x - 1 2 x 2 - 1 3 x 3 - 1 4 x 4 - -x-\frac{1}{2}x^{2}-\frac{1}{3}x^{3}-\frac{1}{4}x^{4}-\cdots\!
  7. ( x - 1 ) - 1 2 ( x - 1 ) 2 + 1 3 ( x - 1 ) 3 - 1 4 ( x - 1 ) 4 + , (x-1)-\frac{1}{2}(x-1)^{2}+\frac{1}{3}(x-1)^{3}-\frac{1}{4}(x-1)^{4}+\cdots,\!
  8. log ( x 0 ) + 1 x 0 ( x - x 0 ) - 1 x 0 2 ( x - x 0 ) 2 2 + . \log(x_{0})+\frac{1}{x_{0}}(x-x_{0})-\frac{1}{x_{0}^{2}}\frac{(x-x_{0})^{2}}{2% }+\cdots.
  9. 1 + x 1 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! + = 1 + x + x 2 2 + x 3 6 + x 4 24 + x 5 120 + = n = 0 x n n ! . 1+\frac{x^{1}}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}+\frac{x^% {5}}{5!}+\cdots=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+\frac{x^{% 5}}{120}+\cdots\!=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}.
  10. f ( x ) = n = 0 a n ( x - b ) n . f(x)=\sum_{n=0}^{\infty}a_{n}(x-b)^{n}.
  11. f ( n ) ( b ) n ! = a n \frac{f^{(n)}(b)}{n!}=a_{n}
  12. sin ( x ) x - x 3 3 ! + x 5 5 ! - x 7 7 ! . \sin\left(x\right)\approx x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}.\!
  13. arcsin x = n = 0 ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 for | x | 1 \arcsin x=\sum^{\infty}_{n=0}\frac{(2n)!}{4^{n}(n!)^{2}(2n+1)}x^{2n+1}\quad\,% \text{ for }|x|\leq 1\!
  14. arccos x = π 2 - arcsin x = π 2 - n = 0 ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 for | x | 1 \arccos x={\pi\over 2}-\arcsin x={\pi\over 2}-\sum^{\infty}_{n=0}\frac{(2n)!}{% 4^{n}(n!)^{2}(2n+1)}x^{2n+1}\quad\,\text{ for }|x|\leq 1\!
  15. arctan x = n = 0 ( - 1 ) n 2 n + 1 x 2 n + 1 for | x | 1 , x ± i \arctan x=\sum^{\infty}_{n=0}\frac{(-1)^{n}}{2n+1}x^{2n+1}\quad\,\text{ for }|% x|\leq 1,x\not=\pm i\!
  16. sinh x = n = 0 x 2 n + 1 ( 2 n + 1 ) ! = x + x 3 3 ! + x 5 5 ! + for all x \sinh x=\sum^{\infty}_{n=0}\frac{x^{2n+1}}{(2n+1)!}=x+\frac{x^{3}}{3!}+\frac{x% ^{5}}{5!}+\cdots\quad\,\text{ for all }x\!
  17. cosh x = n = 0 x 2 n ( 2 n ) ! = 1 + x 2 2 ! + x 4 4 ! + for all x \cosh x=\sum^{\infty}_{n=0}\frac{x^{2n}}{(2n)!}=1+\frac{x^{2}}{2!}+\frac{x^{4}% }{4!}+\cdots\quad\,\text{ for all }x\!
  18. tanh x = n = 1 B 2 n 4 n ( 4 n - 1 ) ( 2 n ) ! x 2 n - 1 = x - 1 3 x 3 + 2 15 x 5 - 17 315 x 7 + for | x | < π 2 \tanh x=\sum^{\infty}_{n=1}\frac{B_{2n}4^{n}(4^{n}-1)}{(2n)!}x^{2n-1}=x-\frac{% 1}{3}x^{3}+\frac{2}{15}x^{5}-\frac{17}{315}x^{7}+\cdots\quad\,\text{ for }|x|<% \frac{\pi}{2}\!
  19. arsinh ( x ) = n = 0 ( - 1 ) n ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 for | x | 1 \mathrm{arsinh}(x)=\sum^{\infty}_{n=0}\frac{(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)% }x^{2n+1}\quad\,\text{ for }|x|\leq 1\!
  20. artanh ( x ) = n = 0 x 2 n + 1 2 n + 1 for | x | 1 , x ± 1 \mathrm{artanh}(x)=\sum^{\infty}_{n=0}\frac{x^{2n+1}}{2n+1}\quad\,\text{ for }% |x|\leq 1,x\not=\pm 1\!
  21. f ( x ) = log cos x , x ( - π / 2 , π / 2 ) f(x)=\log\cos x,\quad x\in(-\pi/2,\pi/2)\!
  22. f ( x ) = log ( 1 + ( cos x - 1 ) ) f(x)=\log(1+(\cos x-1))\!
  23. log ( 1 + x ) = x - x 2 2 + x 3 3 + O ( x 4 ) \log(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+{O}(x^{4})\!
  24. cos x - 1 = - x 2 2 + x 4 24 - x 6 720 + O ( x 8 ) \cos x-1=-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}+{O}(x^{8})\!
  25. f ( x ) = log ( 1 + ( cos x - 1 ) ) = ( cos x - 1 ) - 1 2 ( cos x - 1 ) 2 + 1 3 ( cos x - 1 ) 3 + O ( ( cos x - 1 ) 4 ) = ( - x 2 2 + x 4 24 - x 6 720 + O ( x 8 ) ) - 1 2 ( - x 2 2 + x 4 24 + O ( x 6 ) ) 2 + 1 3 ( - x 2 2 + O ( x 4 ) ) 3 + O ( x 8 ) = - x 2 2 + x 4 24 - x 6 720 - x 4 8 + x 6 48 - x 6 24 + O ( x 8 ) = - x 2 2 - x 4 12 - x 6 45 + O ( x 8 ) . \begin{aligned}\displaystyle f(x)&\displaystyle=\log(1+(\cos x-1))\\ &\displaystyle=\bigl(\cos x-1\bigr)-\frac{1}{2}\bigl(\cos x-1\bigr)^{2}+\frac{% 1}{3}\bigl(\cos x-1\bigr)^{3}+{O}\bigl((\cos x-1)^{4}\bigr)\\ &\displaystyle=\biggl(-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}+{O}(% x^{8})\biggr)-\frac{1}{2}\biggl(-\frac{x^{2}}{2}+\frac{x^{4}}{24}+{O}(x^{6})% \biggr)^{2}+\frac{1}{3}\biggl(-\frac{x^{2}}{2}+O(x^{4})\biggr)^{3}+{O}(x^{8})% \\ &\displaystyle=-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}-\frac{x^{4}% }{8}+\frac{x^{6}}{48}-\frac{x^{6}}{24}+O(x^{8})\\ &\displaystyle=-\frac{x^{2}}{2}-\frac{x^{4}}{12}-\frac{x^{6}}{45}+O(x^{8}).% \end{aligned}\!
  26. g ( x ) = e x cos x . g(x)=\frac{e^{x}}{\cos x}.\!
  27. e x = n = 0 x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + e^{x}=\sum^{\infty}_{n=0}{x^{n}\over n!}=1+x+{x^{2}\over 2!}+{x^{3}\over 3!}+{% x^{4}\over 4!}+\cdots\!
  28. cos x = 1 - x 2 2 ! + x 4 4 ! - \cos x=1-{x^{2}\over 2!}+{x^{4}\over 4!}-\cdots\!
  29. e x cos x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + {e^{x}\over\cos x}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\cdots\!
  30. e x = ( c 0 + c 1 x + c 2 x 2 + c 3 x 3 + ) cos x = ( c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 + ) ( 1 - x 2 2 ! + x 4 4 ! - ) = c 0 - c 0 2 x 2 + c 0 4 ! x 4 + c 1 x - c 1 2 x 3 + c 1 4 ! x 5 + c 2 x 2 - c 2 2 x 4 + c 2 4 ! x 6 + c 3 x 3 - c 3 2 x 5 + c 3 4 ! x 7 + \begin{aligned}\displaystyle e^{x}&\displaystyle=(c_{0}+c_{1}x+c_{2}x^{2}+c_{3% }x^{3}+\cdots)\cos x\\ &\displaystyle=\left(c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots% \right)\left(1-{x^{2}\over 2!}+{x^{4}\over 4!}-\cdots\right)\\ &\displaystyle=c_{0}-{c_{0}\over 2}x^{2}+{c_{0}\over 4!}x^{4}+c_{1}x-{c_{1}% \over 2}x^{3}+{c_{1}\over 4!}x^{5}+c_{2}x^{2}-{c_{2}\over 2}x^{4}+{c_{2}\over 4% !}x^{6}+c_{3}x^{3}-{c_{3}\over 2}x^{5}+{c_{3}\over 4!}x^{7}+\cdots\end{aligned}\!
  31. = c 0 + c 1 x + ( c 2 - c 0 2 ) x 2 + ( c 3 - c 1 2 ) x 3 + ( c 4 - c 2 2 + c 0 4 ! ) x 4 + =c_{0}+c_{1}x+\left(c_{2}-{c_{0}\over 2}\right)x^{2}+\left(c_{3}-{c_{1}\over 2% }\right)x^{3}+\left(c_{4}-{c_{2}\over 2}+{c_{0}\over 4!}\right)x^{4}+\cdots\!
  32. e x cos x = 1 + x + x 2 + 2 x 3 3 + x 4 2 + . \frac{e^{x}}{\cos x}=1+x+x^{2}+{2x^{3}\over 3}+{x^{4}\over 2}+\cdots.\!
  33. ( 1 + x ) e x (1+x)e^{x}
  34. e x = n = 0 x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + . e^{x}=\sum^{\infty}_{n=0}{x^{n}\over n!}=1+x+{x^{2}\over 2!}+{x^{3}\over 3!}+{% x^{4}\over 4!}+\cdots.
  35. ( 1 + x ) e x \displaystyle(1+x)e^{x}
  36. T ( x 1 , , x d ) = n 1 = 0 n 2 = 0 n d = 0 ( x 1 - a 1 ) n 1 ( x d - a d ) n d n 1 ! n d ! ( n 1 + + n d f x 1 n 1 x d n d ) ( a 1 , , a d ) = f ( a 1 , , a d ) + j = 1 d f ( a 1 , , a d ) x j ( x j - a j ) + 1 2 ! j = 1 d k = 1 d 2 f ( a 1 , , a d ) x j x k ( x j - a j ) ( x k - a k ) + 1 3 ! j = 1 d k = 1 d l = 1 d 3 f ( a 1 , , a d ) x j x k x l ( x j - a j ) ( x k - a k ) ( x l - a l ) + \begin{aligned}&\displaystyle T(x_{1},\dots,x_{d})\\ \displaystyle=&\displaystyle\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}% \cdots\sum_{n_{d}=0}^{\infty}\frac{(x_{1}-a_{1})^{n_{1}}\cdots(x_{d}-a_{d})^{n% _{d}}}{n_{1}!\cdots n_{d}!}\,\left(\frac{\partial^{n_{1}+\cdots+n_{d}}f}{% \partial x_{1}^{n_{1}}\cdots\partial x_{d}^{n_{d}}}\right)(a_{1},\dots,a_{d})% \\ \displaystyle=&\displaystyle f(a_{1},\dots,a_{d})+\sum_{j=1}^{d}\frac{\partial f% (a_{1},\dots,a_{d})}{\partial x_{j}}(x_{j}-a_{j})\\ &\displaystyle{}+\frac{1}{2!}\sum_{j=1}^{d}\sum_{k=1}^{d}\frac{\partial^{2}f(a% _{1},\dots,a_{d})}{\partial x_{j}\partial x_{k}}(x_{j}-a_{j})(x_{k}-a_{k})\\ &\displaystyle{}+\frac{1}{3!}\sum_{j=1}^{d}\sum_{k=1}^{d}\sum_{l=1}^{d}\frac{% \partial^{3}f(a_{1},\dots,a_{d})}{\partial x_{j}\partial x_{k}\partial x_{l}}(% x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\dots\end{aligned}
  37. f ( a , b ) + ( x - a ) f x ( a , b ) + ( y - b ) f y ( a , b ) + 1 2 ! [ ( x - a ) 2 f x x ( a , b ) + 2 ( x - a ) ( y - b ) f x y ( a , b ) + ( y - b ) 2 f y y ( a , b ) ] \begin{aligned}\displaystyle f(a,b)&\displaystyle+(x-a)\,f_{x}(a,b)+(y-b)\,f_{% y}(a,b)\\ &\displaystyle+\frac{1}{2!}\left[(x-a)^{2}\,f_{xx}(a,b)+2(x-a)(y-b)\,f_{xy}(a,% b)+(y-b)^{2}\,f_{yy}(a,b)\right]\end{aligned}
  38. T ( 𝐱 ) = f ( 𝐚 ) + ( 𝐱 - 𝐚 ) T D f ( 𝐚 ) + 1 2 ! ( 𝐱 - 𝐚 ) T { D 2 f ( 𝐚 ) } ( 𝐱 - 𝐚 ) + , T(\mathbf{x})=f(\mathbf{a})+(\mathbf{x}-\mathbf{a})^{\mathrm{T}}\mathrm{D}f(% \mathbf{a})+\frac{1}{2!}(\mathbf{x}-\mathbf{a})^{\mathrm{T}}\,\{\mathrm{D}^{2}% f(\mathbf{a})\}\,(\mathbf{x}-\mathbf{a})+\cdots\!\,,
  39. D f ( 𝐚 ) Df(\mathbf{a})\!
  40. f \,f
  41. 𝐱 = 𝐚 \mathbf{x}=\mathbf{a}
  42. D 2 f ( 𝐚 ) D^{2}f(\mathbf{a})\!
  43. T ( 𝐱 ) = | α | 0 ( 𝐱 - 𝐚 ) α α ! ( α f ) ( 𝐚 ) , T(\mathbf{x})=\sum_{|\alpha|\geq 0}\frac{(\mathbf{x}-\mathbf{a})^{\alpha}}{% \alpha!}\,({\mathrm{\partial}^{\alpha}}\,f)(\mathbf{a})\,,
  44. f ( x , y ) = e x log ( 1 + y ) . f(x,y)=e^{x}\log(1+y).\,
  45. f x ( a , b ) = e x log ( 1 + y ) | ( x , y ) = ( 0 , 0 ) = 0 , f_{x}(a,b)=e^{x}\log(1+y)\bigg|_{(x,y)=(0,0)}=0\,,
  46. f y ( a , b ) = e x 1 + y | ( x , y ) = ( 0 , 0 ) = 1 , f_{y}(a,b)=\frac{e^{x}}{1+y}\bigg|_{(x,y)=(0,0)}=1\,,
  47. f x x ( a , b ) = e x log ( 1 + y ) | ( x , y ) = ( 0 , 0 ) = 0 , f_{xx}(a,b)=e^{x}\log(1+y)\bigg|_{(x,y)=(0,0)}=0\,,
  48. f y y ( a , b ) = - e x ( 1 + y ) 2 | ( x , y ) = ( 0 , 0 ) = - 1 , f_{yy}(a,b)=-\frac{e^{x}}{(1+y)^{2}}\bigg|_{(x,y)=(0,0)}=-1\,,
  49. f x y ( a , b ) = f y x ( a , b ) = e x 1 + y | ( x , y ) = ( 0 , 0 ) = 1. f_{xy}(a,b)=f_{yx}(a,b)=\frac{e^{x}}{1+y}\bigg|_{(x,y)=(0,0)}=1.
  50. T ( x , y ) = f ( a , b ) + ( x - a ) f x ( a , b ) + ( y - b ) f y ( a , b ) + 1 2 ! [ ( x - a ) 2 f x x ( a , b ) + 2 ( x - a ) ( y - b ) f x y ( a , b ) + ( y - b ) 2 f y y ( a , b ) ] + , \begin{aligned}\displaystyle T(x,y)=f(a,b)&\displaystyle+(x-a)\,f_{x}(a,b)+(y-% b)\,f_{y}(a,b)\\ &\displaystyle+\frac{1}{2!}\left[(x-a)^{2}\,f_{xx}(a,b)+2(x-a)(y-b)\,f_{xy}(a,% b)+(y-b)^{2}\,f_{yy}(a,b)\right]+\cdots\,,\end{aligned}
  51. T ( x , y ) = 0 + 0 ( x - 0 ) + 1 ( y - 0 ) + 1 2 [ 0 ( x - 0 ) 2 + 2 ( x - 0 ) ( y - 0 ) + ( - 1 ) ( y - 0 ) 2 ] + = y + x y - y 2 2 + . \begin{aligned}\displaystyle T(x,y)&\displaystyle=0+0(x-0)+1(y-0)+\frac{1}{2}% \Big[0(x-0)^{2}+2(x-0)(y-0)+(-1)(y-0)^{2}\Big]+\cdots\\ &\displaystyle=y+xy-\frac{y^{2}}{2}+\cdots.\end{aligned}
  52. a a , b aa,b
  53. f ( x ) f(x)
  54. x = a x=a
  55. f f
  56. a a
  57. f ( x ) f(x)

Tensor.html

  1. σ = [ 𝐓 ( 𝐞 1 ) 𝐓 ( 𝐞 2 ) 𝐓 ( 𝐞 3 ) ] = [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] \begin{aligned}\displaystyle\sigma&\displaystyle=\begin{bmatrix}\mathbf{T}^{(% \mathbf{e}_{1})}\mathbf{T}^{(\mathbf{e}_{2})}\mathbf{T}^{(\mathbf{e}_{3})}\\ \end{bmatrix}\\ &\displaystyle=\begin{bmatrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{bmatrix}\\ \end{aligned}
  2. 𝐞 ^ i \mathbf{\hat{e}}_{i}
  3. 𝐞 j \mathbf{e}_{j}
  4. 𝐞 ^ i = j = 1 n R i j 𝐞 j = R i j 𝐞 j . \mathbf{\hat{e}}_{i}=\sum_{j=1}^{n}R^{j}_{i}\mathbf{e}_{j}=R^{j}_{i}\mathbf{e}% _{j}.
  5. v ^ i = ( R - 1 ) j i v j , \hat{v}^{i}=(R^{-1})^{i}_{j}v^{j},
  6. w ^ i = R i j w j . \hat{w}_{i}=R_{i}^{j}w_{j}.
  7. T ^ j 1 , , j q i 1 , , i p = ( R - 1 ) i 1 i 1 ( R - 1 ) i p i p R j 1 j 1 R j q j q T j 1 , , j q i 1 , , i p . \hat{T}^{i^{\prime}_{1},\ldots,i^{\prime}_{p}}_{j^{\prime}_{1},\ldots,j^{% \prime}_{q}}=(R^{-1})^{i^{\prime}_{1}}_{i_{1}}\cdots(R^{-1})^{i^{\prime}_{p}}_% {i_{p}}R^{j_{1}}_{j^{\prime}_{1}}\cdots R^{j_{q}}_{j^{\prime}_{q}}T^{i_{1},% \ldots,i_{p}}_{j_{1},\ldots,j_{q}}.
  8. T T
  9. T ^ = R - 1 T R \hat{T}=R^{-1}TR
  10. T ^ i 2 i 1 = ( R - 1 ) i 1 i 1 T i 2 i 1 R i 2 i 2 \hat{T}_{i^{\prime}_{2}}^{i^{\prime}_{1}}=(R^{-1})_{i_{1}}^{i^{\prime}_{1}}T_{% i_{2}}^{i_{1}}R_{i^{\prime}_{2}}^{i_{2}}
  11. ( T v ) i (Tv)^{i}
  12. ( T v ) i = T j i v j (Tv)^{i}=T^{i}_{j}v^{j}
  13. ( T v ^ ) i = T ^ j i v ^ j = [ ( R - 1 ) i i T j i R j j ] [ ( R - 1 ) j j v j ) ] = ( R - 1 ) i i ( T v ) i . (\widehat{Tv})^{i^{\prime}}=\hat{T}_{j^{\prime}}^{i^{\prime}}\hat{v}^{j^{% \prime}}=\left[(R^{-1})_{i}^{i^{\prime}}T_{j}^{i}R_{j^{\prime}}^{j}\right]% \left[(R^{-1})^{j^{\prime}}_{j}v^{j})\right]=(R^{-1})_{i}^{i^{\prime}}(Tv)^{i}.
  14. 𝐟 𝐟 R = ( R 1 i 𝐞 i , , R n i 𝐞 i ) \mathbf{f}\mapsto\mathbf{f}\cdot R=\left(R_{1}^{i}\mathbf{e}_{i},\dots,R_{n}^{% i}\mathbf{e}_{i}\right)
  15. T j 1 j q i 1 i p [ 𝐟 R ] = ( R - 1 ) i 1 i 1 ( R - 1 ) i p i p R j 1 j 1 R j q j q T j 1 , , j q i 1 , , i p [ 𝐟 ] . T^{i^{\prime}_{1}\dots i^{\prime}_{p}}_{j^{\prime}_{1}\dots j^{\prime}_{q}}[% \mathbf{f}\cdot R]=(R^{-1})^{i^{\prime}_{1}}_{i_{1}}\cdots(R^{-1})^{i^{\prime}% _{p}}_{i_{p}}R^{j_{1}}_{j^{\prime}_{1}}\cdots R^{j_{q}}_{j^{\prime}_{q}}T^{i_{% 1},\ldots,i_{p}}_{j_{1},\ldots,j_{q}}[\mathbf{f}].
  16. x ¯ i ( x 1 , , x n ) , \bar{x}^{i}(x^{1},\ldots,x^{n}),
  17. T ^ j 1 j q i 1 i p ( x ¯ 1 , , x ¯ n ) = x ¯ i 1 x i 1 x ¯ i p x i p x j 1 x ¯ j 1 x j q x ¯ j q T j 1 j q i 1 i p ( x 1 , , x n ) . \hat{T}^{i^{\prime}_{1}\dots i^{\prime}_{p}}_{j^{\prime}_{1}\dots j^{\prime}_{% q}}(\bar{x}^{1},\ldots,\bar{x}^{n})=\frac{\partial\bar{x}^{i^{\prime}_{1}}}{% \partial x^{i_{1}}}\cdots\frac{\partial\bar{x}^{i^{\prime}_{p}}}{\partial x^{i% _{p}}}\frac{\partial x^{j_{1}}}{\partial\bar{x}^{j^{\prime}_{1}}}\cdots\frac{% \partial x^{j_{q}}}{\partial\bar{x}^{j^{\prime}_{q}}}T^{i_{1}\dots i_{p}}_{j_{% 1}\dots j_{q}}(x^{1},\ldots,x^{n}).
  18. T : V * × × V * p copies × V × × V q copies 𝐑 , T:\underbrace{V^{*}\times\dots\times V^{*}}_{p\,\text{ copies}}\times% \underbrace{V\times\dots\times V}_{q\,\text{ copies}}\rightarrow\mathbf{R},
  19. T j 1 j q i 1 i p T ( ε i 1 , , ε i p , 𝐞 j 1 , , 𝐞 j q ) , T^{i_{1}\dots i_{p}}_{j_{1}\dots j_{q}}\equiv T(\mathbf{\varepsilon}^{i_{1}},% \ldots,\mathbf{\varepsilon}^{i_{p}},\mathbf{e}_{j_{1}},\ldots,\mathbf{e}_{j_{q% }}),
  20. ( p , q ) (p,q)
  21. T V V p copies V * V * q copies . T\in\underbrace{V\otimes\dots\otimes V}_{p\,\text{ copies}}\otimes\underbrace{% V^{*}\otimes\dots\otimes V^{*}}_{q\,\text{ copies}}.
  22. V V
  23. W W
  24. V W V⊗W
  25. T T
  26. V V
  27. T = T j 1 j q i 1 i p 𝐞 i 1 𝐞 i p ε j 1 ε j q . T=T^{i_{1}\dots i_{p}}_{j_{1}\dots j_{q}}\;\mathbf{e}_{i_{1}}\otimes\cdots% \otimes\mathbf{e}_{i_{p}}\otimes\mathbf{\varepsilon}^{j_{1}}\otimes\cdots% \otimes\mathbf{\varepsilon}^{j_{q}}.
  28. ( p , q ) (p,q)
  29. 1 1
  30. 1 1
  31. V V
  32. ( S T ) ( v 1 , , v n , v n + 1 , , v n + m ) = S ( v 1 , , v n ) T ( v n + 1 , , v n + m ) , (S\otimes T)(v_{1},\ldots,v_{n},v_{n+1},\ldots,v_{n+m})=S(v_{1},\ldots,v_{n})T% (v_{n+1},\ldots,v_{n+m}),
  33. ( S T ) j 1 j k j k + 1 j k + m i 1 i l i l + 1 i l + n = S j 1 j k i 1 i l T j k + 1 j k + m i l + 1 i l + n , (S\otimes T)^{i_{1}\ldots i_{l}i_{l+1}\ldots i_{l+n}}_{j_{1}\ldots j_{k}j_{k+1% }\ldots j_{k+m}}=S^{i_{1}\ldots i_{l}}_{j_{1}\ldots j_{k}}T^{i_{l+1}\ldots i_{% l+n}}_{j_{k+1}\ldots j_{k+m}},
  34. T i j T_{i}^{j}
  35. T i i T_{i}^{i}
  36. T V V V * T\in V\otimes V\otimes V^{*}
  37. T = v 1 w 1 α 1 + v 2 w 2 α 2 + + v N w N α N . T=v_{1}\otimes w_{1}\otimes\alpha_{1}+v_{2}\otimes w_{2}\otimes\alpha_{2}+% \cdots+v_{N}\otimes w_{N}\otimes\alpha_{N}.
  38. α 1 ( v 1 ) w 1 + α 2 ( v 2 ) w 2 + + α N ( v N ) w N . \alpha_{1}(v_{1})w_{1}+\alpha_{2}(v_{2})w_{2}+\cdots+\alpha_{N}(v_{N})w_{N}.
  39. P i ε 0 = j χ i j ( 1 ) E j + j k χ i j k ( 2 ) E j E k + j k χ i j k ( 3 ) E j E k E + . \frac{P_{i}}{\varepsilon_{0}}=\sum_{j}\chi^{(1)}_{ij}E_{j}+\sum_{jk}\chi_{ijk}% ^{(2)}E_{j}E_{k}+\sum_{jk\ell}\chi_{ijk\ell}^{(3)}E_{j}E_{k}E_{\ell}+\cdots.\!
  40. χ ( 1 ) \chi^{(1)}
  41. χ ( 2 ) \chi^{(2)}
  42. χ ( 3 ) \chi^{(3)}
  43. V W V⊗W
  44. d d
  45. d d
  46. ( n , m ) (n,m)
  47. n + m n+m
  48. w th w^{\,\text{th}}
  49. w w
  50. det g \sqrt{\det g}
  51. det ( g ) = ( det x x ) 2 det ( g ) \det(g^{\prime})=\left(\det\frac{\partial x}{\partial x^{\prime}}\right)^{2}% \det(g)
  52. B i C i = B 1 C 1 + B 2 C 2 + B n C n B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}+\cdots B_{n}C^{n}

Tensor_product.html

  1. \boxtimes
  2. V V
  3. W W
  4. K K
  5. K K
  6. V W V⊗W
  7. K K
  8. F ( S ) F(S)
  9. S S
  10. F ( S ) F(S)
  11. S S
  12. K K
  13. 2 a + 3 b 2a+3b
  14. 1 x \frac{1−}{x}
  15. x x
  16. S S
  17. K K
  18. m n m≠n
  19. m = n m=n
  20. k k
  21. K K
  22. S S
  23. V V
  24. W W
  25. K K
  26. U U
  27. V V
  28. W W
  29. U = V W U=V⊗W
  30. V × W V×W
  31. F ( V × W ) F(V×W)
  32. K K
  33. V W V⊗W
  34. F ( V × W ) F(V×W)
  35. v , v 1 , v 2 V ; w , w 1 , w 2 W ; c K ; \displaystyle v,v_{1},v_{2}\in V;w,w_{1},w_{2}\in W;c\in K;
  36. V W V⊗W
  37. + : U × U U +:U×U→U
  38. : K × U U ⋅:K×U→U
  39. F ( V × W ) F(V×W)
  40. u ~ 1 , u ~ 2 \tilde{u}_{1},\tilde{u}_{2}
  41. u ~ 1 u 1 , u ~ 2 u 2 ( + ) : ( u 1 , u 2 ) [ u ~ 1 + F u ~ 2 ] \tilde{u}_{1}\in u_{1},\tilde{u}_{2}\in u_{2}\Rightarrow(+):(u_{1},u_{2})% \mapsto[\tilde{u}_{1}+_{F}\tilde{u}_{2}]
  42. u ~ 1 u 1 ( ) : ( c , u 1 ) [ c F u ~ 1 ] \tilde{u}_{1}\in u_{1}\Rightarrow(\cdot):(c,u_{1})\mapsto[c\cdot_{F}\tilde{u}_% {1}]
  43. V W V⊗W
  44. F ( V × W ) / N F(V×W)/N
  45. N N
  46. F ( V × W ) F(V×W)
  47. N = N=∅∅
  48. F ( V × W ) ∅∈F(V×W)
  49. N N
  50. N = { n F ( V × W ) | \displaystyle N=\{n\in F(V\times W)\,|
  51. N N
  52. ( v 1 , w 1 ) + ( v 2 , w 1 ) = ( v 1 + v 2 , w 1 ) , ( v 1 , w 1 ) + ( v 1 , w 2 ) = ( v 1 , w 1 + w 2 ) , c ( v 1 , w 1 ) = ( c v 1 , w 1 ) , c ( v 1 , w 1 ) = ( v 1 , c w 1 ) \begin{aligned}\displaystyle(v_{1},w_{1})+(v_{2},w_{1})&\displaystyle=(v_{1}+v% _{2},w_{1}),\\ \displaystyle(v_{1},w_{1})+(v_{1},w_{2})&\displaystyle=(v_{1},w_{1}+w_{2}),\\ \displaystyle c(v_{1},w_{1})&\displaystyle=(cv_{1},w_{1}),\\ \displaystyle c(v_{1},w_{1})&\displaystyle=(v_{1},cw_{1})\end{aligned}
  53. F ( V × W ) F(V×W)
  54. V W V⊗W
  55. v v
  56. V V
  57. w w
  58. W W
  59. ( v , w ) (v,w)
  60. v w v⊗w
  61. " " "⊗"
  62. v w v⊗w
  63. v w v⊗w
  64. V W V⊗W
  65. v w v⊗w
  66. ( 1 , 1 ) (1,1)
  67. V V
  68. W W
  69. V W V⊗W
  70. m n mn
  71. S : V X S:V→X
  72. T : W Y T:W→Y
  73. S S
  74. T T
  75. S T : V W X Y S\otimes T:V\otimes W\rightarrow X\otimes Y
  76. ( S T ) ( v w ) = S ( v ) T ( w ) . (S\otimes T)(v\otimes w)=S(v)\otimes T(w).
  77. S S
  78. T T
  79. S T S⊗T
  80. S S
  81. T T
  82. S T S⊗T
  83. V V
  84. X X
  85. W W
  86. Y Y
  87. S S
  88. T T
  89. [ a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 ] \begin{bmatrix}a_{1,1}&a_{1,2}\\ a_{2,1}&a_{2,2}\\ \end{bmatrix}
  90. [ b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ] \begin{bmatrix}b_{1,1}&b_{1,2}\\ b_{2,1}&b_{2,2}\\ \end{bmatrix}
  91. [ a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 ] [ b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ] = [ a 1 , 1 [ b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ] a 1 , 2 [ b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ] a 2 , 1 [ b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ] a 2 , 2 [ b 1 , 1 b 1 , 2 b 2 , 1 b 2 , 2 ] ] = [ a 1 , 1 b 1 , 1 a 1 , 1 b 1 , 2 a 1 , 2 b 1 , 1 a 1 , 2 b 1 , 2 a 1 , 1 b 2 , 1 a 1 , 1 b 2 , 2 a 1 , 2 b 2 , 1 a 1 , 2 b 2 , 2 a 2 , 1 b 1 , 1 a 2 , 1 b 1 , 2 a 2 , 2 b 1 , 1 a 2 , 2 b 1 , 2 a 2 , 1 b 2 , 1 a 2 , 1 b 2 , 2 a 2 , 2 b 2 , 1 a 2 , 2 b 2 , 2 ] . \begin{bmatrix}a_{1,1}&a_{1,2}\\ a_{2,1}&a_{2,2}\\ \end{bmatrix}\otimes\begin{bmatrix}b_{1,1}&b_{1,2}\\ b_{2,1}&b_{2,2}\\ \end{bmatrix}=\begin{bmatrix}a_{1,1}\begin{bmatrix}b_{1,1}&b_{1,2}\\ b_{2,1}&b_{2,2}\\ \end{bmatrix}&a_{1,2}\begin{bmatrix}b_{1,1}&b_{1,2}\\ b_{2,1}&b_{2,2}\\ \end{bmatrix}\\ &\\ a_{2,1}\begin{bmatrix}b_{1,1}&b_{1,2}\\ b_{2,1}&b_{2,2}\\ \end{bmatrix}&a_{2,2}\begin{bmatrix}b_{1,1}&b_{1,2}\\ b_{2,1}&b_{2,2}\\ \end{bmatrix}\\ \end{bmatrix}=\begin{bmatrix}a_{1,1}b_{1,1}&a_{1,1}b_{1,2}&a_{1,2}b_{1,1}&a_{1% ,2}b_{1,2}\\ a_{1,1}b_{2,1}&a_{1,1}b_{2,2}&a_{1,2}b_{2,1}&a_{1,2}b_{2,2}\\ a_{2,1}b_{1,1}&a_{2,1}b_{1,2}&a_{2,2}b_{1,1}&a_{2,2}b_{1,2}\\ a_{2,1}b_{2,1}&a_{2,1}b_{2,2}&a_{2,2}b_{2,1}&a_{2,2}b_{2,2}\\ \end{bmatrix}.
  92. v v
  93. w w
  94. φ : V × W V W φ:V×W→V⊗W
  95. Z Z
  96. h : V × W Z h:V×W→Z
  97. [ u o v e r s e t , u , u h ] : V W Z [u^{\prime}overset^{\prime},u^{\prime}~{}^{\prime},u^{\prime}h^{\prime}]:V⊗W→Z
  98. h = [ u o v e r s e t , u , u h ] φ h=[u^{\prime}overset^{\prime},u^{\prime}~{}^{\prime},u^{\prime}h^{\prime}]∘φ
  99. φ φ
  100. V × W V×W
  101. φ : V × W V W φ′:V×W→V⊗′W
  102. k : V W V W k:V⊗W→V⊗′W
  103. φ = k φ φ′=k∘φ
  104. V W W V . V\otimes W\cong W\otimes V.
  105. V × W W V V×W→W⊗V
  106. ( v , w ) (v,w)
  107. w v w⊗v
  108. V W W V V⊗W→W⊗V
  109. W V V W W⊗V→V⊗W
  110. V 1 ( V 2 V 3 ) ( V 1 V 2 ) V 3 . V_{1}\otimes(V_{2}\otimes V_{3})\cong(V_{1}\otimes V_{2})\otimes V_{3}.
  111. n n
  112. n n
  113. V V
  114. n n
  115. V V
  116. V n = def V V n . V^{\otimes n}\;\overset{\mathrm{def}}{=}\;\underbrace{V\otimes\cdots\otimes V}% _{n}.
  117. σ σ
  118. n n
  119. V V
  120. σ : V n V n \sigma:V^{n}\to V^{n}
  121. σ ( v 1 , v 2 , , v n ) = ( v σ 1 , v σ 2 , , v σ n ) . \sigma(v_{1},v_{2},\dots,v_{n})=(v_{\sigma 1},v_{\sigma 2},\dots,v_{\sigma n}).
  122. φ : V n V n \varphi:V^{n}\to V^{\otimes n}
  123. V V
  124. V V
  125. τ σ : V n V n \tau_{\sigma}:V^{\otimes n}\to V^{\otimes n}
  126. φ σ = τ σ φ . \varphi\circ\sigma=\tau_{\sigma}\circ\varphi.
  127. σ σ
  128. r r
  129. s s
  130. ( r , s ) (r,s)
  131. V V
  132. T s r ( V ) = V V r V * V * s = V r V * s . T^{r}_{s}(V)=\underbrace{V\otimes\dots\otimes V}_{r}\otimes\underbrace{V^{*}% \otimes\dots\otimes V^{*}}_{s}=V^{\otimes r}\otimes V^{*\otimes s}.
  133. f f
  134. V V
  135. K K
  136. T s r ( V ) K T s r ( V ) T s + s r + r ( V ) . T^{r}_{s}(V)\otimes_{K}T^{r^{\prime}}_{s^{\prime}}(V)\to T^{r+r^{\prime}}_{s+s% ^{\prime}}(V).
  137. V V
  138. V V
  139. ( v 1 f 1 ) ( v 1 ) = v 1 v 1 f 1 . (v_{1}\otimes f_{1})\otimes(v^{\prime}_{1})=v_{1}\otimes v^{\prime}_{1}\otimes f% _{1}.
  140. V V
  141. T [ u s u , u b = , u s , u p = , u r ] ( V ) T[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}s^{\prime},u^{\prime}p% =^{\prime},u^{\prime}r^{\prime}](V)
  142. F F
  143. G G
  144. m m
  145. n n
  146. F T [ u s u , u b = , u m , u p = 0090 ] F∈T[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}m^{\prime},u^{\prime% }p=\u{2}0090^{\prime}]
  147. G T [ u s u , u b = , u n , u p = 0090 ] G∈T[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime% }p=\u{2}0090^{\prime}]
  148. ( F G ) i 1 i 2 i m + n = F i 1 i 2 i m G i m + 1 i m + 2 i m + 3 i m + n . (F\otimes G)_{i_{1}i_{2}...i_{m+n}}=F_{i_{1}i_{2}...i_{m}}G_{i_{m+1}i_{m+2}i_{% m+3}...i_{m+n}}.
  149. 𝐔 \mathbf{U}
  150. ( 1 , 1 ) (1,1)
  151. 𝐕 \mathbf{V}
  152. ( 1 , 0 ) (1,0)
  153. U α V γ β = ( U V ) α γ β U^{\alpha}{}_{\beta}V^{\gamma}=(U\otimes V)^{\alpha}{}_{\beta}{}^{\gamma}
  154. V μ U ν = σ ( V U ) μ ν . σ V^{\mu}U^{\nu}{}_{\sigma}=(V\otimes U)^{\mu\nu}{}_{\sigma}.
  155. V V
  156. f f
  157. V V
  158. K K
  159. V V * K V\otimes V^{*}\to K
  160. v f f ( v ) . v\otimes f\mapsto f(v).
  161. T s r ( V ) T s - 1 r - 1 ( V ) T^{r}_{s}(V)\to T^{r-1}_{s-1}(V)
  162. r , s > 0 r,s>0
  163. V V
  164. K V V * , λ i λ v i v i * . K\to V\otimes V^{*},\lambda\mapsto\sum_{i}\lambda v_{i}\otimes v^{*}_{i}.
  165. V V
  166. U U
  167. V V
  168. W W
  169. Hom ( U V , W ) Hom ( U , Hom ( V , W ) ) . \mathrm{Hom}(U\otimes V,W)\cong\mathrm{Hom}(U,\mathrm{Hom}(V,W)).
  170. H o m ( - , - ) Hom(-,-)
  171. K K
  172. T s r ( V ) \scriptstyle T^{r}_{s}(V)
  173. E n d ( V ) End(V)
  174. r = s = 1 r=s=1
  175. u E n d ( V ) u∈End(V)
  176. u ( a b ) = u ( a ) b - a u * ( b ) , u(a\otimes b)=u(a)\otimes b-a\otimes u^{*}(b),
  177. u u
  178. u ( a ) , b = a , u * ( b ) \langle u(a),b\rangle=\langle a,u^{*}(b)\rangle
  179. T 1 1 ( V ) End ( V ) \scriptstyle T^{1}_{1}(V)\rightarrow\mathrm{End}(V)
  180. ( a b ) ( x ) = x , b a . (a\otimes b)(x)=\langle x,b\rangle a.
  181. u u
  182. E n d ( V ) End(V)
  183. T 1 1 ( V ) \scriptstyle T^{1}_{1}(V)
  184. E n d ( V ) End(V)
  185. a d ( u ) ad(u)
  186. E n d ( V ) End(V)
  187. A A
  188. B B
  189. R R
  190. A R B := F ( A × B ) / G A\otimes_{R}B:=F(A\times B)/G
  191. F ( A × B ) F(A×B)
  192. R R
  193. G G
  194. R R
  195. a b b a ab≠ba
  196. A A
  197. R R
  198. B B
  199. R R
  200. ( a r , b ) - ( a , r b ) (ar,b)-(a,rb)
  201. R R
  202. R R
  203. ( a , b ) a b (a,b)→a⊗b
  204. ϕ ( a + a , b ) = ϕ ( a , b ) + ϕ ( a , b ) \displaystyle\phi(a+a^{\prime},b)=\phi(a,b)+\phi(a^{\prime},b)
  205. φ φ
  206. A × B A×B
  207. ψ ψ
  208. A × B A×B
  209. f f
  210. ψ = f φ ψ=f∘φ
  211. ϕ \phi
  212. V W V⊗W
  213. V V
  214. W W
  215. V W V⊗W
  216. 𝐙 / n \mathbf{Z}/n
  217. 𝐙 \mathbf{Z}
  218. 𝐙 / n \mathbf{Z}/n
  219. M 𝐙 𝐙 / n = M / n . M\otimes_{\mathbf{Z}}\mathbf{Z}/n=M/n.
  220. R R
  221. M M
  222. j J a j i m i = 0 \sum_{j\in J}a_{ji}m_{i}=0
  223. M R N = coker ( N J N I ) M\otimes_{R}N=\operatorname{coker}(N^{J}\rightarrow N^{I})
  224. n N n∈N
  225. j j
  226. M M
  227. R R
  228. M 1 R N M 2 R N M_{1}\otimes_{R}N\to M_{2}\otimes_{R}N
  229. n n
  230. n : 𝐙 𝐙 n:\mathbf{Z}→\mathbf{Z}
  231. 𝐙 / n \mathbf{Z}/n
  232. 0 : 𝐙 / n 𝐙 / n 0:\mathbf{Z}/n→\mathbf{Z}/n
  233. R R
  234. R R
  235. A A
  236. B B
  237. R R
  238. R R
  239. ( a 1 b 1 ) ( a 2 b 2 ) = ( a 1 a 2 ) ( b 1 b 2 ) . (a_{1}\otimes b_{1})\cdot(a_{2}\otimes b_{2})=(a_{1}\cdot a_{2})\otimes(b_{1}% \cdot b_{2}).
  240. R [ x ] R R [ y ] R [ x , y ] . R[x]\otimes_{R}R[y]\cong R[x,y].
  241. A A
  242. B B
  243. R R
  244. A = R x x / f ( x ) A=Rxx/f(x)
  245. f f
  246. R R
  247. A R B B [ x ] / f ( x ) A\otimes_{R}B\cong B[x]/f(x)
  248. f f
  249. B B
  250. B B
  251. A = B A=B
  252. R R
  253. A R A A [ x ] / f ( x ) A\otimes_{R}A\cong A[x]/f(x)
  254. A A
  255. f ( x 1 , , x k ) \scriptstyle f(x_{1},\dots,x_{k})
  256. g ( x 1 , , x m ) \scriptstyle g(x_{1},\dots,x_{m})
  257. ( f g ) ( x 1 , , x k + m ) = f ( x 1 , , x k ) g ( x k + 1 , , x k + m ) . (f\otimes g)(x_{1},\dots,x_{k+m})=f(x_{1},\dots,x_{k})g(x_{k+1},\dots,x_{k+m}).
  258. V V V\wedge V
  259. V V / ( v v for all v V ) . V\otimes V/(v\otimes v\,\text{ for all }v\in V).
  260. V V / ( v 1 v 2 + v 2 v 1 for all v 1 , v 2 V ) . V\otimes V/(v_{1}\otimes v_{2}+v_{2}\otimes v_{1}\,\text{ for all }v_{1},v_{2}% \in V).
  261. v 1 v 2 v_{1}\otimes v_{2}
  262. v 1 v 2 v_{1}\wedge v_{2}
  263. v 1 v 2 = - v 2 v 1 v_{1}\wedge v_{2}=-v_{2}\wedge v_{1}
  264. V V V\otimes\dots\otimes V
  265. Λ n V \Lambda^{n}V
  266. S y m n V := V V n / ( v i v i + 1 - v i + 1 v i ) Sym^{n}V:=\underbrace{V\otimes\dots\otimes V}_{n}/(\dots\otimes v_{i}\otimes v% _{i+1}\otimes\dots-\dots\otimes v_{i+1}\otimes v_{i}\otimes\dots)
  267. . × \scriptstyle\circ.\times
  268. A . × B \scriptstyle A\circ.\times B
  269. A . × B . × C \scriptstyle A\circ.\times B\circ.\times C
  270. ( mod n ) \;\;(\mathop{{\rm mod}}n)
  271. ( mod n ) \;\;(\mathop{{\rm mod}}n)

Ternary_numeral_system.html

  1. log 2 3 \log_{2}3

Tesla_coil.html

  1. f 1 \scriptstyle f_{1}
  2. f 2 \scriptstyle f_{2}
  3. f 1 = 1 2 π 1 L 1 C 1 f 2 = 1 2 π 1 L 2 C 2 f_{1}={1\over 2\pi}\sqrt{1\over L_{1}C_{1}}\qquad\qquad f_{2}={1\over 2\pi}% \sqrt{1\over L_{2}C_{2}}\,
  4. f = 1 2 π 1 L 1 C 1 = 1 2 π 1 L 2 C 2 f={1\over 2\pi}\sqrt{1\over L_{1}C_{1}}={1\over 2\pi}\sqrt{1\over L_{2}C_{2}}\,
  5. L 1 C 1 = L 2 C 2 L_{1}C_{1}=L_{2}C_{2}\,
  6. W 1 \scriptstyle W_{1}
  7. C 1 \scriptstyle C_{1}
  8. V 1 \scriptstyle V_{1}
  9. W 1 = 1 2 C 1 V 1 2 W_{1}={1\over 2}C_{1}V_{1}^{2}\,
  10. V 2 \scriptstyle V_{2}
  11. C 2 \scriptstyle C_{2}
  12. W 2 = 1 2 C 2 V 2 2 W_{2}={1\over 2}C_{2}V_{2}^{2}\,
  13. W 2 = W 1 \scriptstyle W_{2}\;=\;W_{1}
  14. L 1 C 1 = L 2 C 2 \scriptstyle L_{1}C_{1}\;=\;L_{2}C_{2}
  15. C 2 \scriptstyle C_{2}
  16. L 1 \scriptstyle L_{1}

Tesseract.html

  1. { ( x 1 , x 2 , x 3 , x 4 ) 4 : - 1 x i 1 } \{(x_{1},x_{2},x_{3},x_{4})\in\mathbb{R}^{4}\,:\,-1\leq x_{i}\leq 1\}

Tetrahedron.html

  1. 2 2 2\sqrt{2}
  2. A 0 = 3 4 a 2 A_{0}={\sqrt{3}\over 4}a^{2}\,
  3. A = 4 A 0 = 3 a 2 A=4\,A_{0}={\sqrt{3}}a^{2}\,
  4. h = 6 3 a = 2 3 a h={\sqrt{6}\over 3}a=\sqrt{2\over 3}\,a\,
  5. l = 1 2 a l={1\over\sqrt{2}}\,a\,
  6. V = 1 3 A 0 h = 2 12 a 3 = a 3 6 2 V={1\over 3}A_{0}h={\sqrt{2}\over 12}a^{3}={a^{3}\over{6\sqrt{2}}}\,
  7. arccos ( 1 3 ) = arctan ( 2 ) \arccos\left({1\over\sqrt{3}}\right)=\arctan(\sqrt{2})\,
  8. arccos ( 1 3 ) = arctan ( 2 2 ) \arccos\left({1\over 3}\right)=\arctan(2\sqrt{2})\,
  9. arccos ( - 1 3 ) = 2 arctan ( 2 ) \arccos\left({-1\over 3}\right)=2\arctan(\sqrt{2})\,
  10. arccos ( 23 27 ) \arccos\left({23\over 27}\right)
  11. R = 6 4 a = 3 8 a R={\sqrt{6}\over 4}a=\sqrt{3\over 8}\,a\,
  12. r = 1 3 R = a 24 r={1\over 3}R={a\over\sqrt{24}}\,
  13. r M = r R = a 8 r_{M}=\sqrt{rR}={a\over\sqrt{8}}\,
  14. r E = a 6 r_{E}={a\over\sqrt{6}}\,
  15. d V E = 6 2 a = 3 2 a d_{VE}={\sqrt{6}\over 2}a={\sqrt{3\over 2}}a\,
  16. 2 2 \scriptstyle 2\sqrt{2}
  17. 2 \scriptstyle\sqrt{2}
  18. V = 1 3 A 0 h V=\frac{1}{3}A_{0}\,h\,
  19. V = | ( 𝐚 - 𝐝 ) ( ( 𝐛 - 𝐝 ) × ( 𝐜 - 𝐝 ) ) | 6 . V=\frac{|(\mathbf{a}-\mathbf{d})\cdot((\mathbf{b}-\mathbf{d})\times(\mathbf{c}% -\mathbf{d}))|}{6}.
  20. V = | 𝐚 ( 𝐛 × 𝐜 ) | 6 , V=\frac{|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|}{6},
  21. 6 V = | 𝐚 𝐛 𝐜 | 6\cdot V=\begin{vmatrix}\mathbf{a}&\mathbf{b}&\mathbf{c}\end{vmatrix}
  22. 6 V = | 𝐚 𝐛 𝐜 | 6\cdot V=\begin{vmatrix}\mathbf{a}\\ \mathbf{b}\\ \mathbf{c}\end{vmatrix}
  23. 𝐚 = ( a 1 , a 2 , a 3 ) \mathbf{a}=(a_{1},a_{2},a_{3})\,
  24. 36 V 2 = | 𝐚 𝟐 𝐚 𝐛 𝐚 𝐜 𝐚 𝐛 𝐛 𝟐 𝐛 𝐜 𝐚 𝐜 𝐛 𝐜 𝐜 𝟐 | 36\cdot V^{2}=\begin{vmatrix}\mathbf{a^{2}}&\mathbf{a}\cdot\mathbf{b}&\mathbf{% a}\cdot\mathbf{c}\\ \mathbf{a}\cdot\mathbf{b}&\mathbf{b^{2}}&\mathbf{b}\cdot\mathbf{c}\\ \mathbf{a}\cdot\mathbf{c}&\mathbf{b}\cdot\mathbf{c}&\mathbf{c^{2}}\end{vmatrix}
  25. 𝐚 𝐛 = a b cos γ \mathbf{a}\cdot\mathbf{b}=ab\cos{\gamma}
  26. V = a b c 6 1 + 2 cos α cos β cos γ - cos 2 α - cos 2 β - cos 2 γ , V=\frac{abc}{6}\sqrt{1+2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^{2}{\alpha}-% \cos^{2}{\beta}-\cos^{2}{\gamma}},\,
  27. 288 V 2 = | 0 1 1 1 1 1 0 d 12 2 d 13 2 d 14 2 1 d 12 2 0 d 23 2 d 24 2 1 d 13 2 d 23 2 0 d 34 2 1 d 14 2 d 24 2 d 34 2 0 | 288\cdot V^{2}=\begin{vmatrix}0&1&1&1&1\\ 1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\ 1&d_{12}^{2}&0&d_{23}^{2}&d_{24}^{2}\\ 1&d_{13}^{2}&d_{23}^{2}&0&d_{34}^{2}\\ 1&d_{14}^{2}&d_{24}^{2}&d_{34}^{2}&0\end{vmatrix}
  28. i , j { 1 , 2 , 3 , 4 } i,\,j\in\{1,\,2,\,3,\,4\}
  29. d i j \scriptstyle d_{ij}
  30. volume = ( - a + b + c + d ) ( a - b + c + d ) ( a + b - c + d ) ( a + b + c - d ) 192 u v w \,\text{volume}=\frac{\sqrt{\,(-a+b+c+d)\,(a-b+c+d)\,(a+b-c+d)\,(a+b+c-d)}}{19% 2\,u\,v\,w}
  31. a \displaystyle a
  32. V = d | ( 𝐚 × ( 𝐛 - 𝐜 ) ) | 6 . V=\frac{d|(\mathbf{a}\times\mathbf{(b-c)})|}{6}.
  33. | - 1 cos ( α 12 ) cos ( α 13 ) cos ( α 14 ) cos ( α 12 ) - 1 cos ( α 23 ) cos ( α 24 ) cos ( α 13 ) cos ( α 23 ) - 1 cos ( α 34 ) cos ( α 14 ) cos ( α 24 ) cos ( α 34 ) - 1 | = 0 \begin{vmatrix}-1&\cos{(\alpha_{12})}&\cos{(\alpha_{13})}&\cos{(\alpha_{14})}% \\ \cos{(\alpha_{12})}&-1&\cos{(\alpha_{23})}&\cos{(\alpha_{24})}\\ \cos{(\alpha_{13})}&\cos{(\alpha_{23})}&-1&\cos{(\alpha_{34})}\\ \cos{(\alpha_{14})}&\cos{(\alpha_{24})}&\cos{(\alpha_{34})}&-1\\ \end{vmatrix}=0\,
  34. α i j \alpha_{ij}
  35. 2 2 \scriptstyle 2\sqrt{2}
  36. sin O A B sin O B C sin O C A = sin O A C sin O C B sin O B A . \sin\angle OAB\cdot\sin\angle OBC\cdot\sin\angle OCA=\sin\angle OAC\cdot\sin% \angle OCB\cdot\sin\angle OBA.\,
  37. P A F a + P B F b + P C F c + P D F d 9 V . PA\cdot F_{a}+PB\cdot F_{b}+PC\cdot F_{c}+PD\cdot F_{d}\geq 9V.
  38. P A + P B + P C + P D 3 ( P J + P K + P L + P M ) . PA+PB+PC+PD\geq 3(PJ+PK+PL+PM).
  39. 1 r 1 2 + 1 r 2 2 + 1 r 3 2 + 1 r 4 2 2 r 2 , \frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}+\frac{1}{r_{4}^{2}% }\leq\frac{2}{r^{2}},
  40. arccos ( - 1 3 ) \arccos{\left(-\tfrac{1}{3}\right)}

TeX.html

  1. π \pi
  2. π \pi
  3. e e
  4. n n
  5. 2 n 2^{n}
  6. O ( n 2 ) O(n^{2})
  7. n n

Thales.html

  1. 1 / 3 {1}/{3}
  2. 1 / 3 {1}/{3}

The_Limits_to_Growth.html

  1. ln ( 1 + 0.026 × 418 ) 0.026 95 years \frac{\ln(1+0.026\times 418)}{0.026}\approx\,\text{95 years}
  2. y = ln ( ( r × s ) + 1 ) r y=\frac{\ln((r\times s)+1)}{r}
  3. R = 0 y C e ρ t d t = C ρ ( e ρ y - 1 ) R=\int_{0}^{y}Ce^{\rho t}\ dt=\frac{C}{\rho}\left(e^{\rho y}-1\right)
  4. y = ln ( 1 + ρ R C ) ρ . y=\frac{\ln\left(1+\rho\frac{R}{C}\right)}{\rho}.

The_Number_of_the_Beast_(novel).html

  1. ( 6 6 ) 6 (6^{6})^{6}

Theorem.html

  1. S S
  2. S S
  3. 𝒮 \mathcal{FS}
  4. 𝒮 \mathcal{FS}
  5. 𝒮 \mathcal{FS}
  6. 𝒮 \mathcal{FS}
  7. 𝒮 \mathcal{FS}
  8. 𝒮 . \mathcal{FS}\,.
  9. 𝒮 \mathcal{FS}

Theoretical_ecology.html

  1. d N ( t ) d t = r N ( t ) \frac{dN(t)}{dt}=rN(t)
  2. N ( t ) = N ( 0 ) e r t N(t)=N(0)\ e^{rt}
  3. d N ( t ) d t = ( ( b - a N ( t ) ) - ( d - c N ( t ) ) ) N ( t ) \frac{dN(t)}{dt}=((b-aN(t))-(d-cN(t)))N(t)
  4. d N ( t ) d t = r N ( t ) ( 1 - N K ) \frac{dN(t)}{dt}=rN(t)\left(1-\frac{N}{K}\right)
  5. 𝐍 t + 1 = 𝐋𝐍 t \mathbf{N}_{t+1}=\mathbf{L}\mathbf{N}_{t}
  6. d N ( t ) d t = N ( t ) ( r - α P ( t ) ) \frac{dN(t)}{dt}=N(t)(r-\alpha P(t))
  7. d P ( t ) d t = P ( t ) ( c α N ( t ) - d ) \frac{dP(t)}{dt}=P(t)(c\alpha N(t)-d)
  8. N t + 1 = λ N t [ 1 - f ( N t , P t ) ] N_{t+1}=\lambda\ N_{t}\ [1-f(N_{t},P_{t})]
  9. P t + 1 = c N t f ( N t , p t ) P_{t+1}=c\ N_{t}\ f(N_{t},p_{t})
  10. d N 1 d t = r 1 N 1 K 1 ( K 1 - N 1 + α 12 N 2 ) \frac{dN_{1}}{dt}=\frac{r_{1}N_{1}}{K_{1}}\left(K_{1}-N_{1}+\alpha_{12}N_{2}\right)
  11. d N 2 d t = r 2 N 2 K 2 ( K 2 - N 2 + α 21 N 1 ) \frac{dN_{2}}{dt}=\frac{r_{2}N_{2}}{K_{2}}\left(K_{2}-N_{2}+\alpha_{21}N_{1}\right)
  12. d p d t = m p ( 1 - p ) - e p \frac{dp}{dt}=mp(1-p)-ep
  13. d p 1 d t = m 1 p 1 ( 1 - p 1 ) - e p 1 \frac{dp_{1}}{dt}=m_{1}p_{1}(1-p_{1})-ep_{1}
  14. d p 2 d t = m 2 p 2 ( 1 - p 1 - p 2 ) - e p 2 - m p 1 p 2 \frac{dp_{2}}{dt}=m_{2}p_{2}(1-p_{1}-p_{2})-ep_{2}-mp_{1}p_{2}
  15. p 2 * = e m 1 - m 1 m 2 p^{*}_{2}=\frac{e}{m_{1}}-\frac{m_{1}}{m_{2}}

Theory_of_computation.html

  1. α β \alpha\rightarrow\beta
  2. α A β α γ β \alpha A\beta\rightarrow\alpha\gamma\beta
  3. A γ A\rightarrow\gamma
  4. A a A\rightarrow a
  5. A a B A\rightarrow aB
  6. O ( n ) O(n)
  7. λ \lambda
  8. λ \lambda
  9. f ( x ) f(x)
  10. g ( x ) g(x)
  11. h ( x , y ) h(x,y)
  12. f ( x ) = h ( x , g ( x ) ) f(x)=h(x,g(x))

Thermal_conductivity.html

  1. R h s = Δ T P t h - R s R_{hs}=\frac{\Delta T}{P_{th}}-R_{s}
  2. l l\;
  3. l = V g t l\;=V_{g}t
  4. κ \kappa
  5. Q x = 1 V q , j ω ( n - n 0 ) v x , Q_{x}=\frac{1}{V}\sum_{q,j}{\hslash\omega\left(\left\langle n\right\rangle-{% \left\langle n\right\rangle}^{0}\right)v_{x}}\,\text{,}
  6. d n d t = ( n t ) diff. + ( n t ) decay \frac{d\left\langle n\right\rangle}{dt}={\left(\frac{\partial\left\langle n% \right\rangle}{\partial t}\right)}_{\,\text{diff.}}+{\left(\frac{\partial\left% \langle n\right\rangle}{\partial t}\right)}\text{decay}
  7. ( n t ) decay = - n - n 0 τ , {\left(\frac{\partial\left\langle n\right\rangle}{\partial t}\right)}\text{% decay}=-\,\text{ }\frac{\left\langle n\right\rangle-{\left\langle n\right% \rangle}^{0}}{\tau},
  8. ( ( n ) t ) diff. = - v x ( n ) 0 T T x . {\left(\frac{\partial\left(n\right)}{\partial t}\right)}\text{diff.}=-{v}_{x}% \frac{\partial{\left(n\right)}^{0}}{\partial T}\frac{\partial T}{\partial x}\,% \text{.}
  9. λ L = 1 3 V q , j v ( q , j ) Λ ( q , j ) T ϵ ( ω ( q , j ) , T ) , {\lambda}_{L}=\frac{1}{3V}\sum_{q,j}v\left(q,j\right)\Lambda\left(q,j\right)% \frac{\partial}{\partial T}\epsilon\left(\omega\left(q,j\right),T\right),
  10. T ϵ \frac{\partial}{\partial T}\epsilon
  11. v x 2 = 1 3 v 2 \left\langle v_{x}^{2}\right\rangle=\frac{1}{3}v^{2}
  12. Λ = v τ \Lambda=v\tau
  13. ω 1 = ω 2 + ω 3 \hslash{\omega}_{1}=\hslash{\omega}_{2}+\hslash{\omega}_{3}
  14. q 1 = q 2 + q 3 + G {q}_{1}={q}_{2}+{q}_{3}+G
  15. P e - E / k T P\propto{e}^{-E/kT}
  16. k Θ / 2 \sim k\Theta/2
  17. e - Θ / b T {e}^{-\Theta/bT}
  18. b = 2 b=2
  19. e Θ / b T {e}^{\Theta/bT}
  20. e Θ / b T {e}^{\Theta/bT}
  21. e x x , ( x ) < 1 {e}^{x}\propto x\,\text{ },\,\text{ }\left(x\right)<1
  22. x = Θ / b T x=\Theta/bT
  23. λ E = λ 0 - T σ S 2 {\lambda}_{E}={\lambda}_{0}-T\sigma{S}^{2}
  24. σ = σ 0 I 0 \sigma={\sigma}_{0}{I}_{0}
  25. σ S = ( k e ) σ 0 I 1 \sigma S={\left(\frac{k}{e}\right)\sigma_{0}}{I}_{1}
  26. λ 0 = ( k e ) 2 σ 0 T I 2 {\lambda}_{0}={\left(\frac{k}{e}\right)}^{2}{\sigma}_{0}T{I}_{2}
  27. σ 0 = e 2 / ( a 0 ) {\sigma}_{0}={e}^{2}/\left(\hslash{a}_{0}\right)
  28. I n = - e x ( e x + 1 ) 2 s ( x ) x n d x {I}_{n}=\underset{-\infty}{\overset{\infty}{\int}}\frac{{e}^{x}}{{\left({e}^{x% }+1\right)}^{2}}s\left(x\right){x}^{n}dx
  29. P ( x ) = D ( x ) s ( x ) , D ( x ) = e x ( e x + 1 ) 2 P\left(x\right)=D\left(x\right)s\left(x\right),\,\text{ }D\left(x\right)=\frac% {{e}^{x}}{{\left({e}^{x}+1\right)}^{2}}
  30. λ E = ( k e ) 2 σ 0 T ( I 2 - I 1 2 I 0 ) {\lambda}_{\mathrm{E}}={\left(\frac{k}{e}\right)}^{2}{\sigma}_{0}T\left({I}_{2% }-\frac{{I}_{1}^{2}}{{I}_{0}}\right)
  31. s ( x ) = f ( x ) δ ( x - b ) s\left(x\right)=f\left(x\right)\delta\left(x-b\right)
  32. I 0 = D ( b ) f ( b ) {I}_{0}=D\left(b\right)f\left(b\right)
  33. I 1 = D ( b ) f ( b ) b {I}_{1}=D\left(b\right)f\left(b\right)b
  34. I 2 = D ( b ) f ( b ) b 2 {I}_{2}=D\left(b\right)f\left(b\right){b}^{2}
  35. q = - k T \vec{q}=-k\vec{\nabla}T
  36. q \vec{q}
  37. T \vec{\nabla}T
  38. H = - k A d T d x . H=-kA\frac{\mathrm{d}T}{\mathrm{d}x}.
  39. H L = A T L T H k ( T ) d T HL=A\int_{T_{L}}^{T_{H}}k(T)\mathrm{d}T
  40. I k ( T ) = 0 T k ( T ) d T . I_{k}(T)=\int_{0}^{T}k(T^{\prime})\mathrm{d}T^{\prime}.
  41. H = A L [ I k ( T H ) - I k ( T L ) ] . H=\frac{A}{L}[I_{k}(T_{H})-I_{k}(T_{L})].
  42. H = k A T H - T L L . H=kA\frac{T_{H}-T_{L}}{L}.
  43. H n v c A Δ T . H\propto nvcA\Delta T.
  44. Δ T = l d T d z \Delta T=l\frac{dT}{dz}
  45. H = - 1 3 n v c l A d T d z . H=-\frac{1}{3}nvclA\frac{dT}{dz}.
  46. k = 1 3 n v c l . k=\frac{1}{3}nvcl.
  47. k = 1 3 v l C V V m k=\frac{1}{3}vl\frac{C_{V}}{V_{m}}
  48. l 1 n σ l\propto\frac{1}{n\sigma}
  49. k c σ v . k\propto\frac{c}{\sigma}v.
  50. v = 3 R T M . v=\sqrt{\frac{3RT}{M}}.
  51. k T M . k\propto\sqrt{\frac{T}{M}}.
  52. k = k 0 T (metal at low temperature) k=k_{0}T\,\text{ (metal at low temperature)}

Thermal_diffusivity.html

  1. α = k ρ c p \alpha={k\over{\rho c_{p}}}
  2. k k
  3. ρ \rho
  4. c p c_{p}
  5. ρ c p \rho c_{p}\,
  6. T t = α 2 T \frac{\partial T}{\partial t}=\alpha\nabla^{2}T

Thermal_insulation.html

  1. r c r i t i c a l = k h {r_{critical}}={k\over h}

Thermal_mass.html

  1. Q = C th Δ T Q=C_{\mathrm{th}}\Delta T\,
  2. c ¯ \bar{c}
  3. C t h C_{th}
  4. C t h = m c p C_{th}=mc_{p}
  5. m m
  6. c p c_{p}

Thermistor.html

  1. Δ R = k Δ T \Delta R=k\Delta T\,
  2. Δ R \Delta R
  3. Δ T \Delta T
  4. k k
  5. k k
  6. k k
  7. k k
  8. k k
  9. α T \alpha_{T}
  10. α T = 1 R ( T ) d R d T . \alpha_{T}=\frac{1}{R(T)}\frac{dR}{dT}.
  11. α T \alpha_{T}
  12. a a
  13. 1 T = a + b ln ( R ) + c ( ln ( R ) ) 3 {1\over T}=a+b\,\ln(R)+c\,(\ln(R))^{3}
  14. R = exp [ ( x - 1 2 y ) 1 3 - ( x + 1 2 y ) 1 3 ] R=\mathrm{exp}\left[{{\left(x-{1\over 2}y\right)}^{1\over 3}-{\left(x+{1\over 2% }y\right)}^{1\over 3}}\right]
  15. y = 1 c ( a - 1 T ) x = ( b 3 c ) 3 + ( y 2 ) 2 \begin{aligned}\displaystyle y&\displaystyle={1\over c}\left(a-{1\over T}% \right)\\ \displaystyle x&\displaystyle=\sqrt{\left(\frac{b}{3c}\right)^{3}+\left(\frac{% y}{2}\right)^{2}}\end{aligned}
  16. a \displaystyle a
  17. a = ( 1 / T 0 ) - ( 1 / B ) ln ( R 0 ) a=(1/T_{0})-(1/B)\ln(R_{0})
  18. b = 1 / B b=1/B
  19. c = 0 c=0
  20. 1 T = 1 T 0 + 1 B ln ( R R 0 ) \frac{1}{T}=\frac{1}{T_{0}}+\frac{1}{B}\ln\left(\frac{R}{R_{0}}\right)
  21. R = R 0 e - B ( 1 T 0 - 1 T ) R=R_{0}e^{-B\left(\frac{1}{T_{0}}-\frac{1}{T}\right)}
  22. R = r e B / T R=r_{\infty}e^{B/T}
  23. r = R 0 e - B / T 0 r_{\infty}=R_{0}e^{-{B/T_{0}}}
  24. T = B ln ( R / r ) T={B\over{{\ln{(R/r_{\infty})}}}}
  25. ln R = B / T + ln r \ln R=B/T+\ln r_{\infty}
  26. ln R \ln R
  27. 1 / T 1/T
  28. I = n A v e I=n\cdot A\cdot v\cdot e
  29. I I
  30. n n
  31. A A
  32. v v
  33. e e
  34. e = 1.602 × 10 - 19 e=1.602\times 10^{-19}
  35. P E = I V P_{E}=IV\,
  36. P T = K ( T ( R ) - T 0 ) P_{T}=K(T(R)-T_{0})\,
  37. T 0 T_{0}
  38. P E = P T P_{E}=P_{T}\,
  39. I = V / R I=V/R
  40. T 0 = T ( R ) - V 2 K R T_{0}=T(R)-\frac{V^{2}}{KR}\,

Thermocouple.html

  1. V \scriptstyle V
  2. T sense \scriptstyle T_{\mathrm{sense}}
  3. T ref \scriptstyle T_{\mathrm{ref}}
  4. s y m b o l V \scriptstyle symbol\nabla V
  5. s y m b o l T \scriptstyle symbol\nabla T
  6. s y m b o l V = - S ( T ) s y m b o l T , symbol\nabla V=-S(T)symbol\nabla T,
  7. S ( T ) S(T)
  8. T meter \scriptstyle T_{\mathrm{meter}}
  9. T ref \scriptstyle T_{\mathrm{ref}}
  10. T ref \scriptstyle T_{\mathrm{ref}}
  11. T sense \scriptstyle T_{\mathrm{sense}}
  12. T sense \scriptstyle T_{\mathrm{sense}}
  13. T ref \scriptstyle T_{\mathrm{ref}}
  14. T ref \scriptstyle T_{\mathrm{ref}}
  15. T meter \scriptstyle T_{\mathrm{meter}}
  16. T meter \scriptstyle T_{\mathrm{meter}}
  17. V = T ref T sense ( S + ( T ) - S - ( T ) ) d T , V=\int_{T_{\mathrm{ref}}}^{T_{\mathrm{sense}}}\left(S_{+}(T)-S_{-}(T)\right)\,dT,
  18. S + \scriptstyle S_{+}
  19. S - \scriptstyle S_{-}
  20. E ( T ) \scriptstyle E(T)
  21. V = E ( T sense ) - E ( T ref ) . V=E(T_{\mathrm{sense}})-E(T_{\mathrm{ref}}).
  22. E ( T ) = T S + ( T ) - S - ( T ) d T + const E(T)=\int^{T}S_{+}(T^{\prime})-S_{-}(T^{\prime})dT^{\prime}+\mathrm{const}
  23. E ( 0 C ) = 0 \scriptstyle E(0\,{}^{\circ}{\rm C})=0
  24. E ( T ) \scriptstyle E(T)
  25. T sense \scriptstyle T_{\mathrm{sense}}
  26. V \scriptstyle V
  27. T ref \scriptstyle T_{\mathrm{ref}}
  28. V V
  29. T ref \scriptstyle T_{\mathrm{ref}}
  30. V + E ( T ref ) \scriptstyle V+E(T_{\mathrm{ref}})
  31. E ( T ) \scriptstyle E(T)
  32. T sense \scriptstyle T_{\mathrm{sense}}
  33. E ( T ) \scriptstyle E(T)
  34. V + E ( T ref ) \scriptstyle V+E(T_{\mathrm{ref}})
  35. E ( T ) \scriptstyle E(T)
  36. E ( T ) \scriptstyle E(T)
  37. T h > T c T_{\rm h}>T_{\rm c}
  38. P = B ( V 2 - V 0 2 ) V 0 2 P=\frac{B(V^{2}-V_{0}^{2})}{V_{0}^{2}}

Thermodynamic_free_energy.html

  1. f d x f\cdot dx
  2. d F = - p d V - S d T + i μ i d N i \mathrm{d}F=-p\,\mathrm{d}V-S\mathrm{d}T+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  3. d G = V d p - S d T + i μ i d N i \mathrm{d}G=V\mathrm{d}p-S\mathrm{d}T+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  4. ( d G ) T , p = i μ i d N i (\mathrm{d}G)_{T,p}=\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  5. d W d Q dW\propto dQ

Thermodynamic_temperature.html

  1. k B T / 2 k_{B}T/2
  2. k B k_{B}
  3. 1 / 2 {1}/{2}
  4. E ¯ = 3 2 k B T k \bar{E}\,=\,\frac{3}{2}k_{B}T_{k}
  5. E ¯ \scriptstyle\bar{E}
  6. T k T_{k}
  7. v ¯ = k B T m \bar{v}=\sqrt{\frac{k_{B}T}{m}}
  8. v ¯ \bar{v}
  9. s ¯ = v ¯ 3 \bar{s}=\bar{v}\sqrt{3}
  10. s ¯ \bar{s}
  11. s ¯ \bar{s}
  12. 1 / 2 {1}/{2}
  13. 1 / 1836 {1}/{1836}
  14. 316 K / 296 K {316K}/{296K}
  15. Efficiency = w c y q H = q H - q C q H = 1 - q C q H ( 1 ) \textrm{Efficiency}=\frac{w_{cy}}{q_{H}}=\frac{q_{H}-q_{C}}{q_{H}}=1-\frac{q_{% C}}{q_{H}}\qquad(1)
  16. q C q H = f ( T H , T C ) ( 2 ) . \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})\qquad(2).
  17. f ( T 1 , T 3 ) = q 3 q 1 = q 2 q 3 q 1 q 2 = f ( T 1 , T 2 ) f ( T 2 , T 3 ) . f(T_{1},T_{3})=\frac{q_{3}}{q_{1}}=\frac{q_{2}q_{3}}{q_{1}q_{2}}=f(T_{1},T_{2}% )f(T_{2},T_{3}).
  18. f ( T 1 , T 2 ) = g ( T 2 ) g ( T 1 ) . f(T_{1},T_{2})=\frac{g(T_{2})}{g(T_{1})}.
  19. f ( T 2 , T 3 ) = g ( T 3 ) g ( T 2 ) . f(T_{2},T_{3})=\frac{g(T_{3})}{g(T_{2})}.
  20. f ( T 1 , T 3 ) = g ( T 3 ) g ( T 1 ) = q 3 q 1 . f(T_{1},T_{3})=\frac{g(T_{3})}{g(T_{1})}=\frac{q_{3}}{q_{1}}.
  21. g ( T ) g(T)
  22. g ( T ) = T g(T)=T
  23. q C q H = f ( T H , T C ) = T C T H . ( 3 ) . \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})=\frac{T_{C}}{T_{H}}.\qquad(3).
  24. Efficiency = 1 - q C q H = 1 - T C T H ( 4 ) . \textrm{Efficiency}=1-\frac{q_{C}}{q_{H}}=1-\frac{T_{C}}{T_{H}}\qquad(4).
  25. d S = d q rev T ( 5 ) , dS=\frac{dq_{\mathrm{rev}}}{T}\qquad(5),
  26. T = d q rev d S . T=\frac{dq_{\mathrm{rev}}}{dS}.
  27. 1 T = d S d E , \frac{1}{T}=\frac{dS}{dE},

Thermodynamics.html

  1. T T
  2. S S
  3. p p
  4. V V
  5. μ \mu
  6. N N
  7. i i

Thermometer.html

  1. M M
  2. L L
  3. M M
  4. M M

Thermosphere.html

  1. T = T - ( T - T 0 ) exp { - s ( z - z 0 ) } T=T_{\infty}-(T_{\infty}-T_{0})\exp\{-s(z-z_{0})\}
  2. T 500 + 3.4 F 0 T_{\infty}\simeq 500+3.4F_{0}
  3. T ( φ , λ , t ) = T { 1 + Δ T 2 0 P 2 0 ( φ ) + Δ T 1 0 P 1 0 ( φ ) cos [ ω a ( t - t a ) ] + Δ T 1 1 P 1 1 ( φ ) cos ( τ - τ d ) + } T(\varphi,\lambda,t)=T_{\infty}\{1+\Delta T_{2}^{0}P_{2}^{0}(\varphi)+\Delta T% _{1}^{0}P_{1}^{0}(\varphi)\cos[\omega_{a}(t-t_{a})]+\Delta T_{1}^{1}P_{1}^{1}(% \varphi)\cos(\tau-\tau_{d})+\dots\}

Third-order_intercept_point.html

  1. s ( t ) = V cos ( ω t ) \ s(t)=V\cos(\omega t)
  2. O [ s ( t ) ] = G s ( t ) - D 3 s 3 ( t ) + \ O[s(t)]=Gs(t)-D_{3}s^{3}(t)+\ldots
  3. cos 3 ( x ) = 3 4 cos ( x ) + 1 4 cos ( 3 x ) \ \cos^{3}(x)=\frac{3}{4}\cos(x)+\frac{1}{4}\cos(3x)
  4. O [ s ( t ) ] = ( G V - 3 4 D 3 V 3 ) cos ( ω t ) - ( D 3 V 3 4 ) cos ( 3 ω t ) \ O[s(t)]=(GV-\frac{3}{4}D_{3}V^{3})\cos(\omega t)-(D_{3}\frac{V^{3}}{4})\cos(% 3\omega t)
  5. V 2 = 4 G 3 D 3 \ V^{2}=\frac{4G}{3D_{3}}
  6. 1.122 ( G V - 3 4 D 3 V 3 ) = G V \ 1.122(GV-\frac{3}{4}D_{3}V^{3})=GV
  7. V 2 = 0.10875 × 4 G 3 D 3 \ V^{2}=0.10875\times\frac{4G}{3D_{3}}
  8. V 2 = 0.10875 × TOI \ V^{2}=0.10875\times\mathrm{TOI}

Thorium.html

  1. Th 90 232 + n Th 90 233 + γ β - Pa 91 233 β - U 92 233 {}_{\ 90}^{232}\mathrm{Th}+\mathrm{n}\rightarrow{}_{\ 90}^{233}\mathrm{Th}+% \gamma\ \xrightarrow{\beta^{-}}\ {}_{\ 91}^{233}\mathrm{Pa}\ \xrightarrow{% \beta^{-}}\ {}_{\ 92}^{233}\mathrm{U}

Three-phase_electric_power.html

  1. 2 π / 3 {2π}/{3}
  2. ( 2 / 3 ) × 87 % (2/3)×87\%
  3. V 1 = V LN 0 , V_{1}=V\text{LN}\angle 0^{\circ},
  4. V 2 = V LN - 120 , V_{2}=V\text{LN}\angle{-120}^{\circ},
  5. V 3 = V LN + 120 . V_{3}=V\text{LN}\angle{+120}^{\circ}.
  6. I 1 = V 1 | Z total | ( - θ ) , I_{1}=\frac{V_{1}}{|Z\text{total}|}\angle(-\theta),
  7. I 2 = V 2 | Z total | ( - 120 - θ ) , I_{2}=\frac{V_{2}}{|Z\text{total}|}\angle(-120^{\circ}-\theta),
  8. I 3 = V 3 | Z total | ( 120 - θ ) , I_{3}=\frac{V_{3}}{|Z\text{total}|}\angle(120^{\circ}-\theta),
  9. I 1 + I 2 + I 3 = I N = 0. I_{1}+I_{2}+I_{3}=I\text{N}=0.
  10. V 12 = V 1 - V 2 = ( V LN 0 ) - ( V LN - 120 ) = 3 V LN 30 = 3 V 1 ( ϕ V 1 + 30 ) , V 23 = V 2 - V 3 = ( V LN - 120 ) - ( V LN 120 ) = 3 V LN - 90 = 3 V 2 ( ϕ V 2 + 30 ) , V 31 = V 3 - V 1 = ( V LN 120 ) - ( V LN 0 ) = 3 V LN 150 = 3 V 3 ( ϕ V 3 + 30 ) . \begin{aligned}\displaystyle V_{12}&\displaystyle=V_{1}-V_{2}=(V\text{LN}% \angle 0^{\circ})-(V\text{LN}\angle{-120}^{\circ})\\ &\displaystyle=\sqrt{3}V\text{LN}\angle 30^{\circ}=\sqrt{3}V_{1}\angle(\phi_{V% _{1}}+30^{\circ}),\\ \displaystyle V_{23}&\displaystyle=V_{2}-V_{3}=(V\text{LN}\angle{-120}^{\circ}% )-(V\text{LN}\angle 120^{\circ})\\ &\displaystyle=\sqrt{3}V\text{LN}\angle{-90}^{\circ}=\sqrt{3}V_{2}\angle(\phi_% {V_{2}}+30^{\circ}),\\ \displaystyle V_{31}&\displaystyle=V_{3}-V_{1}=(V\text{LN}\angle 120^{\circ})-% (V\text{LN}\angle 0^{\circ})\\ &\displaystyle=\sqrt{3}V\text{LN}\angle 150^{\circ}=\sqrt{3}V_{3}\angle(\phi_{% V_{3}}+30^{\circ}).\\ \end{aligned}
  11. I 12 = V 12 | Z Δ | ( 30 - θ ) , I_{12}=\frac{V_{12}}{|Z_{\Delta}|}\angle(30^{\circ}-\theta),
  12. I 23 = V 23 | Z Δ | ( - 90 - θ ) , I_{23}=\frac{V_{23}}{|Z_{\Delta}|}\angle(-90^{\circ}-\theta),
  13. I 31 = V 31 | Z Δ | ( 150 - θ ) , I_{31}=\frac{V_{31}}{|Z_{\Delta}|}\angle(150^{\circ}-\theta),
  14. I 1 \displaystyle I_{1}
  15. I 2 = 3 I 23 ( ϕ I 23 - 30 ) = 3 I 23 ( - 120 - θ ) , I_{2}=\sqrt{3}I_{23}\angle(\phi_{I_{23}}-30^{\circ})=\sqrt{3}I_{23}\angle(-120% ^{\circ}-\theta),
  16. I 3 = 3 I 31 ( ϕ I 31 - 30 ) = 3 I 31 ( 120 - θ ) , I_{3}=\sqrt{3}I_{31}\angle(\phi_{I_{31}}-30^{\circ})=\sqrt{3}I_{31}\angle(120^% {\circ}-\theta),
  17. 3 \sqrt{3}
  18. V LL = 3 V LN . V\text{LL}=\sqrt{3}V\text{LN}.
  19. V < s u b > L L = 2 V L N V<sub>LL=2V_{LN}

Throughput.html

  1. λ λ
  2. μ μ
  3. F 3 d B K / T r \ F_{3dB}\approx K/T_{r}

Thrust.html

  1. 𝐓 = 𝐯 d m d t \mathbf{T}=\mathbf{v}\frac{dm}{dt}
  2. d m d t \frac{dm}{dt}
  3. 𝐏 2 𝐓 3 \mathbf{P}^{2}\propto\mathbf{T}^{3}
  4. d m d t = ρ A v \frac{dm}{dt}=\rho A{v}
  5. 𝐓 = d m d t v , 𝐏 = 1 2 d m d t v 2 \mathbf{T}=\frac{dm}{dt}{v},\mathbf{P}=\frac{1}{2}\frac{dm}{dt}{v}^{2}
  6. 𝐓 = ρ A v 2 , 𝐏 = 1 2 ρ A v 3 \mathbf{T}=\rho A{v}^{2},\mathbf{P}=\frac{1}{2}\rho A{v}^{3}
  7. 𝐏 2 = 𝐓 3 4 ρ A \mathbf{P}^{2}=\frac{\mathbf{T}^{3}}{4\rho A}
  8. A A
  9. ρ \rho
  10. 𝐏 = 𝐅 d t \mathbf{P}=\mathbf{F}\frac{d}{t}
  11. 𝐏 = 𝐓 v \mathbf{P}=\mathbf{T}{v}

TI-89_series.html

  1. x 2 - 4 x + 4 x^{2}-4x+4
  2. x 2 - 4 x + 4 x^{2}-4x+4
  3. 3 2 \frac{\sqrt{3}}{2}
  4. π \pi

Tidal_force.html

  1. F g \vec{F}_{g}
  2. F g = - r ^ G M m R 2 \vec{F}_{g}=-\hat{r}~{}G~{}\frac{Mm}{R^{2}}
  3. a g \vec{a}_{g}
  4. a g = - r ^ G M R 2 \vec{a}_{g}=-\hat{r}~{}G~{}\frac{M}{R^{2}}
  5. r ^ \hat{r}
  6. a g = - r ^ G M ( R ± Δ r ) 2 \vec{a}_{g}=-\hat{r}~{}G~{}\frac{M}{(R\pm\Delta r)^{2}}
  7. a g = - r ^ G M R 2 1 ( 1 ± Δ r / R ) 2 \vec{a}_{g}=-\hat{r}~{}G~{}\frac{M}{R^{2}}~{}\frac{1}{(1\pm\Delta r/R)^{2}}
  8. 1 / ( 1 ± x ) 2 1/(1\pm x)^{2}
  9. 1 2 x + 3 x 2 1\mp 2x+3x^{2}\mp\cdots
  10. a g = - r ^ G M R 2 ± r ^ G 2 M R 2 Δ r R + \vec{a}_{g}=-\hat{r}~{}G~{}\frac{M}{R^{2}}\pm\hat{r}~{}G~{}\frac{2M}{R^{2}}~{}% \frac{\Delta r}{R}+\cdots
  11. m m
  12. Δ r \Delta r
  13. a t \vec{a}_{t}
  14. a t \vec{a}_{t}
  15. ± r ^ 2 Δ r G M R 3 ~{}\approx~{}\pm~{}\hat{r}~{}2\Delta r~{}G~{}\frac{M}{R^{3}}
  16. a t \vec{a}_{t}
  17. | a t |\vec{a}_{t}
  18. | / 2 |/2

Tidal_locking.html

  1. t lock w a 6 I Q 3 G m p 2 k 2 R 5 t_{\,\text{lock}}\approx\frac{wa^{6}IQ}{3Gm_{p}^{2}k_{2}R^{5}}
  2. w w\,
  3. a a\,
  4. I I\,
  5. 0.4 m s R 2 \approx 0.4m_{s}R^{2}
  6. Q Q\,
  7. G G\,
  8. m p m_{p}\,
  9. m s m_{s}\,
  10. k 2 k_{2}\,
  11. R R\,
  12. Q Q
  13. k 2 k_{2}
  14. k 2 / Q = 0.0011 k_{2}/Q=0.0011
  15. Q 100 Q\approx 100
  16. k 2 1.5 1 + 19 μ 2 ρ g R , k_{2}\approx\frac{1.5}{1+\frac{19\mu}{2\rho gR}},
  17. ρ \rho\,
  18. g G m s / R 2 g\approx Gm_{s}/R^{2}
  19. μ \mu\,
  20. × 10 1 0 \times 10^{1}0
  21. × 10 9 \times 10^{9}
  22. μ \mu\,
  23. a a
  24. k 2 1 , Q = 100 k_{2}\ll 1\,,Q=100
  25. t lock 6 a 6 R μ m s m p 2 × 10 10 years , t_{\,\text{lock}}\approx 6\ \frac{a^{6}R\mu}{m_{s}m_{p}^{2}}\times 10^{10}\ \,% \text{years},
  26. μ \mu
  27. μ \mu
  28. × 10 1 0 \times 10^{1}0
  29. × 10 9 \times 10^{9}
  30. a a
  31. Q Q
  32. μ \mu
  33. m s m_{s}\,
  34. R R
  35. k 2 / Q k_{2}/Q

Tide.html

  1. 71 / 2 7{1}/{2}
  2. A cos ( ω t + p ) A\cos\,(\omega t+p)
  3. A ( t ) = A ( 1 + A a cos ( ω a t + p a ) ) A(t)=A\bigl(1+A_{a}\cos\,(\omega_{a}t+p_{a})\bigr)
  4. A [ 1 + A a cos ( ω a t + p a ) ] cos ( ω t + p ) A\bigl[1+A_{a}\cos\,(\omega_{a}t+p_{a})\bigr]\cos\,(\omega t+p)
  5. cos x cos y = 1 2 cos ( x + y ) + 1 2 cos ( x - y ) \cos x\cos y={\textstyle\frac{1}{2}}\cos\,(x+y)+{\textstyle\frac{1}{2}}\cos\,(% x-y)
  6. ( 1 + cos x ) cos y (1+\cos x)\cos y
  7. 11 / 4 1{1}/{4}
  8. 121 / 2 12{1}/{2}
  9. 13 / 4 1{3}/{4}

Tietze_extension_theorem.html

  1. f : A f:A\to\mathbb{R}
  2. F : X F:X\to\mathbb{R}
  3. sup { | f ( a ) | : a A } = sup { | F ( x ) | : x X } \sup\{|f(a)|:a\in A\}=\sup\{|F(x)|:x\in X\}

Time-domain_reflectometer.html

  1. ρ = Z t - Z o Z t + Z o \rho=\frac{Z_{t}-Z_{o}}{Z_{t}+Z_{o}}

Timeline_of_geology.html

  1. M L M_{L}
  2. M 0 M_{0}
  3. M W M_{W}

Timeline_of_knowledge_about_galaxies,_clusters_of_galaxies,_and_large-scale_structure.html

  1. D n D_{n}
  2. r g r_{g}

Tire.html

  1. v x v_{x}
  2. v y v_{y}

Titius–Bode_law.html

  1. a a
  2. a = 4 + n a=4+n
  3. n = 0 , 3 , 6 , 12 , 24 , 48... n=0,3,6,12,24,48...
  4. a = 1.5 × 2 ( n - 1 ) + 4 a=1.5\times 2^{(n-1)}+4
  5. n = - , 2 , 3 , 4... n=-\infty,2,3,4...
  6. a = 0.4 + 0.3 × 2 m a=0.4+0.3\times 2^{m}
  7. m = - , 0 , 1 , 2... m=-\infty,0,1,2...

Titration.html

  1. 𝐂 a = 𝐂 t 𝐕 t 𝐌 𝐕 a \mathbf{C}_{a}=\frac{\mathbf{C}_{t}\mathbf{V}_{t}\mathbf{M}}{\mathbf{V}_{a}}

Topological_group.html

  1. G × G G : ( x , y ) x y G\times G\to G:(x,y)\mapsto xy
  2. G G : x x - 1 G\to G:x\mapsto x^{-1}
  3. * G *\to G
  4. \to
  5. x x c x\mapsto xc
  6. x c x x\mapsto cx

Torque.html

  1. τ \tau
  2. s y m b o l τ = 𝐫 × 𝐅 symbol\tau=\mathbf{r}\times\mathbf{F}\,\!
  3. τ = 𝐫 𝐅 sin θ \tau=\|\mathbf{r}\|\,\|\mathbf{F}\|\sin\theta\,\!
  4. s y m b o l τ symbol\tau
  5. τ \tau
  6. s y m b o l τ = 𝐫 × 𝐅 , symbol{\tau}=\mathbf{r}\times\mathbf{F},
  7. τ = r F sin θ , \tau=rF\sin\theta,\!
  8. τ = r F , \tau=rF_{\perp},
  9. s y m b o l τ = d 𝐋 d t symbol{\tau}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}
  10. s y m b o l τ 1 + + s y m b o l τ n = s y m b o l τ net = d 𝐋 d t . symbol{\tau}_{1}+\cdots+symbol{\tau}_{n}=symbol{\tau}_{\mathrm{net}}=\frac{% \mathrm{d}\mathbf{L}}{\mathrm{d}t}.
  11. 𝐋 = I s y m b o l ω , \mathbf{L}=Isymbol{\omega},
  12. I I
  13. s y m b o l τ net = d 𝐋 d t = d ( I s y m b o l ω ) d t = I d s y m b o l ω d t = I s y m b o l α , symbol{\tau}_{\mathrm{net}}=\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}=\frac{% \mathrm{d}(Isymbol{\omega})}{\mathrm{d}t}=I\frac{\mathrm{d}symbol{\omega}}{% \mathrm{d}t}=Isymbol{\alpha},
  14. I I
  15. 𝐋 = 𝐫 × s y m b o l p \mathbf{L}=\mathbf{r}\times symbol{p}
  16. d 𝐋 d t = 𝐫 × d s y m b o l p d t + d 𝐫 d t × s y m b o l p . \frac{d\mathbf{L}}{dt}=\mathbf{r}\times\frac{dsymbol{p}}{dt}+\frac{d\mathbf{r}% }{dt}\times symbol{p}.
  17. 𝐅 = d s y m b o l p d t \mathbf{F}=\frac{dsymbol{p}}{dt}
  18. d 𝐫 d t = 𝐯 \frac{d\mathbf{r}}{dt}=\mathbf{v}
  19. d 𝐋 d t = 𝐫 × 𝐅 + 𝐯 × s y m b o l p . \frac{d\mathbf{L}}{dt}=\mathbf{r}\times\mathbf{F}+\mathbf{v}\times symbol{p}.
  20. s y m b o l p symbol{p}
  21. 𝐯 \mathbf{v}
  22. d 𝐋 d t = 𝐫 × 𝐅 net = s y m b o l τ net . \frac{d\mathbf{L}}{dt}=\mathbf{r}\times\mathbf{F}_{\mathrm{net}}=symbol{\tau}_% {\mathrm{net}}.
  23. E = τ θ E=\tau\theta
  24. τ = ( moment arm ) ( force ) . \tau=(\,\text{moment arm})(\,\text{force}).
  25. τ = ( distance to centre ) ( force ) . \tau=(\,\text{distance to centre})(\,\text{force}).
  26. 𝐅 \mathbf{F}
  27. s y m b o l τ 1 symbol{\tau}_{1}
  28. 𝐫 1 \mathbf{r}_{1}
  29. 𝐫 2 \mathbf{r}_{2}
  30. s y m b o l τ 2 = s y m b o l τ 1 + ( 𝐫 1 - 𝐫 2 ) × 𝐅 symbol{\tau}_{2}=symbol{\tau}_{1}+(\mathbf{r}_{1}-\mathbf{r}_{2})\times\mathbf% {F}
  31. W = θ 1 θ 2 τ d θ , W=\int_{\theta_{1}}^{\theta_{2}}\tau\ \mathrm{d}\theta,
  32. E r = 1 2 I ω 2 , E_{\mathrm{r}}=\tfrac{1}{2}I\omega^{2},
  33. P = s y m b o l τ s y m b o l ω , P=symbol{\tau}\cdot symbol{\omega},
  34. P = τ × 2 π × ω P=\tau\times 2\pi\times\omega
  35. P / W = τ / ( N m ) × 2 π ( rad / rev ) × ω / ( rev / sec ) P/{\rm W}=\tau/{\rm(N\cdot m)}\times 2\pi{\rm(rad/rev)}\times\omega/{\rm(rev/% sec)}
  36. P / W = τ / ( N m ) × 2 π ( rad / rev ) × ω / ( rpm ) 60 P/{\rm W}=\frac{\tau/{\rm(N\cdot m)}\times 2\pi{\rm(rad/rev)}\times\omega/{\rm% (rpm)}}{60}
  37. P / hp = τ / ( lbf ft ) × 2 π ( rad / rev ) × ω / rpm 33 , 000 . P/{\rm hp}=\frac{\tau/{\rm(lbf\cdot ft)}\times 2\pi{\rm(rad/rev)}\times\omega/% {\rm rpm}}{33,000}.
  38. power = force × linear distance time = ( torque r ) × ( r × angular speed × t ) t = torque × angular speed . \mbox{power}~{}=\frac{\mbox{force}~{}\times\mbox{linear distance}~{}}{\mbox{% time}~{}}=\frac{\left(\frac{\mbox{torque}~{}}{\displaystyle{r}}\right)\times(r% \times\mbox{angular speed}~{}\times t)}{t}=\mbox{torque}~{}\times\mbox{angular% speed}~{}.
  39. power = torque × 2 π × rotational speed . \mbox{power}~{}=\mbox{torque}~{}\times 2\pi\times\mbox{rotational speed}~{}.\,
  40. power = torque × 2 π × rotational speed ft lbf min × horsepower 33 , 000 ft lbf min torque × RPM 5 , 252 \mbox{power}~{}=\mbox{torque }~{}\times\ 2\pi\ \times\mbox{ rotational speed}~% {}\cdot\frac{\mbox{ft}~{}\cdot\mbox{lbf}~{}}{\mbox{min}~{}}\times\frac{\mbox{% horsepower}~{}}{33,000\cdot\frac{\mbox{ft }~{}\cdot\mbox{ lbf}~{}}{\mbox{min}~% {}}}\approx\frac{\mbox{torque}~{}\times\mbox{RPM}~{}}{5,252}
  41. 5252.113122 33 , 000 2 π . 5252.113122\approx\frac{33,000}{2\pi}.\,
  42. ( 𝐫 × 𝐅 1 ) + ( 𝐫 × 𝐅 2 ) + = 𝐫 × ( 𝐅 1 + 𝐅 2 + ) . (\mathbf{r}\times\mathbf{F}_{1})+(\mathbf{r}\times\mathbf{F}_{2})+\cdots=% \mathbf{r}\times(\mathbf{F}_{1}+\mathbf{F}_{2}+\cdots).

Total_harmonic_distortion.html

  1. THD F = V 2 2 + V 3 2 + V 4 2 + V 1 \mathrm{THD_{F}}\,=\,\frac{\sqrt{V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots}}{V_{1}}
  2. THD R = V 2 2 + V 3 2 + V 4 2 + V 1 2 + V 2 2 + V 3 2 + = THD F 1 + THD F 2 \mathrm{THD_{R}}\,=\,\frac{\sqrt{V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdots}}{\sqrt{% V_{1}^{2}+V_{2}^{2}+V_{3}^{2}+\cdots}}\,=\,\frac{\mathrm{THD_{F}}}{\sqrt{1+% \mathrm{THD}^{2}_{\mathrm{F}}}}
  3. THD + N = n = 2 harmonics + noise fundamental \mathrm{THD\!\!+\!\!N}=\frac{\displaystyle\sum_{n=2}^{\infty}{\,\text{% harmonics}}+\,\text{noise}}{\,\text{fundamental}}
  4. THD F = π 2 8 - 1 0.483 = 48.3 % \mathrm{THD_{F}}\,=\,\sqrt{\frac{\,\pi^{2}}{8}-1\,}\approx\,0.483\,=\,48.3\%
  5. THD F = π 2 6 - 1 0.803 = 80.3 % \mathrm{THD_{F}}\,=\,\sqrt{\frac{\,\pi^{2}}{6}-1\,}\approx\,0.803\,=\,80.3\%
  6. THD F = π 4 96 - 1 0.121 = 12.1 % \mathrm{THD_{F}}\,=\,\sqrt{\frac{\,\pi^{4}}{96}-1\,}\approx\,0.121\,=\,12.1\%
  7. THD F ( μ ) = μ ( 1 - μ ) π 2 2 sin 2 π μ - 1 , 0 < μ < 1 \mathrm{THD_{F}}\,(\mu)=\sqrt{\frac{\mu(1-\mu)\pi^{2}\,}{2\sin^{2}\pi\mu}-1\;}% \,,\qquad 0<\mu<1
  8. THD F = π 2 3 - π coth π 0.370 = 37.0 % \mathrm{THD_{F}}\,=\,\sqrt{\frac{\,\pi^{2}}{3}-\pi\coth\pi\,}\,\approx\,0.370% \,=\,37.0\%
  9. THD F = π cot π 2 coth 2 π 2 - cot 2 π 2 coth π 2 - cot π 2 - coth π 2 2 ( cot 2 π 2 + coth 2 π 2 ) + π 2 3 - 1 0.181 = 18.1 % \mathrm{THD_{F}}\,=\sqrt{\pi\,\frac{\;\cot\dfrac{\pi}{\sqrt{2\,}}\cdot\coth^{2% \!}\dfrac{\pi}{\sqrt{2\,}}-\cot^{2\!}\dfrac{\pi}{\sqrt{2\,}}\cdot\coth\dfrac{% \pi}{\sqrt{2\,}}-\cot\dfrac{\pi}{\sqrt{2\,}}-\coth\dfrac{\pi}{\sqrt{2\,}}\;}{% \sqrt{2\,}\left(\!\cot^{2\!}\dfrac{\pi}{\sqrt{2\,}}+\coth^{2\!}\dfrac{\pi}{% \sqrt{2\,}}\!\right)}\,+\,\frac{\,\pi^{2}}{3}\,-\,1\;}\;\approx\;0.181\,=\,18.1\%
  10. THD F ( μ , p ) = csc π μ μ ( 1 - μ ) π 2 - sin 2 π μ - π 2 s = 1 2 p cot π z s z s 2 l = 1 l s 2 p 1 z s - z l + π 2 Re s = 1 2 p e i π z s ( 2 μ - 1 ) z s 2 sin π z s l = 1 l s 2 p 1 z s - z l \mathrm{THD_{F}}\,(\mu,p)=\csc\pi\mu\,\cdot\!\sqrt{\mu(1-\mu)\pi^{2}-\,\sin^{2% }\!\pi\mu\,-\,\frac{\,\pi}{2}\sum_{s=1}^{2p}\frac{\cot\pi z_{s}}{z_{s}^{2}}% \prod\limits_{\scriptstyle l=1\atop\scriptstyle l\neq s}^{2p}\!\frac{1}{\,z_{s% }-z_{l}\,}\,+\,\frac{\,\pi}{2}\,\mathrm{Re}\sum_{s=1}^{2p}\frac{e^{i\pi z_{s}(% 2\mu-1)}}{z_{s}^{2}\sin\pi z_{s}}\prod\limits_{\scriptstyle l=1\atop% \scriptstyle l\neq s}^{2p}\!\frac{1}{\,z_{s}-z_{l}\,}\,}

Total_internal_reflection.html

  1. θ c \theta_{c}
  2. n 1 sin θ i = n 2 sin θ t n_{1}\sin\theta_{i}=n_{2}\sin\theta_{t}\quad
  3. sin θ i = n 2 n 1 sin θ t \sin\theta_{i}=\frac{n_{2}}{n_{1}}\sin\theta_{t}
  4. θ i \theta_{i}
  5. θ t = \theta_{t}=
  6. sin θ t = 1 \sin\theta_{t}=1
  7. θ i \theta_{i}
  8. θ c \theta_{c}
  9. θ i \theta_{i}
  10. θ c = θ i = arcsin ( n 2 n 1 ) , \theta_{c}=\theta_{i}=\arcsin\left(\frac{n_{2}}{n_{1}}\right),
  11. θ c = arcsin ( 1.00 1.50 ) = 41.8 \theta_{c}=\arcsin\left(\frac{1.00}{1.50}\right)=41.8{}^{\circ}
  12. n 2 / n 1 {n_{2}}/{n_{1}}
  13. n 2 / n 1 {n_{2}}/{n_{1}}
  14. θ I \theta_{I}
  15. 𝐤 𝐈 \mathbf{k_{I}}
  16. 𝐤 𝐓 = k T sin ( θ T ) x ^ + k T cos ( θ T ) z ^ \mathbf{k_{T}}=k_{T}\sin(\theta_{T})\hat{x}+k_{T}\cos(\theta_{T})\hat{z}
  17. n 1 > n 2 n_{1}>n_{2}
  18. sin ( θ T ) > 1 \sin(\theta_{T})>1
  19. sin ( θ T ) = n 1 n 2 sin ( θ I ) \sin(\theta_{T})=\frac{n_{1}}{n_{2}}\sin(\theta_{I})
  20. n 1 n 2 sin ( θ I ) \frac{n_{1}}{n_{2}}\sin(\theta_{I})
  21. θ I > θ C \theta_{I}>\theta_{C}
  22. cos ( θ T ) \cos(\theta_{T})
  23. cos ( θ T ) = 1 - sin 2 ( θ T ) = i sin 2 ( θ T ) - 1 \cos(\theta_{T})=\sqrt{1-\sin^{2}(\theta_{T})}=i\sqrt{\sin^{2}(\theta_{T})-1}
  24. 𝐄 𝐓 = 𝐄 𝟎 e i ( 𝐤 𝐓 𝐫 - ω t ) \mathbf{E_{T}}=\mathbf{E_{0}}e^{i(\mathbf{k_{T}}\cdot\mathbf{r}-\omega t)}
  25. 𝐄 𝐓 = 𝐄 𝟎 e i ( 𝐤 𝐓 𝐫 - ω t ) = 𝐄 𝟎 e i ( x k T sin ( θ T ) + z k T cos ( θ T ) - ω t ) \mathbf{E_{T}}=\mathbf{E_{0}}e^{i(\mathbf{k_{T}}\cdot\mathbf{r}-\omega t)}=% \mathbf{E_{0}}e^{i(xk_{T}\sin(\theta_{T})+zk_{T}\cos(\theta_{T})-\omega t)}
  26. 𝐄 𝐓 = 𝐄 𝟎 e i ( x k T sin ( θ T ) + z k T i sin 2 ( θ T ) - 1 - ω t ) \mathbf{E_{T}}=\mathbf{E_{0}}e^{i(xk_{T}\sin(\theta_{T})+zk_{T}i\sqrt{\sin^{2}% (\theta_{T})-1}-\omega t)}
  27. k T = ω n 2 c k_{T}=\frac{\omega n_{2}}{c}
  28. 𝐄 𝐓 = 𝐄 𝟎 e - κ z e i ( k x - ω t ) \mathbf{E_{T}}=\mathbf{E_{0}}e^{-\kappa z}e^{i(kx-\omega t)}
  29. κ = ω c ( n 1 sin ( θ I ) ) 2 - n 2 2 \kappa=\frac{\omega}{c}\sqrt{(n_{1}\sin(\theta_{I}))^{2}-n^{2}_{2}}
  30. k = ω n 1 c sin ( θ I ) k=\frac{\omega n_{1}}{c}\sin(\theta_{I})

Total_order.html

  1. { a b , a b } = { a , b } \{a\vee b,a\wedge b\}=\{a,b\}
  2. a = a b a=a\wedge b
  3. ( A 1 , 1 ) (A_{1},\leq_{1})
  4. ( A 2 , 2 ) (A_{2},\leq_{2})
  5. + \leq_{+}
  6. A 1 A 2 A_{1}\cup A_{2}
  7. A 1 + A 2 A_{1}+A_{2}
  8. x , y A 1 A 2 x,y\in A_{1}\cup A_{2}
  9. x + y x\leq_{+}y
  10. x , y A 1 x,y\in A_{1}
  11. x 1 y x\leq_{1}y
  12. x , y A 2 x,y\in A_{2}
  13. x 2 y x\leq_{2}y
  14. x A 1 x\in A_{1}
  15. y A 2 y\in A_{2}
  16. ( I , ) (I,\leq)
  17. i I i\in I
  18. ( A i , i ) (A_{i},\leq_{i})
  19. A i A_{i}
  20. i A i \bigcup_{i}A_{i}
  21. x , y i I A i x,y\in\bigcup_{i\in I}A_{i}
  22. x y x\leq y
  23. i I i\in I
  24. x i y x\leq_{i}y
  25. i < j i<j
  26. I I
  27. x A i x\in A_{i}
  28. y A j y\in A_{j}

Tower_of_Hanoi.html

  1. 2 h - 1 2^{h}-1
  2. T h - 1 T_{h-1}
  3. T h = 2 T h - 1 + 1 T_{h}=2T_{h-1}+1
  4. 466 885 2 n - 1 3 - 3 5 ( 1 3 ) n + ( 12 29 + 18 1003 17 ) ( 5 + 17 18 ) n + ( 12 29 - 18 1003 17 ) ( 5 - 17 18 ) n . \frac{466}{885}\cdot 2^{n}-\frac{1}{3}-\frac{3}{5}\cdot\left(\frac{1}{3}\right% )^{n}+\left(\frac{12}{29}+\frac{18}{1003}\sqrt{17}\right)\left(\frac{5+\sqrt{1% 7}}{18}\right)^{n}+\left(\frac{12}{29}-\frac{18}{1003}\sqrt{17}\right)\left(% \frac{5-\sqrt{17}}{18}\right)^{n}.
  5. 466 / 885 2 n - 1 / 3 + o ( 1 ) 466/885\cdot 2^{n}-1/3+o(1)
  6. n n\to\infty
  7. 466 / 885 52.6 % 466/885\approx 52.6\%
  8. 2 n - 1 2^{n}-1
  9. n n
  10. r r
  11. T ( n , r ) T(n,r)
  12. k k
  13. 1 k < n 1\leq k<n
  14. k k
  15. T ( k , r ) T(k,r)
  16. k k
  17. n - k n-k
  18. r - 1 r-1
  19. T ( n - k , r - 1 ) T(n-k,r-1)
  20. k k
  21. T ( k , r ) T(k,r)
  22. 2 T ( k , r ) + T ( n - k , r - 1 ) 2T(k,r)+T(n-k,r-1)
  23. k k
  24. k k
  25. 2 n + 1 \sqrt{2n+1}
  26. k k

Trace_(linear_algebra).html

  1. tr ( A ) = a 11 + a 22 + + a n n = i = 1 n a i i \operatorname{tr}(A)=a_{11}+a_{22}+\dots+a_{nn}=\sum_{i=1}^{n}a_{ii}
  2. A = ( a b c d e f g h i ) A=\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}
  3. tr ( A ) = a + e + i \operatorname{tr}(A)=a+e+i
  4. tr ( A + B ) = tr ( A ) + tr ( B ) \operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B)
  5. tr ( c A ) = c tr ( A ) \operatorname{tr}(cA)=c\operatorname{tr}(A)
  6. tr ( A ) = tr ( A T ) \operatorname{tr}(A)=\operatorname{tr}(A^{\mathrm{T}})
  7. tr ( X T Y ) = tr ( X Y T ) = tr ( Y T X ) = tr ( Y X T ) = i , j X i j Y i j \operatorname{tr}(X^{\mathrm{T}}Y)=\operatorname{tr}(XY^{\mathrm{T}})=% \operatorname{tr}(Y^{\mathrm{T}}X)=\operatorname{tr}(YX^{\mathrm{T}})=\sum_{i,% j}X_{ij}Y_{ij}
  8. tr ( X T Y ) = i ( X Y ) i i \operatorname{tr}(X^{\mathrm{T}}Y)=\sum_{i}(X\circ Y)_{ii}
  9. tr ( X T Y ) = vec ( Y ) T vec ( X ) = vec ( X ) T vec ( Y ) \operatorname{tr}(X^{\mathrm{T}}Y)=\operatorname{vec}(Y)^{\mathrm{T}}% \operatorname{vec}(X)=\operatorname{vec}(X)^{\mathrm{T}}\operatorname{vec}(Y)
  10. tr ( A B ) = tr ( B A ) \operatorname{tr}(AB)=\operatorname{tr}(BA)
  11. tr ( A B C D ) = tr ( B C D A ) = tr ( C D A B ) = tr ( D A B C ) \operatorname{tr}(ABCD)=\operatorname{tr}(BCDA)=\operatorname{tr}(CDAB)=% \operatorname{tr}(DABC)
  12. tr ( A B C ) tr ( A C B ) \operatorname{tr}(ABC)\neq\operatorname{tr}(ACB)
  13. tr ( X Y ) = tr ( X ) tr ( Y ) \operatorname{tr}(X\otimes Y)=\operatorname{tr}(X)\operatorname{tr}(Y)
  14. tr ( A + B ) = tr ( A ) + tr ( B ) \operatorname{tr}(A+B)=\operatorname{tr}(A)+\operatorname{tr}(B)
  15. tr ( c A ) = c tr ( A ) \operatorname{tr}(cA)=c\cdot\operatorname{tr}(A)
  16. tr ( A B ) = tr ( B A ) \operatorname{tr}(AB)=\operatorname{tr}(BA)
  17. f ( e i j ) = 0 f(e_{ij})=0
  18. i j i\neq j
  19. f ( e j j ) = f ( e 11 ) f(e_{jj})=f(e_{11})
  20. e i j e_{ij}
  21. f ( A ) = i , j [ A ] i j f ( e i j ) = i [ A ] i i f ( e 11 ) = f ( e 11 ) tr ( A ) f(A)=\sum_{i,j}[A]_{ij}f(e_{ij})=\sum_{i}[A]_{ii}f(e_{11})=f(e_{11})% \operatorname{tr}(A)
  22. 𝑔𝑙 n = 𝑠𝑙 n k \mathit{gl}_{n}=\mathit{sl}_{n}\oplus k
  23. tr ( [ A , B ] ) = 0 \operatorname{tr}([A,B])=0
  24. tr ( P - 1 A P ) = tr ( P - 1 ( A P ) ) = tr ( ( A P ) P - 1 ) = tr ( A ( P P - 1 ) ) = tr ( A ) \operatorname{tr}(P^{-1}AP)=\operatorname{tr}(P^{-1}(AP))=\operatorname{tr}((% AP)P^{-1})=\operatorname{tr}(A(PP^{-1}))=\operatorname{tr}(A)
  25. tr ( A B ) = 0 \operatorname{tr}(AB)=0
  26. tr ( A ) = d 1 λ 1 + + d k λ k \operatorname{tr}(A)=d_{1}\lambda_{1}+\cdots+d_{k}\lambda_{k}
  27. g l n k gl_{n}\to k
  28. tr ( x k ) = 0 \operatorname{tr}(x^{k})=0
  29. k k
  30. x x
  31. P X = X ( X T X ) - 1 X T P_{X}=X\left(X^{\mathrm{T}}X\right)^{-1}X^{\mathrm{T}}
  32. tr ( P X ) = rank ( X ) \operatorname{tr}\left(P_{X}\right)=\operatorname{rank}\left(X\right)
  33. etr ( A ) := exp ( tr ( A ) ) \operatorname{etr}(A):=\exp(\operatorname{tr}(A))
  34. tr ( A ) = i λ i \operatorname{tr}(A)=\sum_{i}\lambda_{i}
  35. A A
  36. det ( A ) = i λ i \operatorname{det}(A)=\prod_{i}\lambda_{i}
  37. tr ( A k ) = i λ i k \operatorname{tr}(A^{k})=\sum_{i}\lambda_{i}^{k}
  38. tr = det I \operatorname{tr}=\operatorname{det}^{\prime}_{I}
  39. det ( exp ( A ) ) = exp ( tr ( A ) ) \det(\exp(A))=\exp(\operatorname{tr}(A))
  40. R θ = ( cos θ - sin θ sin θ cos θ ) R_{\theta}=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)
  41. A = ( 0 - 1 1 0 ) A=\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right)
  42. d tr ( X ) = tr ( d X ) \operatorname{d}\operatorname{tr}(X)=\operatorname{tr}(\operatorname{d\!}X)
  43. A , B : G 𝐺𝐿 ( V ) A,B:G\to\mathit{GL}(V)
  44. tr A ( g ) = tr B ( g ) \operatorname{tr}A(g)=\operatorname{tr}B(g)
  45. tr : 𝑔𝑙 n k \operatorname{tr}\colon\mathit{gl}_{n}\to k
  46. tr ( [ A , B ] ) = 0 \operatorname{tr}([A,B])=0
  47. 𝑔𝑙 n = 𝑠𝑙 n k \mathit{gl}_{n}=\mathit{sl}_{n}\oplus k
  48. A 1 n tr ( A ) I A\mapsto\textstyle{\frac{1}{n}}\operatorname{tr}(A)\cdot I
  49. k 𝑔𝑙 n k\to\mathit{gl}_{n}
  50. 𝑔𝑙 n 𝑔𝑙 n \mathit{gl}_{n}\to\mathit{gl}_{n}
  51. 0 𝑠𝑙 n 𝑔𝑙 n tr k 0 0\to\mathit{sl}_{n}\to\mathit{gl}_{n}\overset{\operatorname{tr}}{\to}k\to 0
  52. 1 𝑆𝐿 n 𝐺𝐿 n det K * 1 1\to\mathit{SL}_{n}\to\mathit{GL}_{n}\overset{\operatorname{det}}{\to}K^{*}\to 1
  53. 1 n \textstyle{\frac{1}{n}}
  54. 𝑔𝑙 n = s l n k \mathit{gl}_{n}=sl_{n}\oplus k
  55. 𝐺𝐿 n S L n × K * . \mathit{GL}_{n}\neq SL_{n}\times K^{*}.
  56. B ( x , y ) = tr ( ad ( x ) ad ( y ) ) where ad ( x ) y = [ x , y ] = x y - y x B(x,y)=\operatorname{tr}(\operatorname{ad}(x)\operatorname{ad}(y))\,\text{ % where }\operatorname{ad}(x)y=[x,y]=xy-yx
  57. ( x , y ) tr ( x y ) (x,y)\mapsto\operatorname{tr}(xy)
  58. tr ( x [ y , z ] ) = tr ( [ x , y ] z ) \operatorname{tr}(x[y,z])=\operatorname{tr}([x,y]z)
  59. 𝔰 𝔩 n \mathfrak{sl}_{n}
  60. tr ( x y ) = 0 \operatorname{tr}(xy)=0
  61. tr ( A * A ) 0 \operatorname{tr}(A^{*}A)\geq 0
  62. A , B = tr ( B * A ) \langle A,B\rangle=\operatorname{tr}(B^{*}A)
  63. 0 tr ( A B ) 2 tr ( A 2 ) tr ( B 2 ) tr ( A ) 2 tr ( B ) 2 0\leq\operatorname{tr}(AB)^{2}\leq\operatorname{tr}(A^{2})\operatorname{tr}(B^% {2})\leq\operatorname{tr}(A)^{2}\operatorname{tr}(B)^{2}
  64. Z Z
  65. A B A\otimes B
  66. A A
  67. B B
  68. tr ( Z ) = tr A ( tr B ( Z ) ) = tr B ( tr A ( Z ) ) \operatorname{tr}(Z)=\operatorname{tr}_{A}(\operatorname{tr}_{B}(Z))=% \operatorname{tr}_{B}(\operatorname{tr}_{A}(Z))
  69. V V * V\otimes V^{*}
  70. v h = ( w h ( w ) v ) v\otimes h=(w\mapsto h(w)v)
  71. t : V × V * F t\colon V\times V^{*}\to F
  72. t ( v , w * ) := w * ( v ) F t(v,w^{*}):=w^{*}(v)\in F
  73. t : V V * F t\colon V\otimes V^{*}\to F
  74. tr ( A B ) = tr ( B A ) \operatorname{tr}(AB)=\operatorname{tr}(BA)
  75. tr ( A B ) tr ( A ) tr ( B ) \operatorname{tr}(AB)\neq\operatorname{tr}(A)\operatorname{tr}(B)
  76. End ( V ) V V * \operatorname{End}(V)\cong V\otimes V^{*}
  77. End ( V ) × End ( V ) End ( V ) \operatorname{End}(V)\times\operatorname{End}(V)\to\operatorname{End}(V)
  78. ( V V * ) × ( V V * ) ( V V * ) (V\otimes V^{*})\times(V\otimes V^{*})\to(V\otimes V^{*})
  79. V * × V F V^{*}\times V\to F
  80. ( A B ) i k = j a i j b j k \textstyle{(AB)_{ik}=\sum_{j}a_{ij}b_{jk}}
  81. tr ( A B ) = i j a i j b j i \textstyle{\operatorname{tr}(AB)=\sum_{ij}a_{ij}b_{ji}}
  82. tr ( B A ) = i j b i j a j i \textstyle{\operatorname{tr}(BA)=\sum_{ij}b_{ij}a_{ji}}
  83. tr ( A ) tr ( B ) = i a i i j b j j \textstyle{\operatorname{tr}(A)\cdot\operatorname{tr}(B)=\sum_{i}a_{ii}\cdot% \sum_{j}b_{jj}}
  84. V V
  85. { e i } \{e_{i}\}
  86. { e i } \{e^{i}\}
  87. e i e j e_{i}\otimes e^{j}
  88. A A
  89. A = a i j e i e j A=a_{ij}e_{i}\otimes e^{j}
  90. t t
  91. t ( A ) = a i j t ( e i e j ) t(A)=a_{ij}t(e_{i}\otimes e^{j})
  92. t ( A ) t(A)
  93. F * = F V V * End ( V ) F^{*}=F\to V\otimes V^{*}\cong\operatorname{End}(V)
  94. F 𝐼 End ( V ) tr F F\overset{I}{\to}\operatorname{End}(V)\overset{\operatorname{tr}}{\to}F
  95. tr ( A B ) = i = 1 m ( A B ) i i = i = 1 m j = 1 n A i j B j i = j = 1 n i = 1 m B j i A i j = j = 1 n ( B A ) j j = tr ( B A ) \operatorname{tr}(AB)=\sum_{i=1}^{m}\left(AB\right)_{ii}=\sum_{i=1}^{m}\sum_{j% =1}^{n}A_{ij}B_{ji}=\sum_{j=1}^{n}\sum_{i=1}^{m}B_{ji}A_{ij}=\sum_{j=1}^{n}% \left(BA\right)_{jj}=\operatorname{tr}(BA)
  96. 𝔰 𝔩 n \mathfrak{sl}_{n}
  97. tr ( A * A ) = 0 \operatorname{tr}(A^{*}A)=0
  98. A = 0 A=0