wpmath0000002_9

Homogeneous_coordinates.html

  1. f ( λ x , λ y , λ z ) = λ k f ( x , y , z ) . f(\lambda x,\lambda y,\lambda z)=\lambda^{k}f(x,y,z).\,
  2. f ( x , y , z ) = 0 f ( λ x , λ y , λ z ) = λ k f ( x , y , z ) = 0. f(x,y,z)=0\iff f(\lambda x,\lambda y,\lambda z)=\lambda^{k}f(x,y,z)=0.
  3. f ( x , y , z ) = z k g ( x / z , y / z ) . f(x,y,z)=z^{k}g(x/z,y/z).\,
  4. g ( x , y ) = f ( x , y , 1 ) . g(x,y)=f(x,y,1).\,
  5. A = ( a b c d e f g h i ) , A=\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix},
  6. ( X Y Z ) = A ( x y z ) . \begin{pmatrix}X\\ Y\\ Z\end{pmatrix}=A\begin{pmatrix}x\\ y\\ z\end{pmatrix}.
  7. X = a x + b y + c , Y = d x + e y + f , Z = g x + h y + i X=ax+by+c,\,Y=dx+ey+f,\,Z=gx+hy+i
  8. a x + b y + c = 0 , d x + e y + f = 0 , g x + h y + i = 0. ax+by+c=0,\,dx+ey+f=0,\,gx+hy+i=0.
  9. a x + b y + c z = 0 , d x + e y + f z = 0 , g x + h y + i z = 0 ax+by+cz=0,\,dx+ey+fz=0,\,gx+hy+iz=0
  10. X = 0 , Y = 0 , Z = 0 X=0,\,Y=0,\,Z=0
  11. ( 1 0 0 0 0 1 0 0 0 0 1 0 ) \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{pmatrix}

Homological_algebra.html

  1. ( C , d ) (C_{\bullet},d_{\bullet})
  2. C : C n + 1 d n + 1 C n d n C n - 1 d n - 1 , d n d n + 1 = 0. C_{\bullet}:\cdots\longrightarrow C_{n+1}\stackrel{d_{n+1}}{\longrightarrow}C_% {n}\stackrel{d_{n}}{\longrightarrow}C_{n-1}\stackrel{d_{n-1}}{\longrightarrow}% \cdots,\quad d_{n}\circ d_{n+1}=0.
  3. B n Z n C n . B_{n}\subseteq Z_{n}\subseteq C_{n}.
  4. H n ( C ) = Z n / B n = Ker d n / Im d n + 1 . H_{n}(C)=Z_{n}/B_{n}=\operatorname{Ker}\,d_{n}/\operatorname{Im}\,d_{n+1}.
  5. C ( X ) C_{\bullet}(X)
  6. H ( C ) H_{\bullet}(C)
  7. G 0 f 1 G 1 f 2 G 2 f 3 f n G n G_{0}\;\xrightarrow{f_{1}}\;G_{1}\;\xrightarrow{f_{2}}\;G_{2}\;\xrightarrow{f_% {3}}\;\cdots\;\xrightarrow{f_{n}}\;G_{n}
  8. im ( f k ) = ker ( f k + 1 ) . \mathrm{im}(f_{k})=\mathrm{ker}(f_{k+1}).\!
  9. A 𝑓 B 𝑔 C A\;\overset{f}{\hookrightarrow}\;B\;\overset{g}{\twoheadrightarrow}\;C
  10. C B / f ( A ) . C\cong B/f(A).
  11. 0 A 𝑓 B 𝑔 C 0 0\;\xrightarrow{}\;A\;\xrightarrow{f}\;B\;\xrightarrow{g}\;C\;\xrightarrow{}\;0
  12. ker a \color G r a y ker b \color G r a y ker c 𝑑 coker a \color G r a y coker b \color G r a y coker c \ker a\;{\color{Gray}\longrightarrow}\ker b\;{\color{Gray}\longrightarrow}\ker c% \;\overset{d}{\longrightarrow}\operatorname{coker}a\;{\color{Gray}% \longrightarrow}\operatorname{coker}b\;{\color{Gray}\longrightarrow}% \operatorname{coker}c
  13. Ext R n ( A , B ) = ( R n T ) ( B ) . \operatorname{Ext}_{R}^{n}(A,B)=(R^{n}T)(B).
  14. 0 B I 0 I 1 , 0\rightarrow B\rightarrow I^{0}\rightarrow I^{1}\rightarrow\dots,
  15. 0 Hom R ( A , I 0 ) Hom R ( A , I 1 ) . 0\rightarrow\operatorname{Hom}_{R}(A,I^{0})\rightarrow\operatorname{Hom}_{R}(A% ,I^{1})\rightarrow\dots.
  16. Ext R n ( A , B ) = ( R n G ) ( A ) . \operatorname{Ext}_{R}^{n}(A,B)=(R^{n}G)(A).
  17. P 1 P 0 A 0 , \dots\rightarrow P^{1}\rightarrow P^{0}\rightarrow A\rightarrow 0,
  18. 0 Hom R ( P 0 , B ) Hom R ( P 1 , B ) . 0\rightarrow\operatorname{Hom}_{R}(P^{0},B)\rightarrow\operatorname{Hom}_{R}(P% ^{1},B)\rightarrow\dots.
  19. Tor n R ( A , B ) = ( L n T ) ( A ) \mathrm{Tor}_{n}^{R}(A,B)=(L_{n}T)(A)
  20. P 2 P 1 P 0 A 0 \cdots\rightarrow P_{2}\rightarrow P_{1}\rightarrow P_{0}\rightarrow A\rightarrow 0
  21. P 2 R B P 1 R B P 0 R B 0 \cdots\rightarrow P_{2}\otimes_{R}B\rightarrow P_{1}\otimes_{R}B\rightarrow P_% {0}\otimes_{R}B\rightarrow 0
  22. E r p , q E_{r}^{p,q}
  23. F : C D F:C_{\bullet}\to D_{\bullet}
  24. H ( F ) H_{\bullet}(F)
  25. C ( X ) C_{\bullet}(X)
  26. C ( Y ) , C_{\bullet}(Y),
  27. C ( X ) C_{\bullet}(X)
  28. C ( Z ) C_{\bullet}(Z)
  29. H ( C ) H_{\bullet}(C)
  30. L , M , N L_{\bullet},M_{\bullet},N_{\bullet}
  31. f : L M , g : M N , f:L_{\bullet}\to M_{\bullet},g:M_{\bullet}\to N_{\bullet},
  32. 0 L f M g N 0 , 0\longrightarrow L_{\bullet}\stackrel{f}{\longrightarrow}M_{\bullet}\stackrel{% g}{\longrightarrow}N_{\bullet}\longrightarrow 0,
  33. 0 L n f n M n g n N n 0 0\longrightarrow L_{n}\stackrel{f_{n}}{\longrightarrow}M_{n}\stackrel{g_{n}}{% \longrightarrow}N_{n}\longrightarrow 0
  34. H n ( L ) H n ( f ) H n ( M ) H n ( g ) H n ( N ) δ n H n - 1 ( L ) H n - 1 ( f ) H n - 1 ( M ) , \ldots\longrightarrow H_{n}(L)\stackrel{H_{n}(f)}{\longrightarrow}H_{n}(M)% \stackrel{H_{n}(g)}{\longrightarrow}H_{n}(N)\stackrel{\delta_{n}}{% \longrightarrow}H_{n-1}(L)\stackrel{H_{n-1}(f)}{\longrightarrow}H_{n-1}(M)% \longrightarrow\ldots,

Homology_(mathematics).html

  1. H 0 ( X ) , H 1 ( X ) , H 2 ( X ) , H_{0}(X),H_{1}(X),H_{2}(X),\ldots
  2. k th k^{\rm th}
  3. H k ( X ) H_{k}(X)
  4. H 0 ( X ) H_{0}(X)
  5. S 1 S^{1}
  6. H k ( S 1 ) = { k = 0 , 1 { 0 } otherwise H_{k}(S^{1})=\begin{cases}\mathbb{Z}&k=0,1\\ \{0\}&\,\text{otherwise}\end{cases}
  7. \mathbb{Z}
  8. { 0 } \{0\}
  9. H 1 ( S 1 ) = H_{1}(S^{1})=\mathbb{Z}
  10. S 2 S^{2}
  11. H k ( S 2 ) = { k = 0 , 2 { 0 } otherwise H_{k}(S^{2})=\begin{cases}\mathbb{Z}&k=0,2\\ \{0\}&\,\text{otherwise}\end{cases}
  12. H k ( S n ) = { k = 0 , n { 0 } otherwise H_{k}(S^{n})=\begin{cases}\mathbb{Z}&k=0,n\\ \{0\}&\,\text{otherwise}\end{cases}
  13. H 0 ( B 1 ) = H_{0}(B^{1})=\mathbb{Z}
  14. H k ( B n ) = { k = 0 { 0 } otherwise H_{k}(B^{n})=\begin{cases}\mathbb{Z}&k=0\\ \{0\}&\,\text{otherwise}\end{cases}
  15. T = S 1 × S 1 T=S^{1}\times S^{1}
  16. H k ( T ) = { k = 0 , 2 × k = 1 { 0 } otherwise H_{k}(T)=\begin{cases}\mathbb{Z}&k=0,2\\ \mathbb{Z}\times\mathbb{Z}&k=1\\ \{0\}&\,\text{otherwise}\end{cases}
  17. × \mathbb{Z}\times\mathbb{Z}
  18. n : C n C n - 1 , \partial_{n}:C_{n}\to C_{n-1},
  19. n + 1 C n n C n - 1 n - 1 2 C 1 1 C 0 0 0 \cdots\overset{\partial_{n+1}}{\longrightarrow\,}C_{n}\overset{\partial_{n}}{% \longrightarrow\,}C_{n-1}\overset{\partial_{n-1}}{\longrightarrow\,}\cdots% \overset{\partial_{2}}{\longrightarrow\,}C_{1}\overset{\partial_{1}}{% \longrightarrow\,}C_{0}\overset{\partial_{0}}{\longrightarrow\,}0
  20. C i 0 C_{i}\equiv 0
  21. im ( n + 1 ) ker ( n ) \mathrm{im}(\partial_{n+1})\subseteq\ker(\partial_{n})
  22. im ( n + 1 ) \mathrm{im}(\partial_{n+1})
  23. ker ( n ) \ker(\partial_{n})
  24. B n ( X ) = im ( n + 1 ) B_{n}(X)=\mathrm{im}(\partial_{n+1})
  25. Z n ( X ) = ker ( n ) Z_{n}(X)=\ker(\partial_{n})
  26. im ( n + 1 ) \mathrm{im}(\partial_{n+1})
  27. ker ( n ) \ker(\partial_{n})
  28. im ( n + 1 ) \mathrm{im}(\partial_{n+1})
  29. ker ( n ) \ker(\partial_{n})
  30. H n ( X ) := ker ( n ) / im ( n + 1 ) = Z n ( X ) / B n ( X ) , H_{n}(X):=\ker(\partial_{n})/\mathrm{im}(\partial_{n+1})=Z_{n}(X)/B_{n}(X),
  31. n + 1 C n n C n - 1 n - 1 2 C 1 1 C 0 ϵ \Z 0 \cdots\overset{\partial_{n+1}}{\longrightarrow\,}C_{n}\overset{\partial_{n}}{% \longrightarrow\,}C_{n-1}\overset{\partial_{n-1}}{\longrightarrow\,}\cdots% \overset{\partial_{2}}{\longrightarrow\,}C_{1}\overset{\partial_{1}}{% \longrightarrow\,}C_{0}\overset{\epsilon}{\longrightarrow\,}\Z{\longrightarrow% \,}0
  32. ϵ \epsilon
  33. ϵ ( i n i σ i ) = i n i \epsilon\left(\sum_{i}n_{i}\sigma_{i}\right)=\sum_{i}n_{i}
  34. H ~ i ( X ) \tilde{H}_{i}(X)
  35. H i ( X ) H_{i}(X)
  36. \Z \Z
  37. [ ] X [\emptyset]\longrightarrow X
  38. Z n ( X ) Z_{n}(X)
  39. B n ( X ) B_{n}(X)
  40. ker ( d n ) = Z n ( X ) \ker(d^{n})=Z^{n}(X)
  41. im ( d n - 1 ) = B n ( X ) \mathrm{im}(d^{n-1})=B^{n}(X)
  42. H n ( X ) = Z n ( X ) / B n ( X ) , H^{n}(X)=Z^{n}(X)/B^{n}(X),
  43. ( a [ 0 ] , a [ 1 ] , , a [ n ] ) (a[0],a[1],\dots,a[n])
  44. i = 0 n ( - 1 ) i ( a [ 0 ] , , a [ i - 1 ] , a [ i + 1 ] , , a [ n ] ) \sum_{i=0}^{n}(-1)^{i}\left(a[0],\dots,a[i-1],a[i+1],\dots,a[n]\right)
  45. f n - 1 d n = e n f n f_{n-1}\circ d_{n}=e_{n}\circ f_{n}
  46. χ = ( - 1 ) n rank ( A n ) \chi=\sum(-1)^{n}\,\mathrm{rank}(A_{n})
  47. χ = ( - 1 ) n rank ( H n ) \chi=\sum(-1)^{n}\,\mathrm{rank}(H_{n})
  48. 0 A B C 0 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0
  49. H n ( A ) H n ( B ) H n ( C ) H n - 1 ( A ) H n - 1 ( B ) H n - 1 ( C ) H n - 2 ( A ) \cdots\to H_{n}(A)\to H_{n}(B)\to H_{n}(C)\to H_{n-1}(A)\to H_{n-1}(B)\to H_{n% -1}(C)\to H_{n-2}(A)\to\cdots

Homoscedasticity.html

  1. N ( μ i , Σ i ) N(\mu_{i},\Sigma_{i})
  2. Σ i = Σ j , i , j \Sigma_{i}=\Sigma_{j},\ \forall i,j

Homotopy_group.html

  1. f g = { f ( 2 t ) if t [ 0 , 1 / 2 ] g ( 2 t - 1 ) , if t [ 1 / 2 , 1 ] f\ast g=\begin{cases}f(2t)&\,\text{if }t\in[0,1/2]\\ g(2t-1),&\,\text{if }t\in[1/2,1]\end{cases}

Hölder's_inequality.html

  1. ( S , Σ , μ ) (S,Σ,μ)
  2. p , q p,q∈
  3. [ 1 , ] [1,∞]
  4. 1 / p + 1 / q = 1 1/p+1/q=1
  5. f f
  6. g g
  7. S S
  8. f g 1 f p g q . \|fg\|_{1}\leq\|f\|_{p}\|g\|_{q}.
  9. p , q p,q∈
  10. ( 1 , ) (1,∞)
  11. α , β 0 α,β≥0
  12. μ μ
  13. p p
  14. q q
  15. p = q = 2 p=q=2
  16. f f
  17. g g
  18. f g fg
  19. p p∈
  20. [ 1 , ) [1,∞)
  21. 1 / 1/ ∞
  22. p , q p,q∈
  23. [ 1 , ) [1,∞)
  24. ( S | f | p d μ ) 1 p \displaystyle\left(\int_{S}|f|^{p}\,\mathrm{d}\mu\right)^{\frac{1}{p}}
  25. p = p=∞
  26. | f | |f|
  27. 1 p 1≤p≤∞
  28. f f
  29. f f
  30. μ μ
  31. a > 0 a>0
  32. f f
  33. g g
  34. S S
  35. f f
  36. g g
  37. μ μ
  38. | S f g ¯ d μ | S | f g | d μ = f g 1 \biggl|\int_{S}f\bar{g}\,\mathrm{d}\mu\biggr|\leq\int_{S}|fg|\,\mathrm{d}\mu=% \|fg\|_{1}
  39. f g fg
  40. f f
  41. g g
  42. p = q = 2 p=q=2
  43. | f , g | f 2 g 2 , |\langle f,g\rangle|\leq\|f\|_{2}\|g\|_{2},
  44. f f
  45. g g
  46. p = 2 p=2
  47. | f | |f|
  48. | g | |g|
  49. f f
  50. g g
  51. ( S , Σ , μ ) (S, Σ,μ)
  52. p , q p,q∈
  53. [ 1 , ] [1,∞]
  54. 1 / p + 1 / q 1 1/p+1/q≤1
  55. f g 1 f p g q \|fg\|_{1}\leq\|f\|_{p}\|g\|_{q}
  56. f f
  57. g g
  58. S S
  59. p p
  60. q q
  61. ( 1 , ) (1,∞)
  62. 1 / p + 1 / q = 1 1/p+1/q=1
  63. n n
  64. S S
  65. 1 , , n {1,...,n}
  66. k = 1 n | x k y k | ( k = 1 n | x k | p ) 1 p ( k = 1 n | y k | q ) 1 q for all ( x 1 , , x n ) , ( y 1 , , y n ) n or n . \sum_{k=1}^{n}|x_{k}\,y_{k}|\leq\biggl(\sum_{k=1}^{n}|x_{k}|^{p}\biggr)^{\frac% {1}{p}}\biggl(\sum_{k=1}^{n}|y_{k}|^{q}\biggr)^{\frac{1}{q}}\,\text{ for all }% (x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}\,\text{ or }\mathbb% {C}^{n}.
  67. S = 𝐍 S=\mathbf{N}
  68. k = 1 | x k y k | ( k = 1 | x k | p ) 1 p ( k = 1 | y k | q ) 1 q for all ( x k ) k , ( y k ) k or . \sum_{k=1}^{\infty}|x_{k}\,y_{k}|\leq\biggl(\sum_{k=1}^{\infty}|x_{k}|^{p}% \biggr)^{\frac{1}{p}}\left(\sum_{k=1}^{\infty}|y_{k}|^{q}\right)^{\frac{1}{q}}% \,\text{ for all }(x_{k})_{k\in\mathbb{N}},(y_{k})_{k\in\mathbb{N}}\in\mathbb{% R}^{\mathbb{N}}\,\text{ or }\mathbb{C}^{\mathbb{N}}.
  69. S S
  70. f f
  71. g g
  72. S S
  73. S | f ( x ) g ( x ) | d x ( S | f ( x ) | p d x ) 1 p ( S | g ( x ) | q d x ) 1 q . \int_{S}\bigl|f(x)g(x)\bigr|\,\mathrm{d}x\leq\biggl(\int_{S}|f(x)|^{p}\,% \mathrm{d}x\biggr)^{\frac{1}{p}}\biggl(\int_{S}|g(x)|^{q}\,\mathrm{d}x\biggr)^% {\frac{1}{q}}.
  74. ( Ω , , ) (Ω,\mathcal{F}, ℙ)
  75. 𝐄 \mathbf{E}
  76. X X
  77. Y Y
  78. Ω Ω
  79. 𝔼 [ | X Y | ] ( 𝔼 [ | X | p ] ) 1 p ( 𝔼 [ | Y | q ] ) 1 q . \mathbb{E}[|XY|]\leq\bigl(\mathbb{E}\bigl[|X|^{p}\bigr]\bigr)^{\frac{1}{p}}% \bigl(\mathbb{E}\bigl[|Y|^{q}\bigr]\bigr)^{\frac{1}{q}}.
  80. q = p / ( p 1 ) q=p/(p− 1)
  81. p p
  82. 𝔼 [ | X | r ] ( 𝔼 [ | X | s ] ) r / s . \mathbb{E}\bigl[|X|^{r}\bigr]\leq\bigl(\mathbb{E}\bigl[|X|^{s}\bigr]\bigr)^{r/% s}.
  83. s s
  84. r r
  85. S = S 1 × S 2 , Σ = Σ 1 Σ 2 , μ = μ 1 μ 2 , S=S_{1}\times S_{2},\quad\Sigma=\Sigma_{1}\otimes\Sigma_{2},\quad\mu=\mu_{1}% \otimes\mu_{2},
  86. S S
  87. μ μ
  88. f f
  89. g g
  90. S S
  91. S 1 S 2 | f ( x , y ) g ( x , y ) | μ 2 ( d y ) μ 1 ( d x ) ( S 1 S 2 | f ( x , y ) | p μ 2 ( d y ) μ 1 ( d x ) ) 1 p ( S 1 S 2 | g ( x , y ) | q μ 2 ( d y ) μ 1 ( d x ) ) 1 q . \int_{S_{1}}\int_{S_{2}}|f(x,y)\,g(x,y)|\,\mu_{2}(\mathrm{d}y)\,\mu_{1}(% \mathrm{d}x)\leq\left(\int_{S_{1}}\int_{S_{2}}|f(x,y)|^{p}\,\mu_{2}(\mathrm{d}% y)\,\mu_{1}(\mathrm{d}x)\right)^{\frac{1}{p}}\left(\int_{S_{1}}\int_{S_{2}}|g(% x,y)|^{q}\,\mu_{2}(\mathrm{d}y)\,\mu_{1}(\mathrm{d}x)\right)^{\frac{1}{q}}.
  92. ( S , Σ , μ ) (S,Σ,μ)
  93. Σ Σ
  94. S S
  95. n n
  96. 1 , , n {1,...,n}
  97. S k = 1 n | f k ( x ) g k ( x ) | μ ( d x ) ( S k = 1 n | f k ( x ) | p μ ( d x ) ) 1 p ( S k = 1 n | g k ( x ) | q μ ( d x ) ) 1 q . \int_{S}\sum_{k=1}^{n}|f_{k}(x)\,g_{k}(x)|\,\mu(\mathrm{d}x)\leq\left(\int_{S}% \sum_{k=1}^{n}|f_{k}(x)|^{p}\,\mu(\mathrm{d}x)\right)^{\frac{1}{p}}\left(\int_% {S}\sum_{k=1}^{n}|g_{k}(x)|^{q}\,\mu(\mathrm{d}x)\right)^{\frac{1}{q}}.
  98. α , β 0 α,β≥0
  99. α ( | f 1 ( x ) | p , , | f n ( x ) | p ) = β ( | g 1 ( x ) | q , , | g n ( x ) | q ) , \alpha\left(|f_{1}(x)|^{p},\ldots,|f_{n}(x)|^{p}\right)=\beta\left(|g_{1}(x)|^% {q},\ldots,|g_{n}(x)|^{q}\right),
  100. μ μ
  101. x x
  102. S S
  103. f f
  104. g g
  105. f f
  106. μ μ
  107. f g fg
  108. μ μ
  109. ( 0 , ) (0, ∞)
  110. p = p=∞
  111. q = 1 q=1
  112. p = 1 p=1
  113. q = q=∞
  114. p , q p,q∈
  115. ( 1 , ) (1, ∞)
  116. f f
  117. g g
  118. f p = g q = 1. \|f\|_{p}=\|g\|_{q}=1.
  119. a b a p p + b q q ab\leq\frac{a^{p}}{p}+\frac{b^{q}}{q}
  120. a a
  121. b b
  122. | f ( s ) g ( s ) | | f ( s ) | p p + | g ( s ) | q q , s S . |f(s)g(s)|\leq\frac{|f(s)|^{p}}{p}+\frac{|g(s)|^{q}}{q},\qquad s\in S.
  123. f g 1 1 p + 1 q = 1 , \|fg\|_{1}\leq\frac{1}{p}+\frac{1}{q}=1,
  124. p p∈
  125. ( 1 , ) (1, ∞)
  126. ( 0 , ) (0, ∞)
  127. α , β > 0 α,β>0
  128. α = g q q , β = f p p , \alpha=\|g\|_{q}^{q},\qquad\beta=\|f\|_{p}^{p},
  129. α | f | p = β | g | q \alpha|f|^{p}=\beta|g|^{q}\,
  130. β = 0 β=0
  131. α = 0 α=0
  132. x p x^{p}
  133. p 1 p\geq 1
  134. h d ν ( h p d ν ) 1 p \int hd\nu\leq\left(\int h^{p}d\nu\right)^{\frac{1}{p}}
  135. ν ν
  136. h h
  137. ν ν
  138. μ μ
  139. ν ν
  140. μ μ
  141. g q g^{q}
  142. d ν = g q g q d μ d μ d\nu=\frac{g^{q}}{\int g^{q}d\mu}d\mu
  143. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  144. p ( 1 - q ) + q = 0 p(1-q)+q=0
  145. h = f g 1 - q h=fg^{1-q}
  146. f g d μ = ( g q d μ ) f g 1 - q h g q g q d μ d μ d ν ( g q d μ ) ( f p g p ( 1 - q ) h p g q g q d μ d μ d ν ) 1 p = ( g q d μ ) ( f p g q d μ d μ ) 1 p \int fgd\mu=\left(\int g^{q}d\mu\right)\int\underbrace{fg^{1-q}}_{h}% \underbrace{\frac{g^{q}}{\int g^{q}d\mu}d\mu}_{d\nu}\leq\left(\int g^{q}d\mu% \right)\left(\int\underbrace{f^{p}g^{p(1-q)}}_{h^{p}}\underbrace{\frac{g^{q}}{% \int g^{q}d\mu}d\mu}_{d\nu}\right)^{\frac{1}{p}}=\left(\int g^{q}d\mu\right)% \left(\int\frac{f^{p}}{\int g^{q}d\mu}d\mu\right)^{\frac{1}{p}}
  147. f g d μ ( f p d μ ) 1 p ( g q d μ ) 1 q \int fgd\mu\leq\left(\int f^{p}d\mu\right)^{\frac{1}{p}}\left(\int g^{q}d\mu% \right)^{\frac{1}{q}}
  148. f , g f,g
  149. f , g f,g
  150. f p , g q \|f\|_{p},\|g\|_{q}
  151. p , q > 1 p,q>1
  152. q q
  153. f p = max { | S f g d μ | : g L q ( μ ) , g q 1 } , \|f\|_{p}=\max\left\{\left|\int_{S}fg\,\mathrm{d}\mu\right|:g\in L^{q}(\mu),\|% g\|_{q}\leq 1\right\},
  154. g g
  155. p = p=∞
  156. A A
  157. Σ Σ
  158. μ ( A ) = μ(A)=∞
  159. B Σ B∈Σ
  160. f = sup { | S f g d μ | : g L 1 ( μ ) , g 1 1 } . \|f\|_{\infty}=\sup\left\{\left|\int_{S}fg\,\mathrm{d}\mu\right|:g\in L^{1}(% \mu),\|g\|_{1}\leq 1\right\}.
  161. 1 p 1≤p≤∞
  162. | S f g d μ | S | f g | d μ f p , \left|\int_{S}fg\,\mathrm{d}\mu\right|\leq\int_{S}|fg|\,\mathrm{d}\mu\leq\|f\|% _{p},
  163. 1 p 1≤p≤∞
  164. g g
  165. S S
  166. g ( x ) = { f p 1 - p | f ( x ) | p / f ( x ) if f ( x ) 0 , 0 otherwise. g(x)=\begin{cases}\|f\|_{p}^{1-p}\,|f(x)|^{p}/f(x)&\,\text{if }f(x)\not=0,\\ 0&\,\text{otherwise.}\end{cases}
  167. p = 1 p=1
  168. S f g d μ = f p . \int_{S}fg\,\mathrm{d}\mu=\|f\|_{p}.
  169. p = p=∞
  170. ε ε∈
  171. ( 0 , 1 ) (0, 1)
  172. A = { x S : | f ( x ) | > ( 1 - ε ) f } . A=\left\{x\in S:|f(x)|>(1-\varepsilon)\|f\|_{\infty}\right\}.
  173. f f
  174. A Σ A∈Σ
  175. f f
  176. μ ( A ) > 0 μ(A)>0
  177. Σ Σ
  178. B Σ B∈Σ
  179. A A
  180. S S
  181. g ( x ) = { 1 - ε μ ( B ) f f ( x ) if x B , 0 otherwise. g(x)=\begin{cases}\frac{1-\varepsilon}{\mu(B)}\frac{\|f\|_{\infty}}{f(x)}&\,% \text{if }x\in B,\\ 0&\,\text{otherwise.}\end{cases}
  182. g g
  183. | g ( x ) | 1 / μ ( B ) |g(x)|≤1/μ(B)
  184. x B x∈B
  185. | S f g d μ | = B 1 - ε μ ( B ) f d μ = ( 1 - ε ) f . \left|\int_{S}fg\,\mathrm{d}\mu\right|=\int_{B}\frac{1-\varepsilon}{\mu(B)}\|f% \|_{\infty}\,\mathrm{d}\mu=(1-\varepsilon)\|f\|_{\infty}.
  186. p = p=∞
  187. A A
  188. Σ Σ
  189. μ ( A ) = μ(A)=∞
  190. B Σ B∈Σ
  191. Σ Σ
  192. S S
  193. μ μ
  194. μ ( S ) = μ(S)=∞
  195. μ μ
  196. A A
  197. g g
  198. p = p=∞
  199. S S
  200. Σ Σ
  201. S S
  202. μ μ
  203. f ( n ) = ( n 1 ) / n f(n)=(n−1)/n
  204. n n
  205. g ( m ) 0 g(m)≠0
  206. | S f g d μ | m - 1 m | g ( m ) | + n = m + 1 | g ( n ) | = g 1 - | g ( m ) | m < 1. \Bigl|\int_{S}fg\,\mathrm{d}\mu\Bigr|\leq\frac{m-1}{m}|g(m)|+\sum_{n=m+1}^{% \infty}|g(n)|=\|g\|_{1}-\frac{|g(m)|}{m}<1.
  207. κ f ( g ) = S f g d μ , g L q ( μ ) . \kappa_{f}(g)=\int_{S}fg\,\mathrm{d}\mu,\qquad g\in L^{q}(\mu).
  208. f f
  209. r r∈
  210. ( 0 , ) (0, ∞)
  211. ( 0 , ] (0, ∞]
  212. k = 1 n 1 p k = 1 r . \sum_{k=1}^{n}\frac{1}{p_{k}}=\frac{1}{r}.
  213. S S
  214. k = 1 n f k r k = 1 n f k p k . \left\|\prod_{k=1}^{n}f_{k}\right\|_{r}\leq\prod_{k=1}^{n}\|f_{k}\|_{p_{k}}.
  215. f k L p k ( μ ) k { 1 , , n } k = 1 n f k L r ( μ ) . f_{k}\in L^{p_{k}}(\mu)\;\;\forall k\in\{1,\ldots,n\}\implies\prod_{k=1}^{n}f_% {k}\in L^{r}(\mu).
  216. r ( 0 , 1 ) r∈(0,1)
  217. n = 1 n=1
  218. n 1 n−1
  219. n n
  220. k = 1 n - 1 1 p k = 1 r . \sum_{k=1}^{n-1}\frac{1}{p_{k}}=\frac{1}{r}.
  221. f 1 f n r f 1 f n - 1 r f n f 1 p 1 f n - 1 p n - 1 f n . \begin{aligned}\displaystyle\left\|f_{1}\cdots f_{n}\right\|_{r}&\displaystyle% \leq\|f_{1}\cdots f_{n-1}\|_{r}\|f_{n}\|_{\infty}\\ &\displaystyle\leq\|f_{1}\|_{p_{1}}\cdots\|f_{n-1}\|_{p_{n-1}}\|f_{n}\|_{% \infty}.\end{aligned}
  222. k = 1 n - 1 1 p k = 1 r - 1 p n = p n - r r p n = 1 p r , \sum_{k=1}^{n-1}\frac{1}{p_{k}}=\frac{1}{r}-\frac{1}{p_{n}}=\frac{p_{n}-r}{rp_% {n}}=\frac{1}{pr},
  223. ( 0 , ] (0, ∞]
  224. p p
  225. 1 p = k = 1 n θ k p k . \frac{1}{p}=\sum_{k=1}^{n}\frac{\theta_{k}}{p_{k}}.
  226. f f
  227. S S
  228. f k = | f | θ k , k { 1 , , n } . f_{k}=|f|^{\theta_{k}},\quad k\in\{1,\ldots,n\}.
  229. f p = k = 1 n f k p k = 1 n f k p k / θ k = k = 1 n f p k θ k . \|f\|_{p}=\biggl\|\prod_{k=1}^{n}f_{k}\biggr\|_{p}\leq\prod_{k=1}^{n}\|f_{k}\|% _{p_{k}/\theta_{k}}=\prod_{k=1}^{n}\|f\|_{p_{k}}^{\theta_{k}}.
  230. n = 2 n=2
  231. f p θ f p 1 θ f p 0 1 - θ , \|f\|_{p_{\theta}}\leq\|f\|_{p_{1}}^{\theta}\cdot\|f\|_{p_{0}}^{1-\theta},
  232. θ θ∈
  233. ( 0 , 1 ) (0, 1)
  234. 1 p θ = θ p 1 + 1 - θ p 0 . \frac{1}{p}_{\theta}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{0}}.
  235. p = θ p 0 + ( 1 - θ ) p 1 p=\theta p_{0}+(1-\theta)p_{1}
  236. θ ( 0 , 1 ) \theta\in(0,1)
  237. f p p f p 0 p 0 θ f p 1 p 1 ( 1 - θ ) . \|f\|_{p}^{p}\leq\|f\|_{p_{0}}^{p_{0}\cdot\theta}\cdot\|f\|_{p_{1}}^{p_{1}(1-% \theta)}.
  238. f L p 0 L p 1 f\in L^{p_{0}}\cap L^{p_{1}}
  239. f L p f\in L^{p}
  240. p 0 < p < p 1 p_{0}<p<p_{1}
  241. p ( 1 , ) p∈(1,∞)
  242. ( S , Σ , μ ) (S,Σ,μ)
  243. μ ( S ) > 0 μ(S)>0
  244. f f
  245. g g
  246. S S
  247. g ( s ) 0 g(s)≠0
  248. s S s∈S
  249. f g 1 f 1 p g - 1 / ( p - 1 ) . \|fg\|_{1}\geq\|f\|_{\frac{1}{p}}\,\|g\|_{-1/(p-1)}.
  250. ( S | g | - 1 / ( p - 1 ) d μ ) - ( p - 1 ) . \biggl(\int_{S}|g|^{-1/(p-1)}\,\mathrm{d}\mu\biggr)^{-(p-1)}.
  251. p p
  252. q := p p - 1 ( 1 , ) q:=\frac{p}{p-1}\in(1,\infty)
  253. | f | 1 p 1 = | f g | 1 p | g | - 1 p 1 | f g | 1 p p | g | - 1 p q = f g 1 1 p | g | - 1 / ( p - 1 ) 1 ( p - 1 ) / p . \begin{aligned}\displaystyle\left\||f|^{\frac{1}{p}}\right\|_{1}&\displaystyle% =\left\||fg|^{\frac{1}{p}}\,|g|^{-\frac{1}{p}}\right\|_{1}\\ &\displaystyle\leq\bigl\||fg|^{\frac{1}{p}}\bigr\|_{p}\,\bigl\||g|^{-\frac{1}{% p}}\bigr\|_{q}=\|fg\|_{1}^{\frac{1}{p}}\,\bigl\||g|^{-1/(p-1)}\bigr\|_{1}^{(p-% 1)/p}.\end{aligned}
  254. p p
  255. f g < s u b > 1 \|fg\|<sub>1
  256. ( Ω , , ) (Ω,\mathcal{F}, ℙ)
  257. 𝒢 \mathcal{G}⊂\mathcal{F}
  258. p , q p,q∈
  259. ( 1 , ) (1, ∞)
  260. 1 / p + 1 / q = 1 1/p+1/q=1
  261. X X
  262. Y Y
  263. Ω Ω
  264. 𝔼 [ | X Y | | 𝒢 ] ( 𝔼 [ | X | p | 𝒢 ] ) 1 p ( 𝔼 [ | Y | q | 𝒢 ] ) 1 q -almost surely. \mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]\leq\bigl(\mathbb{E}\bigl[|X|^{p}% \big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{p}}\,\bigl(\mathbb{E}\bigl[|Y|^{q}% \big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{q}}\qquad\mathbb{P}\,\text{-almost % surely.}
  265. Z Z
  266. 𝔼 [ Z | 𝒢 ] = sup n 𝔼 [ min { Z , n } | 𝒢 ] a.s. \mathbb{E}[Z|\mathcal{G}]=\sup_{n\in\mathbb{N}}\,\mathbb{E}[\min\{Z,n\}|% \mathcal{G}]\quad\,\text{a.s.}
  267. a > 0 a>0
  268. U = ( 𝔼 [ | X | p | 𝒢 ] ) 1 p , V = ( 𝔼 [ | Y | q | 𝒢 ] ) 1 q U=\bigl(\mathbb{E}\bigl[|X|^{p}\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{p}},% \qquad V=\bigl(\mathbb{E}\bigl[|Y|^{q}\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}% {q}}
  269. 𝔼 [ | X | p 1 { U = 0 } ] = 𝔼 [ 1 { U = 0 } 𝔼 [ | X | p | 𝒢 ] = U p ] = 0 , \mathbb{E}\bigl[|X|^{p}1_{\{U=0\}}\bigr]=\mathbb{E}\bigl[1_{\{U=0\}}% \underbrace{\mathbb{E}\bigl[|X|^{p}\big|\,\mathcal{G}\bigr]}_{=\,U^{p}}\bigr]=0,
  270. | X | = 0 |X|=0
  271. U = 0 {U=0}
  272. | Y | = 0 |Y|=0
  273. V = 0 {V=0}
  274. 𝔼 [ | X Y | | 𝒢 ] = 0 a.s. on { U = 0 } { V = 0 } \mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]=0\qquad\,\text{a.s. on }\{U=0\}% \cup\{V=0\}
  275. { U = , V > 0 } { U > 0 , V = } \{U=\infty,V>0\}\cup\{U>0,V=\infty\}
  276. 𝔼 [ | X Y | | 𝒢 ] U V 1 a.s. on the set H := { 0 < U < , 0 < V < } . \frac{\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]}{UV}\leq 1\qquad\,\text{a.s% . on the set }H:=\{0<U<\infty,\,0<V<\infty\}.
  277. G 𝒢 , G H . G\in\mathcal{G},\quad G\subset H.
  278. 1 / p + 1 / q = 1 1/p+1/q=1
  279. 𝔼 [ 𝔼 [ | X Y | | 𝒢 ] U V 1 G ] = 𝔼 [ 𝔼 [ | X Y | U V 1 G | 𝒢 ] ] = 𝔼 [ | X | U 1 G | Y | V 1 G ] ( 𝔼 [ | X | p U p 1 G ] ) 1 p ( 𝔼 [ | Y | q V q 1 G ] ) 1 q = ( 𝔼 [ 𝔼 [ | X | p | 𝒢 ] U p = 1 a.s. on G 1 G ] ) 1 p ( 𝔼 [ 𝔼 [ | Y | q | 𝒢 ] V p = 1 a.s. on G 1 G ] ) 1 q = 𝔼 [ 1 G ] . \begin{aligned}\displaystyle\mathbb{E}\biggl[\frac{\mathbb{E}\bigl[|XY|\big|\,% \mathcal{G}\bigr]}{UV}1_{G}\biggr]&\displaystyle=\mathbb{E}\biggl[\mathbb{E}% \biggl[\frac{|XY|}{UV}1_{G}\bigg|\,\mathcal{G}\biggr]\biggr]\\ &\displaystyle=\mathbb{E}\biggl[\frac{|X|}{U}1_{G}\cdot\frac{|Y|}{V}1_{G}% \biggr]\\ &\displaystyle\leq\biggl(\mathbb{E}\biggl[\frac{|X|^{p}}{U^{p}}1_{G}\biggr]% \biggr)^{\frac{1}{p}}\biggl(\mathbb{E}\biggl[\frac{|Y|^{q}}{V^{q}}1_{G}\biggr]% \biggr)^{\frac{1}{q}}\\ &\displaystyle=\biggl(\mathbb{E}\biggl[\underbrace{\frac{\mathbb{E}\bigl[|X|^{% p}\big|\,\mathcal{G}\bigr]}{U^{p}}}_{=\,1\,\text{ a.s. on }G}1_{G}\biggr]% \biggr)^{\frac{1}{p}}\biggl(\mathbb{E}\biggl[\underbrace{\frac{\mathbb{E}\bigl% [|Y|^{q}\big|\,\mathcal{G}\bigr]}{V^{p}}}_{=\,1\,\text{ a.s. on }G}1_{G}\biggr% ]\biggr)^{\frac{1}{q}}\\ &\displaystyle=\mathbb{E}\bigl[1_{G}\bigr].\end{aligned}
  280. S S
  281. F ( S , 𝐂 ) F(S,\mathbf{C})
  282. S S
  283. N N
  284. F ( S , 𝐂 ) F(S,\mathbf{C})
  285. f f
  286. g g
  287. F ( S , 𝐂 ) F(S,\mathbf{C})
  288. f ( s ) g ( s ) 0 f(s)≥g(s)≥0
  289. s S s∈S
  290. N ( f ) N ( g ) N(f)≥N(g)
  291. p , q p,q∈
  292. ( 1 , ) (1,∞)
  293. 1 / p + 1 / q = 1 1/p+1/q=1
  294. f , g F ( S , 𝐂 ) f,g∈F(S,\mathbf{C})
  295. N ( | f g | ) ( N ( | f | p ) ) 1 p ( N ( | g | q ) ) 1 q . N(|fg|)\leq\bigl(N(|f|^{p})\bigr)^{\frac{1}{p}}\bigl(N(|g|^{q})\bigr)^{\frac{1% }{q}}.
  296. ( S , Σ , μ ) (S,Σ,μ)
  297. N ( f ) N(f)
  298. | f | |f|
  299. N N

HSL_and_HSV.html

  1. M = max ( R , G , B ) m = min ( R , G , B ) C = M - m \begin{aligned}\displaystyle M&\displaystyle=\operatorname{max}(R,G,B)\\ \displaystyle m&\displaystyle=\operatorname{min}(R,G,B)\\ \displaystyle C&\displaystyle=M-m\end{aligned}
  2. H = { undefined , if C = 0 G - B C mod 6 , if M = R B - R C + 2 , if M = G R - G C + 4 , if M = B H = 60 × H \begin{aligned}\displaystyle H^{\prime}&\displaystyle=\begin{cases}\mathrm{% undefined},&\mbox{if }~{}C=0\\ \frac{G-B}{C}\;\bmod 6,&\mbox{if }~{}M=R\\ \frac{B-R}{C}+2,&\mbox{if }~{}M=G\\ \frac{R-G}{C}+4,&\mbox{if }~{}M=B\end{cases}\\ \displaystyle H&\displaystyle=60^{\circ}\times H^{\prime}\end{aligned}
  3. α = 1 2 ( 2 R - G - B ) β = 3 2 ( G - B ) H 2 = atan2 ( β , α ) C 2 = α 2 + β 2 \begin{aligned}\displaystyle\alpha&\displaystyle=\textstyle{\frac{1}{2}}(2R-G-% B)\\ \displaystyle\beta&\displaystyle=\textstyle{\frac{\sqrt{3}}{2}}(G-B)\\ \displaystyle H_{2}&\displaystyle=\operatorname{atan2}(\beta,\alpha)\\ \displaystyle C_{2}&\displaystyle=\sqrt{\alpha^{2}+\beta^{2}}\end{aligned}
  4. I = 1 3 ( R + G + B ) I=\textstyle{\frac{1}{3}}(R+G+B)\,\!
  5. V = M V=M\,\!
  6. L = 1 2 ( M + m ) L=\textstyle{\frac{1}{2}}(M+m)\,\!
  7. Y 601 = 0.30 R + 0.59 G + 0.11 B Y^{\prime}_{601}=0.30R+0.59G+0.11B\,\!
  8. S H S V = { 0 , if V = 0 C V , otherwise S H S L = { 0 , if C = 0 C 1 - | 2 L - 1 | , otherwise \begin{aligned}\displaystyle S_{HSV}&\displaystyle=\begin{cases}0,&\mbox{if }~% {}V=0\\ \frac{C}{V},&\mbox{otherwise}\end{cases}\\ \displaystyle S_{HSL}&\displaystyle=\begin{cases}0,&\mbox{if }~{}C=0\\ \frac{C}{1-|2L-1|},&\mbox{otherwise}\end{cases}\end{aligned}
  9. S H S I = { 0 , if C = 0 1 - m I , otherwise S_{HSI}=\begin{cases}0,&\mbox{if }~{}C=0\\ 1-\frac{m}{I},&\mbox{otherwise}\end{cases}
  10. C = V × S H S V C=V\times S_{HSV}\,\!
  11. H \displaystyle H^{\prime}
  12. ( R 1 , G 1 , B 1 ) = { ( 0 , 0 , 0 ) if H is undefined ( C , X , 0 ) if 0 H < 1 ( X , C , 0 ) if 1 H < 2 ( 0 , C , X ) if 2 H < 3 ( 0 , X , C ) if 3 H < 4 ( X , 0 , C ) if 4 H < 5 ( C , 0 , X ) if 5 H < 6 (R_{1},G_{1},B_{1})=\begin{cases}(0,0,0)&\mbox{if }~{}H\mbox{ is undefined}\\ (C,X,0)&\mbox{if }~{}0\leq H^{\prime}<1\\ (X,C,0)&\mbox{if }~{}1\leq H^{\prime}<2\\ (0,C,X)&\mbox{if }~{}2\leq H^{\prime}<3\\ (0,X,C)&\mbox{if }~{}3\leq H^{\prime}<4\\ (X,0,C)&\mbox{if }~{}4\leq H^{\prime}<5\\ (C,0,X)&\mbox{if }~{}5\leq H^{\prime}<6\end{cases}
  13. m = V - C \displaystyle m=V-C
  14. C = ( 1 - | 2 L - 1 | ) × S H S L C=\begin{aligned}\displaystyle(1-\left|2L-1\right|)\times S_{HSL}\end{aligned}
  15. H \displaystyle H^{\prime}
  16. ( R 1 , G 1 , B 1 ) = { ( 0 , 0 , 0 ) if H is undefined ( C , X , 0 ) if 0 H < 1 ( X , C , 0 ) if 1 H < 2 ( 0 , C , X ) if 2 H < 3 ( 0 , X , C ) if 3 H < 4 ( X , 0 , C ) if 4 H < 5 ( C , 0 , X ) if 5 H < 6 (R_{1},G_{1},B_{1})=\begin{cases}(0,0,0)&\mbox{if }~{}H\mbox{ is undefined}\\ (C,X,0)&\mbox{if }~{}0\leq H^{\prime}<1\\ (X,C,0)&\mbox{if }~{}1\leq H^{\prime}<2\\ (0,C,X)&\mbox{if }~{}2\leq H^{\prime}<3\\ (0,X,C)&\mbox{if }~{}3\leq H^{\prime}<4\\ (X,0,C)&\mbox{if }~{}4\leq H^{\prime}<5\\ (C,0,X)&\mbox{if }~{}5\leq H^{\prime}<6\end{cases}
  17. m = L - 1 2 C \displaystyle m=L-\textstyle{\frac{1}{2}}C
  18. H \displaystyle H^{\prime}
  19. ( R 1 , G 1 , B 1 ) = { ( 0 , 0 , 0 ) if H is undefined ( C , X , 0 ) if 0 H < 1 ( X , C , 0 ) if 1 H < 2 ( 0 , C , X ) if 2 H < 3 ( 0 , X , C ) if 3 H < 4 ( X , 0 , C ) if 4 H < 5 ( C , 0 , X ) if 5 H < 6 (R_{1},G_{1},B_{1})=\begin{cases}(0,0,0)&\mbox{if }~{}H\mbox{ is undefined}\\ (C,X,0)&\mbox{if }~{}0\leq H^{\prime}<1\\ (X,C,0)&\mbox{if }~{}1\leq H^{\prime}<2\\ (0,C,X)&\mbox{if }~{}2\leq H^{\prime}<3\\ (0,X,C)&\mbox{if }~{}3\leq H^{\prime}<4\\ (X,0,C)&\mbox{if }~{}4\leq H^{\prime}<5\\ (C,0,X)&\mbox{if }~{}5\leq H^{\prime}<6\end{cases}
  20. m = Y 601 - ( .30 R 1 + .59 G 1 + .11 B 1 ) \displaystyle m=Y^{\prime}_{601}-(.30R_{1}+.59G_{1}+.11B_{1})
  21. 3 \sqrt{3}

Hubbert_curve.html

  1. x = e - t ( 1 + e - t ) 2 = 1 2 + 2 cosh t = 1 4 s e c h 2 t 2 . x={e^{-t}\over(1+e^{-t})^{2}}={1\over 2+2\cosh t}={1\over 4}sech^{2}{t\over 2}.

Huge_cardinal.html

  1. M j ( κ ) M . {}^{j(\kappa)}M\subset M.\!
  2. j n ( κ ) {j^{n}(\kappa)}
  3. M < j n ( κ ) M . {}^{<j^{n}(\kappa)}M\subset M.\!
  4. M < j n ( κ ) M . {}^{<j^{n}(\kappa)}M\subset M.\!
  5. M j n ( κ ) M . {}^{j^{n}(\kappa)}M\subset M.\!
  6. M j n ( κ ) M . {}^{j^{n}(\kappa)}M\subset M.\!

Human_Development_Index.html

  1. = LE - 20 85 - 20 =\frac{\textrm{LE}-20}{85-20}
  2. = MYSI + EYSI 2 =\frac{{\textrm{MYSI}+\textrm{EYSI}}}{2}
  3. = MYS 15 =\frac{\textrm{MYS}}{15}
  4. = EYS 18 =\frac{\textrm{EYS}}{18}
  5. = ln ( GNIpc ) - ln ( 100 ) ln ( 75 , 000 ) - ln ( 100 ) =\frac{\ln(\textrm{GNIpc})-\ln(100)}{\ln(75,000)-\ln(100)}
  6. HDI = LEI EI II 3 . \textrm{HDI}=\sqrt[3]{\textrm{LEI}\cdot\textrm{EI}\cdot\textrm{II}}.
  7. x x
  8. x index = x - min ( x ) max ( x ) - min ( x ) x\,\text{ index}=\frac{x-\min\left(x\right)}{\max\left(x\right)-\min\left(x% \right)}
  9. min ( x ) \min\left(x\right)
  10. max ( x ) \max\left(x\right)
  11. x x
  12. L E - 25 85 - 25 \frac{LE-25}{85-25}
  13. 2 3 × A L I + 1 3 × G E I \frac{2}{3}\times ALI+\frac{1}{3}\times GEI
  14. A L R - 0 100 - 0 \frac{ALR-0}{100-0}
  15. C G E R - 0 100 - 0 \frac{CGER-0}{100-0}
  16. log ( G D P p c ) - log ( 100 ) log ( 40000 ) - log ( 100 ) \frac{\log\left(GDPpc\right)-\log\left(100\right)}{\log\left(40000\right)-\log% \left(100\right)}

Humidex.html

  1. Humidex = Air temperature + 0.5555 × ( 6.11 × e 5417.7530 × ( 1 273.16 - 1 dewpoint in kelvins ) - 10 ) \,\text{Humidex}=\,\text{Air temperature}\ +\ 0.5555\times(6.11\times e^{5417.% 7530\times\left(\frac{1}{273.16}-\frac{1}{\,\text{dewpoint in kelvins}}\right)% }-10)

Huzita–Hatori_axioms.html

  1. F ( s ) = p 1 + s ( p 2 - p 1 ) . F(s)=p_{1}+s(p_{2}-p_{1}).
  2. P ( s ) = p 1 + s ( p 2 - p 1 ) P(s)=p_{1}+s(p_{2}-p_{1})
  3. F ( s ) = p mid + s 𝐯 perp . F(s)=p_{\mathrm{mid}}+s\cdot\mathbf{v}^{\mathrm{perp}}.
  4. 𝐮 = ( p 2 - p 1 ) / | ( p 2 - p 1 ) | \mathbf{u}=(p_{2}-p_{1})/\left|(p_{2}-p_{1})\right|
  5. 𝐯 = ( q 2 - q 1 ) / | ( q 2 - q 1 ) | . \mathbf{v}=(q_{2}-q_{1})/\left|(q_{2}-q_{1})\right|.
  6. p int = p 1 + s int 𝐮 p_{\mathrm{int}}=p_{1}+s_{\mathrm{int}}\cdot\mathbf{u}
  7. s i n t = - 𝐯 ( p 1 - q 1 ) 𝐯 𝐮 . s_{int}=-\frac{\mathbf{v}^{\perp}\cdot(p_{1}-q_{1})}{\mathbf{v}^{\perp}\cdot% \mathbf{u}}.
  8. 𝐰 = | 𝐮 | 𝐯 + | 𝐯 | 𝐮 | 𝐮 | + | 𝐯 | . \mathbf{w}=\frac{\left|\mathbf{u}\right|\mathbf{v}+\left|\mathbf{v}\right|% \mathbf{u}}{\left|\mathbf{u}\right|+\left|\mathbf{v}\right|}.
  9. F ( s ) = p int + s 𝐰 . F(s)=p_{\mathrm{int}}+s\cdot\mathbf{w}.
  10. F ( s ) = p 1 + s 𝐯 . F(s)=p_{1}+s\cdot\mathbf{v}.
  11. x = x 1 + s ( x 2 - x 1 ) x=x_{1}+s(x_{2}-x_{1})\,
  12. y = y 1 + s ( y 2 - y 1 ) . y=y_{1}+s(y_{2}-y_{1}).\,
  13. r = | p 1 - p 2 | r=\left|p_{1}-p_{2}\right|
  14. ( x - x c ) 2 + ( y - y c ) 2 = r 2 . (x-x_{c})^{2}+(y-y_{c})^{2}=r^{2}.\,
  15. ( x 1 + s ( x 2 - x 1 ) - x c ) 2 + ( y 1 + s ( y 2 - y 1 ) - y c ) 2 = r 2 . (x_{1}+s(x_{2}-x_{1})-x_{c})^{2}+(y_{1}+s(y_{2}-y_{1})-y_{c})^{2}=r^{2}.\,
  16. a s 2 + b s + c = 0 as^{2}+bs+c=0\,
  17. a = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 a=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}\,
  18. b = 2 ( x 2 - x 1 ) ( x 1 - x c ) + 2 ( y 2 - y 1 ) ( y 1 - y c ) b=2(x_{2}-x_{1})(x_{1}-x_{c})+2(y_{2}-y_{1})(y_{1}-y_{c})\,
  19. c = x c 2 + y c 2 + x 1 2 + y 1 2 - 2 ( x c x 1 + y c y 1 ) - r 2 . c=x_{c}^{2}+y_{c}^{2}+x_{1}^{2}+y_{1}^{2}-2(x_{c}x_{1}+y_{c}y_{1})-r^{2}.\,
  20. - b ± b 2 - 4 a c 2 a . \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  21. m 1 = p 1 d 1 ¯ m_{1}=\overline{p_{1}d_{1}}\,
  22. m 2 = p 1 d 2 ¯ . m_{2}=\overline{p_{1}d_{2}}.\,
  23. F 1 ( s ) \displaystyle F_{1}(s)
  24. ( α , β ) (\alpha,\beta)
  25. α \alpha
  26. β \beta
  27. 1 + α 2 \sqrt{1+\alpha^{2}}
  28. α \alpha
  29. 2 a 3 b ρ 3 2^{a}3^{b}\rho\geq 3
  30. ρ \rho
  31. 2 a ϕ 3 2^{a}\phi\geq 3
  32. ϕ \phi

Hydraulic_jump.html

  1. q , q,
  2. ρ v 0 h 0 \rho v_{0}h_{0}
  3. ρ v 1 h 1 \rho v_{1}h_{1}
  4. v 0 h 0 = v 1 h 1 = q v_{0}h_{0}=v_{1}h_{1}=q
  5. v 1 = v 0 h 0 h 1 , v_{1}=v_{0}{h_{0}\over h_{1}},
  6. ρ \rho
  7. v 0 v_{0}
  8. v 1 v_{1}
  9. h 0 h_{0}
  10. h 1 h_{1}
  11. ρ v 0 2 h 0 + 1 2 ρ g h 0 2 = ρ v 1 2 h 1 + 1 2 ρ g h 1 2 . \rho v_{0}^{2}h_{0}+{1\over 2}\rho gh_{0}^{2}=\rho v_{1}^{2}h_{1}+{1\over 2}% \rho gh_{1}^{2}.
  12. ρ \rho
  13. v 0 2 ( h 0 - h 0 2 h 1 ) + g 2 ( h 0 2 - h 1 2 ) = 0. v_{0}^{2}\left(h_{0}-{h_{0}^{2}\over h_{1}}\right)+{g\over 2}(h_{0}^{2}-h_{1}^% {2})=0.
  14. 1 2 h 1 h 0 ( h 1 h 0 + 1 ) - F r 2 = 0 , {1\over 2}{h_{1}\over h_{0}}\left({h_{1}\over h_{0}}+1\right)-Fr^{2}=0,
  15. F r 2 = v 0 2 g h 0 . Fr^{2}={v_{0}^{2}\over gh_{0}}.
  16. F r Fr
  17. h 1 h 0 = - 1 ± 1 + 8 v 0 2 g h 0 2 . {h_{1}\over h_{0}}=\frac{-1\pm{\sqrt{1+{\frac{8v_{0}^{2}}{gh_{0}}}}}}{2}.
  18. h 1 h 0 = - 1 + 1 + 8 v 0 2 g h 0 2 {h_{1}\over h_{0}}=\frac{-1+{\sqrt{1+{\frac{8v_{0}^{2}}{gh_{0}}}}}}{2}
  19. h 1 h 0 = 1 + < m t p l > 8 F r 2 - 1 2 , {h_{1}\over h_{0}}=\frac{{\sqrt{1+<mtpl>{{8Fr^{2}}}}-1}}{2},
  20. v 0 2 g h 0 = 1 \frac{v_{0}^{2}}{gh_{0}}=1
  21. h 1 h 0 = 1 {h_{1}\over h_{0}}=1
  22. v 0 2 g h 0 < 1 \frac{v_{0}^{2}}{gh_{0}}<1
  23. h 1 h 0 < 1 {h_{1}\over h_{0}}<1
  24. v 0 2 g h 0 > 1 \frac{v_{0}^{2}}{gh_{0}}>1
  25. h 1 h 0 > 1 {h_{1}\over h_{0}}>1
  26. F r > 1 \ Fr>1
  27. g h 0 \ \sqrt{gh_{0}}
  28. F r > 1 Fr>1
  29. Δ E = ( y 2 - y 1 ) 3 4 y 1 y 2 \Delta E=\frac{(y_{2}-y_{1})^{3}}{4y_{1}y_{2}}
  30. h 0 h_{0}
  31. h 0 = h 1 2 ( - 1 + 1 + 8 F r 2 2 ) h_{0}={h_{1}\over 2}\left({-1+\sqrt{1+8Fr_{2}^{2}}}\right)
  32. F r Fr
  33. h 0 h_{0}
  34. h 1 h_{1}

Hyperbolic_geometry.html

  1. 2 π r 2\pi r
  2. R = 1 - K R=\frac{1}{\sqrt{-K}}
  3. K K
  4. 2 π R sinh r R . 2\pi R\sinh\frac{r}{R}\,.
  5. 2 π R 2 ( cosh r R - 1 ) . 2\pi R^{2}(\cosh\frac{r}{R}-1)\,.
  6. x 2 + y 2 + z 2 = 1 , z > 0 x^{2}+y^{2}+z^{2}=1,z>0
  7. ( 0 , 0 , - 1 ) (0,0,-1)
  8. z = 0 z=0
  9. ( 0 , 0 , - 1 ) (0,0,-1)
  10. x 2 + y 2 - z 2 = - 1 , z > 0 x^{2}+y^{2}-z^{2}=-1,z>0
  11. ( - 1 , 0 , 0 ) (-1,0,0)
  12. x = 1 x=1
  13. z = C z=C
  14. z = 1 z=1
  15. O ( 1 , n ) / ( O ( 1 ) × O ( n ) ) . \mathrm{O}(1,n)/(\mathrm{O}(1)\times\mathrm{O}(n)).

Hyperbolic_spiral.html

  1. r = a θ r=\frac{a}{\theta}
  2. x = r cos θ , y = r sin θ , x=r\cos\theta,\qquad y=r\sin\theta,
  3. x = a cos t t , y = a sin t t , x=a{\cos t\over t},\qquad y=a{\sin t\over t},
  4. lim t 0 x = a lim t 0 cos t t = , \lim_{t\to 0}x=a\lim_{t\to 0}{\cos t\over t}=\infty,
  5. lim t 0 y = a lim t 0 sin t t = a 1 = a . \lim_{t\to 0}y=a\lim_{t\to 0}{\sin t\over t}=a\cdot 1=a.
  6. κ = r 2 + 2 r θ 2 - r r θ θ ( r 2 + r θ 2 ) 3 / 2 \kappa={r^{2}+2r_{\theta}^{2}-rr_{\theta\theta}\over(r^{2}+r^{2}_{\theta})^{3/% 2}}
  7. r θ = d r d θ = - a θ 2 r_{\theta}={dr\over d\theta}={-a\over\theta^{2}}
  8. r θ θ = d 2 r d θ 2 = 2 a θ 3 . r_{\theta\theta}={d^{2}r\over d\theta^{2}}={2a\over\theta^{3}}.
  9. θ \theta
  10. κ ( θ ) = θ 4 a ( θ 2 + 1 ) 3 / 2 . \kappa(\theta)={\theta^{4}\over a(\theta^{2}+1)^{3/2}}.
  11. θ \theta
  12. θ \theta
  13. ϕ ( θ ) = - tan - 1 θ . \phi(\theta)=-\tan^{-1}\theta.

Hyperboloid.html

  1. x 2 a 2 + y 2 b 2 - z 2 c 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-{z^{2}\over c^{2}}=1
  2. x 2 a 2 + y 2 b 2 - z 2 c 2 = - 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-{z^{2}\over c^{2}}=-1
  3. x 2 a 2 + y 2 b 2 - z 2 c 2 = 0 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-{z^{2}\over c^{2}}=0
  4. x = a cosh v cos θ x=a\cosh v\cos\theta
  5. y = b cosh v sin θ y=b\cosh v\sin\theta
  6. z = c sinh v z=c\sinh v
  7. x = a sinh v cos θ x=a\sinh v\cos\theta
  8. y = b sinh v sin θ y=b\sinh v\sin\theta
  9. z = ± c cosh v z=\pm c\cosh v
  10. ( 𝐱 - 𝐯 ) T A ( 𝐱 - 𝐯 ) = 1 , (\mathbf{x-v})^{\mathrm{T}}A(\mathbf{x-v})=1,
  11. 1 / a 2 {1/a^{2}}
  12. 1 / b 2 {1/b^{2}}
  13. 1 / c 2 {1/c^{2}}
  14. q ( x ) = ( x 1 2 + + x k 2 ) - ( x k + 1 2 + + x n 2 ) , k < n . q(x)=\left(x_{1}^{2}+\cdots+x_{k}^{2}\right)-\left(x_{k+1}^{2}+\cdots+x_{n}^{2% }\right),\,\quad k<n.
  15. { x : q ( x ) = c } \{x\ :\ q(x)=c\}
  16. ( y 1 , y 4 ) , (y_{1},...y_{4}),
  17. y 1 2 + y 2 2 + y 3 2 - y 4 2 = - 1 , y_{1}^{2}+y_{2}^{2}+y_{3}^{2}-y_{4}^{2}=-1,
  18. y 1 2 + y 2 2 - y 3 2 = - 1 y_{1}^{2}+y_{2}^{2}-y_{3}^{2}=-1
  19. - 1 \sqrt{-1}
  20. \parallel
  21. \parallel
  22. p = ( w , x , y , z ) R 4 p=(w,x,y,z)\in R^{4}
  23. P = { p : w 2 = x 2 + y 2 + z 2 } P=\{p\ :\ w^{2}=x^{2}+y^{2}+z^{2}\}
  24. H r = { p : w = r } , H_{r}=\{p\ :\ w=r\},
  25. P H r P\cap H_{r}
  26. Q = { p : w 2 + z 2 = x 2 + y 2 } Q=\{p\ :\ w^{2}+z^{2}=x^{2}+y^{2}\}
  27. Q H r Q\cap H_{r}

Hypercomputation.html

  1. Σ 1 0 \Sigma^{0}_{1}
  2. Π 1 0 \Pi^{0}_{1}
  3. Δ 2 0 \Delta^{0}_{2}
  4. Σ 2 0 \Sigma^{0}_{2}
  5. Σ 1 0 , Π 1 0 \Sigma^{0}_{1},\Pi^{0}_{1}
  6. Δ 2 0 \Delta^{0}_{2}
  7. Δ k + 1 0 \Delta^{0}_{k+1}
  8. Δ 1 0 [ f ] \Delta^{0}_{1}[f]
  9. T ( Σ 1 1 ) \geq T(\Sigma^{1}_{1})
  10. Σ 1 0 Π 1 0 \Sigma^{0}_{1}\cup\Pi^{0}_{1}
  11. Δ 1 1 \Delta^{1}_{1}
  12. Π 1 1 \Pi^{1}_{1}

Hypergeometric_distribution.html

  1. ( K k ) ( N - K n - k ) ( N n ) {{{K\choose k}{{N-K}\choose{n-k}}}\over{N\choose n}}
  2. 1 - ( n k + 1 ) ( N - n K - k - 1 ) ( N K ) 3 F 2 [ 1 , k + 1 - K , k + 1 - n k + 2 , N + k + 2 - K - n ; 1 ] 1-{{{n\choose{k+1}}{{N-n}\choose{K-k-1}}}\over{N\choose K}}\,_{3}F_{2}\!\!% \left[\begin{array}[]{c}1,\ k+1-K,\ k+1-n\\ k+2,\ N+k+2-K-n\end{array};1\right]
  3. n K N n{K\over N}
  4. ( n + 1 ) ( K + 1 ) N + 2 \left\lfloor\frac{(n+1)(K+1)}{N+2}\right\rfloor
  5. n K N ( N - K ) N N - n N - 1 n{K\over N}{(N-K)\over N}{N-n\over N-1}
  6. ( N - 2 K ) ( N - 1 ) 1 2 ( N - 2 n ) [ n K ( N - K ) ( N - n ) ] 1 2 ( N - 2 ) \frac{(N-2K)(N-1)^{\frac{1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac{1}{2}}(N-2)}
  7. 1 n K ( N - K ) ( N - n ) ( N - 2 ) ( N - 3 ) \left.\frac{1}{nK(N-K)(N-n)(N-2)(N-3)}\cdot\right.
  8. [ ( N - 1 ) N 2 ( N ( N + 1 ) - 6 K ( N - K ) - 6 n ( N - n ) ) + \Big[(N-1)N^{2}\Big(N(N+1)-6K(N-K)-6n(N-n)\Big)+
  9. 6 n K ( N - K ) ( N - n ) ( 5 N - 6 ) ] 6nK(N-K)(N-n)(5N-6)\Big]
  10. ( N - K n ) F 1 2 ( - n , - K ; N - K - n + 1 ; e t ) < m t p l > ( N n ) \frac{{N-K\choose n}\scriptstyle{\,{}_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{<}mtpl>{% {N\choose n}}\,\!
  11. ( N - K n ) F 1 2 ( - n , - K ; N - K - n + 1 ; e i t ) < m t p l > ( N n ) \frac{{N-K\choose n}\scriptstyle{\,{}_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{<}mtpl>% {{N\choose n}}
  12. k k
  13. n n
  14. N N
  15. K K
  16. k k
  17. n n
  18. k k
  19. n n
  20. X X
  21. P ( X = k ) = ( K k ) ( N - K n - k ) ( N n ) P(X=k)=\frac{{\left({{K}\atop{k}}\right)}{\left({{N-K}\atop{n-k}}\right)}}{{% \left({{N}\atop{n}}\right)}}
  22. N N
  23. K K
  24. n n
  25. k k
  26. ( a b ) \textstyle{a\choose b}
  27. max ( 0 , n + K - N ) k min ( K , n ) \max(0,n+K-N)\leq k\leq\min(K,n)
  28. ( k + 1 ) ( N - K - ( n - k - 1 ) ) P ( X = k + 1 ) = ( K - k ) ( n - k ) P ( X = k ) (k+1)(N-K-(n-k-1))P(X=k+1)=(K-k)(n-k)P(X=k)
  29. P ( X = 0 ) = ( N - K n ) ( N n ) P(X=0)=\frac{{\left({{N-K}\atop{n}}\right)}}{{\left({{N}\atop{n}}\right)}}
  30. 0 k n ( K k ) ( N - K n - k ) ( N n ) = 1 \sum_{0\leq k\leq n}{{K\choose k}{N-K\choose n-k}\over{N\choose n}}=1
  31. ( K k ) ( N - K n - k ) ( N n ) = ( n k ) ( N - n K - k ) ( N K ) . {{{K\choose k}{{N-K}\choose{n-k}}}\over{N\choose n}}={{{n\choose k}{{N-n}% \choose{K-k}}}\over{N\choose K}}.
  32. P ( X = k ) = f ( k ; N , K , n ) = ( K k ) ( N - K n - k ) ( N n ) . P(X=k)=f(k;N,K,n)={{{K\choose k}{{N-K}\choose{n-k}}}\over{N\choose n}}.
  33. P ( X = 4 ) = f ( 4 ; 50 , 5 , 10 ) = ( 5 4 ) ( 45 6 ) ( 50 10 ) = 5 8145060 10272278170 = 0.003964583 . P(X=4)=f(4;50,5,10)={{{5\choose 4}{{45}\choose{6}}}\over{50\choose 10}}={5% \cdot 8145060\over 10272278170}=0.003964583\dots.
  34. P ( X = 5 ) = f ( 5 ; 50 , 5 , 10 ) = ( 5 5 ) ( 45 5 ) ( 50 10 ) = 1 1221759 10272278170 = 0.0001189375 , P(X=5)=f(5;50,5,10)={{{5\choose 5}{{45}\choose{5}}}\over{50\choose 10}}={1% \cdot 1221759\over 10272278170}=0.0001189375\dots,
  35. 52 - 5 = 47 52-5=47
  36. k = 1 , n = 2 , K = 9 k=1,n=2,K=9
  37. N = 47 N=47
  38. k = 2 , n = 2 , K = 9 k=2,n=2,K=9
  39. N = 47 N=47
  40. k = 0 , n = 2 , K = 9 k=0,n=2,K=9
  41. N = 47 N=47
  42. f ( k ; N , K , n ) = f ( n - k ; N , N - K , n ) f(k;N,K,n)=f(n-k;N,N-K,n)
  43. f ( k ; N , K , n ) = f ( K - k ; N , K , N - n ) f(k;N,K,n)=f(K-k;N,K,N-n)
  44. f ( k ; N , K , n ) = f ( k ; N , n , K ) f(k;N,K,n)=f(k;N,n,K)
  45. k k
  46. n n
  47. N N
  48. K K
  49. k k
  50. n n
  51. k k
  52. i th i^{\,\text{th}}
  53. P ( W i ) = K N . P(W_{i})={\frac{K}{N}}.
  54. K K
  55. N N
  56. n n
  57. p = K / N p=K/N
  58. n = 1 n=1
  59. X X
  60. p p
  61. Y Y
  62. n n
  63. p p
  64. N N
  65. K K
  66. n n
  67. p p
  68. X X
  69. Y Y
  70. P ( X k ) P ( Y k ) P(X\leq k)\approx P(Y\leq k)
  71. n n
  72. N N
  73. K K
  74. n n
  75. p p
  76. P ( X k ) Φ ( k - n p n p ( 1 - p ) ) P(X\leq k)\approx\Phi\left(\frac{k-np}{\sqrt{np(1-p)}}\right)
  77. Φ \Phi
  78. X X
  79. E ( X i ) = n K i N E(X_{i})=\frac{nK_{i}}{N}
  80. Var ( X i ) = K i N ( 1 - K i N ) n N - n N - 1 \,\text{Var}(X_{i})=\frac{K_{i}}{N}\left(1-\frac{K_{i}}{N}\right)n\frac{N-n}{N% -1}
  81. Cov ( X i , X j ) = - n K i K j N 2 N - n N - 1 \,\text{Cov}(X_{i},X_{j})=-\frac{nK_{i}K_{j}}{N^{2}}\frac{N-n}{N-1}
  82. N = i = 1 c K i N=\sum_{i=1}^{c}K_{i}
  83. P ( 2 black , 2 white , 2 red ) = ( 5 2 ) ( 10 2 ) ( 15 2 ) ( 30 6 ) = 0.079575596816976 P(2\,\text{ black},2\,\text{ white},2\,\text{ red})={{{5\choose 2}{10\choose 2% }{15\choose 2}}\over{30\choose 6}}=0.079575596816976

Hypergraph.html

  1. X = { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 } X=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}
  2. E = { e 1 , e 2 , e 3 , e 4 } = E=\{e_{1},e_{2},e_{3},e_{4}\}=
  3. { { v 1 , v 2 , v 3 } , \{\{v_{1},v_{2},v_{3}\},
  4. { v 2 , v 3 } , \{v_{2},v_{3}\},
  5. { v 3 , v 5 , v 6 } , \{v_{3},v_{5},v_{6}\},
  6. { v 4 } } \{v_{4}\}\}
  7. H H
  8. H = ( X , E ) H=(X,E)
  9. X X
  10. E E
  11. X X
  12. E E
  13. 𝒫 ( X ) { } \mathcal{P}(X)\setminus\{\emptyset\}
  14. 𝒫 ( X ) \mathcal{P}(X)
  15. X X
  16. H = ( X , E ) H=(X,E)
  17. X = { x i | i I v } , X=\{x_{i}|i\in I_{v}\},
  18. E = { e i | i I e , e i X } , E=\{e_{i}|i\in I_{e},e_{i}\subseteq X\},
  19. I v I_{v}
  20. I e I_{e}
  21. H A H_{A}
  22. A A
  23. X X
  24. H A = ( A , { e i A | e i A } ) . H_{A}=\left(A,\{e_{i}\cap A|e_{i}\cap A\neq\varnothing\}\right).
  25. J I e J\subset I_{e}
  26. J J
  27. ( X , { e i | i J } ) . \left(X,\{e_{i}|i\in J\}\right).
  28. A X A\subseteq X
  29. H × A = ( A , { e i | i I e , e i A } ) . H\times A=\left(A,\{e_{i}|i\in I_{e},e_{i}\subseteq A\}\right).
  30. H * H^{*}
  31. H H
  32. { e i } \{e_{i}\}
  33. { X m } \{X_{m}\}
  34. X m = { x m | x m e i } . X_{m}=\{x_{m}|x_{m}\in e_{i}\}.
  35. ( H * ) * = H . \left(H^{*}\right)^{*}=H.
  36. v v v\neq v^{\prime}
  37. f f f\neq f^{\prime}
  38. v , v f v,v^{\prime}\in f
  39. v , v f v,v^{\prime}\in f^{\prime}
  40. H = ( X , E ) H=(X,E)
  41. G = ( Y , F ) G=(Y,F)
  42. H G H\simeq G
  43. ϕ : X Y \phi:X\to Y
  44. π \pi
  45. I I
  46. ϕ ( e i ) = f π ( i ) \phi(e_{i})=f_{\pi(i)}
  47. ϕ \phi
  48. H G H\simeq G
  49. H * G * H^{*}\simeq G^{*}
  50. H H
  51. G G
  52. H G H\cong G
  53. H H
  54. G G
  55. H G H\equiv G
  56. ϕ \phi
  57. ϕ ( x n ) = y n \phi(x_{n})=y_{n}
  58. ϕ ( e i ) = f π ( i ) \phi(e_{i})=f_{\pi(i)}
  59. H G H\equiv G
  60. H * G * H^{*}\cong G^{*}
  61. π \pi
  62. H H
  63. G G
  64. H = G H=G
  65. ( H * ) * = H \left(H^{*}\right)^{*}=H
  66. H H
  67. H = { e 1 = { a , b } , e 2 = { b , c } , e 3 = { c , d } , e 4 = { d , a } , e 5 = { b , d } , e 6 = { a , c } } H=\{e_{1}=\{a,b\},e_{2}=\{b,c\},e_{3}=\{c,d\},e_{4}=\{d,a\},e_{5}=\{b,d\},e_{6% }=\{a,c\}\}
  68. G = { f 1 = { α , β } , f 2 = { β , γ } , f 3 = { γ , δ } , f 4 = { δ , α } , f 5 = { α , γ } , f 6 = { β , δ } } G=\{f_{1}=\{\alpha,\beta\},f_{2}=\{\beta,\gamma\},f_{3}=\{\gamma,\delta\},f_{4% }=\{\delta,\alpha\},f_{5}=\{\alpha,\gamma\},f_{6}=\{\beta,\delta\}\}
  69. H H
  70. G G
  71. ϕ ( a ) = α \phi(a)=\alpha
  72. H H
  73. a a
  74. e 1 e 4 e 6 = { a } e_{1}\cap e_{4}\cap e_{6}=\{a\}
  75. G G
  76. f 1 f 4 f 6 = f_{1}\cap f_{4}\cap f_{6}=\varnothing
  77. H H
  78. G G
  79. H G H\equiv G
  80. H * G * H^{*}\cong G^{*}
  81. r ( H ) r(H)
  82. H H
  83. ϕ ( x ) = y \phi(x)=y
  84. e i e_{i}
  85. e j e_{j}
  86. ϕ ( e i ) = e j \phi(e_{i})=e_{j}
  87. T X T\subseteq X
  88. V = { v 1 , v 2 , , v n } V=\{v_{1},v_{2},~{}\ldots,~{}v_{n}\}
  89. E = { e 1 , e 2 , e m } E=\{e_{1},e_{2},~{}\ldots~{}e_{m}\}
  90. n × m n\times m
  91. A = ( a i j ) A=(a_{ij})
  92. a i j = { 1 if v i e j 0 otherwise . a_{ij}=\left\{\begin{matrix}1&\mathrm{if}~{}v_{i}\in e_{j}\\ 0&\mathrm{otherwise}.\end{matrix}\right.
  93. A t A^{t}
  94. H * = ( V * , E * ) H^{*}=(V^{*},\ E^{*})
  95. H H
  96. V * V^{*}
  97. E * E^{*}
  98. V * V^{*}
  99. v j * V * v^{*}_{j}\in V^{*}
  100. e i * E * , v j * e i * e^{*}_{i}\in E^{*},~{}v^{*}_{j}\in e^{*}_{i}
  101. a i j = 1 a_{ij}=1
  102. { 1 , 2 , 3 , λ } \{1,2,3,...\lambda\}
  103. H = ( X , E ) H=(X,E)
  104. ( X 1 , X 2 , , X K ) (X_{1},X_{2},\cdots,X_{K})
  105. X X
  106. H X k H_{X_{k}}
  107. X k X_{k}
  108. 1 k K 1\leq k\leq K
  109. k = 1 K r ( H X k ) = r ( H ) \sum_{k=1}^{K}r\left(H_{X_{k}}\right)=r(H)
  110. r ( H ) r(H)
  111. V = { a , b } V=\{a,b\}
  112. e 1 = { a , b } e_{1}=\{a,b\}
  113. e 2 = { a , e 1 } e_{2}=\{a,e_{1}\}
  114. b e 1 b\in e_{1}
  115. e 1 e 2 e_{1}\in e_{2}
  116. b e 2 b\in e_{2}
  117. e 1 e_{1}
  118. e 2 e_{2}
  119. e 1 = { e 2 } e_{1}=\{e_{2}\}
  120. e 2 = { e 1 } e_{2}=\{e_{1}\}
  121. [ 0 1 1 0 ] . \left[\begin{matrix}0&1\\ 1&0\end{matrix}\right].

Hypericum_perforatum.html

  1. log P \log{P}

Hyperplane.html

  1. a i a_{i}
  2. a 1 x 1 + a 2 x 2 + + a n x n = b . a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}=b.
  3. a 1 x 1 + a 2 x 2 + + a n x n < b a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}<b
  4. a 1 x 1 + a 2 x 2 + + a n x n > b . a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}>b.
  5. H P H\cap P\neq\varnothing

Hypotenuse.html

  1. c = a 2 + b 2 . c=\sqrt{a^{2}+b^{2}}.
  2. c 2 = a 2 + b 2 - 2 a b cos 90 = a 2 + b 2 c = a 2 + b 2 . c^{2}=a^{2}+b^{2}-2ab\cos 90^{\circ}=a^{2}+b^{2}\therefore c=\sqrt{a^{2}+b^{2}}.
  3. α \alpha\,
  4. β \beta\,
  5. c c\,
  6. b b\,
  7. b c = sin ( β ) \frac{b}{c}=\sin(\beta)\,
  8. β = arcsin ( b c ) \beta\ =\arcsin\left(\frac{b}{c}\right)\,
  9. β \beta\,
  10. b b\,
  11. b b\,
  12. α \alpha\,
  13. β \beta\,
  14. β \beta\,
  15. β = arccos ( a c ) \beta\ =\arccos\left(\frac{a}{c}\right)\,
  16. a a\,

Hypothetical_syllogism.html

  1. P Q , Q R P R \frac{P\to Q,Q\to R}{\therefore P\to R}
  2. P Q P\to Q
  3. Q R Q\to R
  4. P R P\to R
  5. ( P Q ) , ( Q R ) ( P R ) (P\to Q),(Q\to R)\vdash(P\to R)
  6. \vdash
  7. P R P\to R
  8. P Q P\to Q
  9. Q R Q\to R
  10. ( ( P Q ) and ( Q R ) ) ( P R ) ((P\to Q)\and(Q\to R))\to(P\to R)
  11. P P
  12. Q Q
  13. R R

Hysteresis.html

  1. X ( t ) \displaystyle X(t)
  2. Y ( t ) = χ i X ( t ) + 0 Φ d ( τ ) X ( t - τ ) d τ , Y(t)=\chi\text{i}X(t)+\int_{0}^{\infty}\Phi\text{d}(\tau)X(t-\tau)\,\mathrm{d}\tau,
  3. χ i \chi\text{i}
  4. Φ d ( τ ) \Phi_{d}(\tau)
  5. τ \tau
  6. Φ d \Phi_{d}
  7. < v a r > X ( t ) <var>X(t)

Icosian_calculus.html

  1. ι 2 \displaystyle\iota^{2}
  2. λ = ι κ . \lambda=\iota\kappa.\,\!
  3. B B
  4. C C
  5. B C BC
  6. ι \iota
  7. C C
  8. B B
  9. C B CB
  10. κ \kappa
  11. B C BC
  12. D C DC
  13. λ \lambda
  14. B C BC
  15. C D CD

Ideal_class_group.html

  1. R R
  2. N ( a + b - 5 ) = a 2 + 5 b 2 N(a+b\sqrt{-5})=a^{2}+5b^{2}
  3. N ( u v ) = N ( u ) N ( v ) N(uv)=N(u)N(v)
  4. N ( u ) = 1 N(u)=1
  5. u u
  6. R R
  7. J R J\neq R
  8. R R
  9. ( 1 + - 5 ) (1+\sqrt{-5})
  10. Z / 6 Z Z/6Z
  11. R R
  12. J J
  13. Z / 2 Z Z/2Z
  14. N ( x ) N(x)
  15. N ( 2 ) = 4 N(2)=4
  16. N ( 1 + 5 ) = 6 N(1+\sqrt{5})=6
  17. N ( x ) = 1 N(x)=1
  18. x x
  19. J = R J=R
  20. N ( x ) N(x)
  21. b 2 + 5 c 2 = 2 b^{2}+5c^{2}=2

Identity_(mathematics).html

  1. sin 2 θ + cos 2 θ 1 \sin^{2}\theta+\cos^{2}\theta\equiv 1\,
  2. θ \theta
  3. \mathbb{C}
  4. cos θ = 1 , \cos\theta=1,\,
  5. θ \theta
  6. θ = 0 , \theta=0,\,
  7. θ = 2 \theta=2\,
  8. b m + n \displaystyle b^{m+n}
  9. b p q = b ( p q ) ( b p ) q = b ( p q ) = b p q . b^{p^{q}}=b^{(p^{q})}\neq(b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}.
  10. log b ( x y ) = log b ( x ) + log b ( y ) \log_{b}(xy)=\log_{b}(x)+\log_{b}(y)\,
  11. log 3 ( 243 ) = log 3 ( 9 27 ) = log 3 ( 9 ) + log 3 ( 27 ) = 2 + 3 = 5 \log_{3}(243)=\log_{3}(9\cdot 27)=\log_{3}(9)+\log_{3}(27)=2+3=5\,
  12. log b ( x y ) = log b ( x ) - log b ( y ) \log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)\,
  13. log 2 ( 16 ) = log 2 ( 64 4 ) = log 2 ( 64 ) - log 2 ( 4 ) = 6 - 2 = 4 \log_{2}(16)=\log_{2}\!\left(\frac{64}{4}\right)=\log_{2}(64)-\log_{2}(4)=6-2=4
  14. log b ( x p ) = p log b ( x ) \log_{b}(x^{p})=p\log_{b}(x)\,
  15. log 2 ( 64 ) = log 2 ( 2 6 ) = 6 log 2 ( 2 ) = 6 \log_{2}(64)=\log_{2}(2^{6})=6\log_{2}(2)=6\,
  16. log b x p = log b ( x ) p \log_{b}\sqrt[p]{x}=\frac{\log_{b}(x)}{p}\,
  17. log 10 1000 = 1 2 log 10 1000 = 3 2 = 1.5 \log_{10}\sqrt{1000}=\frac{1}{2}\log_{10}1000=\frac{3}{2}=1.5
  18. log b ( x ) = log k ( x ) log k ( b ) . \log_{b}(x)=\frac{\log_{k}(x)}{\log_{k}(b)}.\,
  19. log b ( x ) = log 10 ( x ) log 10 ( b ) = log e ( x ) log e ( b ) . \log_{b}(x)=\frac{\log_{10}(x)}{\log_{10}(b)}=\frac{\log_{e}(x)}{\log_{e}(b)}.\,
  20. b = x 1 log b ( x ) . b=x^{\frac{1}{\log_{b}(x)}}.

Igloo.html

  1. S α = γ d 2 24 h 1 + cos α + cos 2 α ( 1 + cos α ) cos 2 α \mathrm{S}\alpha=\frac{\gamma\mathrm{d}^{2}}{24h}\cdot\frac{1+\cos\alpha+\cos^% {2}\alpha}{(1+\cos\alpha)\cos^{2}\alpha}

Imagery_intelligence.html

  1. sin θ = 1.22 λ D \sin\theta=1.22\frac{\lambda}{D}
  2. sin θ = s i z e d i s t a n c e \sin\theta=\frac{size}{distance}
  3. s i z e = 1.22 λ D d i s t a n c e size=1.22\frac{\lambda}{D}distance
  4. λ 550 \lambda\approx 550

Implicit_function.html

  1. x 2 + y 2 - 1 = 0. x^{2}+y^{2}-1=0.
  2. x 2 + [ f ( x ) ] 2 - 1 = 0. x^{2}+[f(x)]^{2}-1=0.
  3. y = f ( x ) y=f(x)
  4. x = f - 1 ( y ) . x=f^{-1}(y).
  5. R ( x , y ) = y - f ( x ) = 0. R(x,y)=y-f(x)=0.\,
  6. a n ( x ) y n + a n - 1 ( x ) y n - 1 + + a 0 ( x ) = 0 a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots+a_{0}(x)=0\,
  7. x 2 + y 2 - 1 = 0. x^{2}+y^{2}-1=0.\,
  8. y = ± 1 - x 2 . y=\pm\sqrt{1-x^{2}}.\,
  9. y 5 + 2 y 4 - 7 y 3 + 3 y 2 - 6 y - x = 0. y^{5}+2y^{4}-7y^{3}+3y^{2}-6y-x=0.\,
  10. y + x + 5 = 0. y+x+5=0.
  11. y = - x - 5 , y=-x-5\,,
  12. d y d x + d x d x + d d x ( 5 ) = 0 ; \frac{dy}{dx}+\frac{dx}{dx}+\frac{d}{dx}(5)=0;
  13. d y d x + 1 = 0. \frac{dy}{dx}+1=0.
  14. d y d x = - 1 , \frac{dy}{dx}=-1,
  15. x 4 + 2 y 2 = 8. x^{4}+2y^{2}=8.
  16. y = f ( x ) = ± 8 - x 4 2 , y=f(x)=\pm\sqrt{\frac{8-x^{4}}{2}},
  17. d y d x = - 4 x 3 4 y = - x 3 y . \frac{dy}{dx}=\frac{-4x^{3}}{4y}=\frac{-x^{3}}{y}.
  18. y 5 - y = x . y^{5}-y=x.
  19. 5 y 4 d y d x - d y d x = d x d x 5y^{4}\frac{dy}{dx}-\frac{dy}{dx}=\frac{dx}{dx}
  20. d y d x ( 5 y 4 - 1 ) = 1 \frac{dy}{dx}(5y^{4}-1)=1
  21. d y d x = 1 5 y 4 - 1 , \frac{dy}{dx}=\frac{1}{5y^{4}-1},
  22. y ± 1 5 4 . y\neq\pm\frac{1}{\sqrt[4]{5}}.
  23. R ( x , y ) = 0 , R(x,y)=0,
  24. d y d x = - R / x R / y = - R x R y , \frac{dy}{dx}=-\frac{\partial R/\partial x}{\partial R/\partial y}=-\frac{R_{x% }}{R_{y}},
  25. R x d x d x + R y d y d x = 0 , \frac{\partial R}{\partial x}\frac{dx}{dx}+\frac{\partial R}{\partial y}\frac{% dy}{dx}=0,
  26. R x + R y d y d x = 0. \frac{\partial R}{\partial x}+\frac{\partial R}{\partial y}\frac{dy}{dx}=0.
  27. R ( x , y ) R(x,y)
  28. M M
  29. ( a , b ) (a,b)
  30. R / y 0 ∂R/∂y≠0
  31. M M
  32. ( a , b ) (a,b)
  33. ( x , f ( x ) ) (x,f(x))
  34. f f
  35. R ( x , y ) = 0 R(x,y)=0
  36. R R

Implied_volatility.html

  1. C = f ( σ , ) C=f(\sigma,\cdot)\,
  2. f ( σ , ) f(\sigma,\cdot)\,
  3. σ C ¯ = g ( C ¯ , ) \sigma_{\bar{C}}=g(\bar{C},\cdot)\,
  4. C ¯ \scriptstyle\bar{C}\,
  5. σ C ¯ \sigma_{\bar{C}}\,
  6. C ¯ \scriptstyle\bar{C}\,
  7. C X Y Z C_{XYZ}
  8. C X Y Z C_{XYZ}
  9. C X Y Z C_{XYZ}
  10. σ C ¯ = g ( C ¯ , ) = 18.7 % \sigma_{\bar{C}}=g(\bar{C},\cdot)=18.7\%
  11. C t h e o = f ( σ C ¯ , ) = $ 2.0004 C_{theo}=f(\sigma_{\bar{C}},\cdot)=\$2.0004
  12. f ( σ C ¯ , ) - C ¯ = 0 f(\sigma_{\bar{C}},\cdot)-\bar{C}=0\,
  13. C σ \frac{\partial C}{\partial\sigma}\,

Improper_rotation.html

  1. n ¯ \bar{n}

Impulse_(physics).html

  1. J = F a v e r a g e ( t 2 - t 1 ) J=F_{average}(t_{2}-t_{1})
  2. J = F d t J=\int Fdt
  3. 𝐉 = t 1 t 2 𝐅 d t \mathbf{J}=\int_{t_{1}}^{t_{2}}\mathbf{F}\,dt
  4. 𝐅 = d 𝐩 d t \mathbf{F}=\frac{d\mathbf{p}}{dt}
  5. 𝐉 = t 1 t 2 d 𝐩 d t d t = p 1 p 2 d 𝐩 = 𝐩 𝟐 - 𝐩 𝟏 = Δ 𝐩 \begin{aligned}\displaystyle\mathbf{J}&\displaystyle=\int_{t_{1}}^{t_{2}}\frac% {d\mathbf{p}}{dt}\,dt\\ &\displaystyle=\int_{p_{1}}^{p_{2}}d\mathbf{p}\\ &\displaystyle=\mathbf{p_{2}}-\mathbf{p_{1}}=\Delta\mathbf{p}\end{aligned}
  6. 𝐉 = t 1 t 2 𝐅 d t = Δ 𝐩 = m 𝐯 𝟐 - m 𝐯 𝟏 \mathbf{J}=\int_{t_{1}}^{t_{2}}\mathbf{F}\,dt=\Delta\mathbf{p}=m\mathbf{v_{2}}% -m\mathbf{v_{1}}

Inaccessible_cardinal.html

  1. 0 \aleph_{0}
  2. 0 \aleph_{0}
  3. 0 \aleph_{0}
  4. ( V κ , , U ) (V_{\kappa},\in,U)
  5. Π n 0 \Pi_{n}^{0}
  6. \models

Incidence_algebra.html

  1. ( f * g ) ( a , b ) = a x b f ( a , x ) g ( x , b ) . (f*g)(a,b)=\sum_{a\leq x\leq b}f(a,x)g(x,b).
  2. δ ( a , b ) = { 1 if a = b 0 if a < b . \delta(a,b)=\begin{cases}1&\,\text{if }a=b\\ 0&\,\text{if }a<b.\end{cases}
  3. μ ( x , y ) = { 1 if x = y - z : x z < y μ ( x , z ) for x < y 0 otherwise . \mu(x,y)=\begin{cases}{}\qquad 1&\textrm{if}\quad x=y\\ \displaystyle-\sum_{z\,:\,x\leq z<y}\mu(x,z)&\textrm{for}\quad x<y\\ {}\qquad 0&\textrm{otherwise}.\end{cases}
  4. μ ( S , T ) = ( - 1 ) | T S | \mu(S,T)=(-1)^{\left|T\setminus S\right|}
  5. 2 E = { 0 , 1 } E . 2^{E}=\{0,1\}^{E}.
  6. μ ( x , y ) = { 1 if y - x = 0 , - 1 if y - x = 1 , 0 if y - x > 1 , \mu(x,y)=\begin{cases}1&\,\text{if }y-x=0,\\ -1&\,\text{if }y-x=1,\\ 0&\,\text{if }y-x>1,\end{cases}
  7. ( 1 - z ) - 1 = 1 + z + z 2 + z 3 + (1-z)^{-1}=1+z+z^{2}+z^{3}+\cdots
  8. μ G ( H 1 , H 2 ) = ( - 1 ) k p ( k 2 ) \mu_{G}(H_{1},H_{2})=(-1)^{k}p^{{\left({{k}\atop{2}}\right)}}
  9. H 1 H_{1}
  10. H 2 H_{2}
  11. H 2 / H 1 ( 𝐙 / p 𝐙 ) k H_{2}/H_{1}\cong({\mathbf{Z}}/p{\mathbf{Z}})^{k}
  12. μ ( S , T ) = { 0 if T S is a proper multiset (has repeated elements) ( - 1 ) | T S | if T S is a set (has no repeated elements) . \mu(S,T)=\begin{cases}0&\,\text{if }T\setminus S\,\text{ is a proper multiset % (has repeated elements)}\\ (-1)^{\left|T\setminus S\right|}&\,\text{if }T\setminus S\,\text{ is a set (% has no repeated elements)}.\end{cases}
  13. { 2 , 2 , 3 } . \{2,2,3\}.
  14. { 1 , 1 , 1 } . \{1,1,1\}.
  15. μ ( σ , τ ) = ( - 1 ) n - r ( 2 ! ) r 3 ( 3 ! ) r 4 ( ( n - 1 ) ! ) r n \mu(\sigma,\tau)=(-1)^{n-r}(2!)^{r_{3}}(3!)^{r_{4}}\cdots((n-1)!)^{r_{n}}

Incircle_and_excircles_of_a_triangle.html

  1. A B C \triangle ABC
  2. A C I \angle AC^{\prime}I
  3. I A B \triangle IAB
  4. I A B \triangle IAB
  5. 1 2 c r \tfrac{1}{2}cr
  6. I A C \triangle IAC
  7. 1 2 b r \tfrac{1}{2}br
  8. I B C \triangle IBC
  9. 1 2 a r \tfrac{1}{2}ar
  10. A B C \triangle ABC
  11. Δ = 1 2 ( a + b + c ) r = s r , \Delta=\frac{1}{2}(a+b+c)r=sr,
  12. r = Δ s , r=\frac{\Delta}{s},
  13. Δ \Delta
  14. A B C \triangle ABC
  15. s = 1 2 ( a + b + c ) s=\frac{1}{2}(a+b+c)
  16. I C A \triangle IC^{\prime}A
  17. r cot A 2 r\cot\frac{\angle A}{2}
  18. I B A \triangle IB^{\prime}A
  19. Δ = r 2 ( cot A 2 + cot B 2 + cot C 2 ) \Delta=r^{2}\cdot(\cot\frac{\angle A}{2}+\cot\frac{\angle B}{2}+\cot\frac{% \angle C}{2})
  20. r c r_{c}
  21. I c I_{c}
  22. I c G I_{c}G
  23. A C I c \triangle ACI_{c}
  24. A C I c \triangle ACI_{c}
  25. 1 2 b r c \tfrac{1}{2}br_{c}
  26. B C I c \triangle BCI_{c}
  27. 1 2 a r c \tfrac{1}{2}ar_{c}
  28. A B I c \triangle ABI_{c}
  29. 1 2 c r c \tfrac{1}{2}cr_{c}
  30. Δ = 1 2 ( a + b - c ) r c = ( s - c ) r c \Delta=\frac{1}{2}(a+b-c)r_{c}=(s-c)r_{c}
  31. Δ = s r = ( s - a ) r a = ( s - b ) r b = ( s - c ) r c \Delta=sr=(s-a)r_{a}=(s-b)r_{b}=(s-c)r_{c}
  32. cos A = b 2 + c 2 - a 2 2 b c \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}
  33. sin 2 A + cos 2 A = 1 \sin^{2}A+\cos^{2}A=1
  34. sin A = - a 4 - b 4 - c 4 + 2 a 2 b 2 + 2 b 2 c 2 + 2 a 2 c 2 2 b c \sin A=\frac{\sqrt{-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}}{2bc}
  35. Δ = 1 2 b c sin A \Delta=\tfrac{1}{2}bc\sin A
  36. Δ = 1 4 - a 4 - b 4 - c 4 + 2 a 2 b 2 + 2 b 2 c 2 + 2 a 2 c 2 = 1 4 ( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c ) = s ( s - a ) ( s - b ) ( s - c ) , \begin{aligned}\displaystyle\Delta&\displaystyle=\frac{1}{4}\sqrt{-a^{4}-b^{4}% -c^{4}+2a^{2}b^{2}+2b^{2}c^{2}+2a^{2}c^{2}}\\ &\displaystyle=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\\ &\displaystyle=\sqrt{s(s-a)(s-b)(s-c)},\end{aligned}
  37. s r = Δ sr=\Delta
  38. r 2 = Δ 2 s 2 = ( s - a ) ( s - b ) ( s - c ) s . r^{2}=\frac{\Delta^{2}}{s^{2}}=\frac{(s-a)(s-b)(s-c)}{s}.
  39. ( s - a ) r a = Δ (s-a)r_{a}=\Delta
  40. r a 2 = s ( s - b ) ( s - c ) s - a r_{a}^{2}=\frac{s(s-b)(s-c)}{s-a}
  41. r a = s ( s - b ) ( s - c ) s - a . r_{a}=\sqrt{\frac{s(s-b)(s-c)}{s-a}}.
  42. Δ = r r a r b r c . \Delta=\sqrt{rr_{a}r_{b}r_{c}}.
  43. π 3 3 \frac{\pi}{3\sqrt{3}}
  44. vertex A = 0 : sec 2 ( B 2 ) : sec 2 ( C 2 ) \,\text{vertex}\,A=0:\sec^{2}\left(\frac{B}{2}\right):\sec^{2}\left(\frac{C}{2% }\right)
  45. vertex B = sec 2 ( A 2 ) : 0 : sec 2 ( C 2 ) \,\text{vertex}\,B=\sec^{2}\left(\frac{A}{2}\right):0:\sec^{2}\left(\frac{C}{2% }\right)
  46. vertex C = sec 2 ( A 2 ) : sec 2 ( B 2 ) : 0 \,\text{vertex}\,C=\sec^{2}\left(\frac{A}{2}\right):\sec^{2}\left(\frac{B}{2}% \right):0
  47. sec 2 ( A 2 ) : sec 2 ( B 2 ) : sec 2 ( C 2 ) \sec^{2}\left(\frac{A}{2}\right):\sec^{2}\left(\frac{B}{2}\right):\sec^{2}% \left(\frac{C}{2}\right)
  48. b c b + c - a : c a c + a - b : a b a + b - c \frac{bc}{b+c-a}:\frac{ca}{c+a-b}:\frac{ab}{a+b-c}
  49. vertex A = 0 : csc 2 ( B 2 ) : csc 2 ( C 2 ) \,\text{vertex}\,A=0:\csc^{2}\left(\frac{B}{2}\right):\csc^{2}\left(\frac{C}{2% }\right)
  50. vertex B = csc 2 ( A 2 ) : 0 : csc 2 ( C 2 ) \,\text{vertex}\,B=\csc^{2}\left(\frac{A}{2}\right):0:\csc^{2}\left(\frac{C}{2% }\right)
  51. vertex C = csc 2 ( A 2 ) : csc 2 ( B 2 ) : 0 \,\text{vertex}\,C=\csc^{2}\left(\frac{A}{2}\right):\csc^{2}\left(\frac{B}{2}% \right):0
  52. csc 2 ( A 2 ) : csc 2 ( B 2 ) : csc 2 ( C 2 ) \csc^{2}\left(\frac{A}{2}\right):\csc^{2}\left(\frac{B}{2}\right):\csc^{2}% \left(\frac{C}{2}\right)
  53. b + c - a a : c + a - b b : a + b - c c \frac{b+c-a}{a}:\frac{c+a-b}{b}:\frac{a+b-c}{c}
  54. vertex A = 0 : 1 : 1 \ \,\text{vertex}\,A=0:1:1
  55. vertex B = 1 : 0 : 1 \ \,\text{vertex}\,B=1:0:1
  56. vertex C = 1 : 1 : 0 \ \,\text{vertex}\,C=1:1:0
  57. vertex A = - 1 : 1 : 1 \,\text{vertex}\,A=-1:1:1
  58. vertex B = 1 : - 1 : 1 \,\text{vertex}\,B=1:-1:1
  59. vertex C = 1 : - 1 : - 1 \,\text{vertex}\,C=1:-1:-1
  60. u 2 x 2 + v 2 y 2 + w 2 z 2 - 2 v w y z - 2 w u z x - 2 u v x y = 0 \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0
  61. ± x cos A 2 ± y cos B 2 ± z cos C 2 = 0 \pm\sqrt{x}\cos\frac{A}{2}\pm\sqrt{y}\cos\frac{B}{2}\pm\sqrt{z}\cos\frac{C}{2}=0
  62. u 2 x 2 + v 2 y 2 + w 2 z 2 - 2 v w y z + 2 w u z x + 2 u v x y = 0 \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz+2wuzx+2uvxy=0
  63. ± - x cos A 2 ± y cos B 2 ± z cos C 2 = 0 \pm\sqrt{-x}\cos\frac{A}{2}\pm\sqrt{y}\cos\frac{B}{2}\pm\sqrt{z}\cos\frac{C}{2% }=0
  64. u 2 x 2 + v 2 y 2 + w 2 z 2 + 2 v w y z - 2 w u z x + 2 u v x y = 0 \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz-2wuzx+2uvxy=0
  65. ± x cos A 2 ± - y cos B 2 ± z cos C 2 = 0 \pm\sqrt{x}\cos\frac{A}{2}\pm\sqrt{-y}\cos\frac{B}{2}\pm\sqrt{z}\cos\frac{C}{2% }=0
  66. u 2 x 2 + v 2 y 2 + w 2 z 2 + 2 v w y z + 2 w u z x - 2 u v x y = 0 \ u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}+2vwyz+2wuzx-2uvxy=0
  67. ± x cos A 2 ± y cos B 2 ± - z cos C 2 = 0 \pm\sqrt{x}\cos\frac{A}{2}\pm\sqrt{y}\cos\frac{B}{2}\pm\sqrt{-z}\cos\frac{C}{2% }=0
  68. ( R - r i n ) 2 = d 2 + r i n 2 , (R-r_{in})^{2}=d^{2}+r_{in}^{2},
  69. ( R + r e x ) 2 = d 2 + r e x 2 , (R+r_{ex})^{2}=d^{2}+r_{ex}^{2},
  70. r = x y z x + y + z r=\sqrt{\frac{xyz}{x+y+z}}
  71. K = x y z ( x + y + z ) . K=\sqrt{xyz(x+y+z)}.
  72. r = 1 h a - 1 + h b - 1 + h c - 1 . r=\frac{1}{h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1}}.
  73. r R = a b c 2 ( a + b + c ) . rR=\frac{abc}{2(a+b+c)}.
  74. a b + b c + c a = s 2 + ( 4 R + r ) r , ab+bc+ca=s^{2}+(4R+r)r,
  75. a 2 + b 2 + c 2 = 2 s 2 - 2 ( 4 R + r ) r . a^{2}+b^{2}+c^{2}=2s^{2}-2(4R+r)r.
  76. I A I A C A A B + I B I B A B B C + I C I C B C C A = 1 \frac{IA\cdot IA}{CA\cdot AB}+\frac{IB\cdot IB}{AB\cdot BC}+\frac{IC\cdot IC}{% BC\cdot CA}=1
  77. I A I B I C = 4 R r 2 . IA\cdot IB\cdot IC=4Rr^{2}.
  78. B T A = B T C = B C + A B - A C 2 . BT_{A}=BT_{C}=\frac{BC+AB-AC}{2}.
  79. O I 2 = R ( R - 2 r ) , OI^{2}=R(R-2r),
  80. I N = 1 2 ( R - 2 r ) < 1 2 R . IN=\frac{1}{2}(R-2r)<\frac{1}{2}R.
  81. r 2 + s 2 4 r \frac{r^{2}+s^{2}}{4r}
  82. r a + r b + r c = 4 R + r , r_{a}+r_{b}+r_{c}=4R+r,
  83. r a r b + r b r c + r c r a = s 2 , r_{a}r_{b}+r_{b}r_{c}+r_{c}r_{a}=s^{2},
  84. r a 2 + r b 2 + r c 2 = ( 4 R + r ) 2 - 2 s 2 , r_{a}^{2}+r_{b}^{2}+r_{c}^{2}=(4R+r)^{2}-2s^{2},
  85. r a + r b + r c + r = A H + B H + C H + 2 R , r_{a}+r_{b}+r_{c}+r=AH+BH+CH+2R,
  86. r a 2 + r b 2 + r c 2 + r 2 = A H 2 + B H 2 + C H 2 + ( 2 R ) 2 . r_{a}^{2}+r_{b}^{2}+r_{c}^{2}+r^{2}=AH^{2}+BH^{2}+CH^{2}+(2R)^{2}.

Income_statement.html

  1. Earnings per share = Net income - Preferred stock dividends Weighted average of common stock shares outstanding \,\text{Earnings per share}=\frac{\,\text{Net income}-\,\text{Preferred stock % dividends}}{\,\text{Weighted average of common stock shares outstanding}}

Incremental_cost-effectiveness_ratio.html

  1. I C E R = ( C 1 - C 0 ) ( E 1 - E 0 ) ICER=\frac{(C_{1}-C_{0})}{(E_{1}-E_{0})}
  2. C 1 C_{1}
  3. E 1 E_{1}
  4. C 0 C_{0}
  5. E 0 E_{0}

Indefinite_orthogonal_group.html

  1. z j i z j z_{j}\mapsto iz_{j}
  2. [ cosh ( α ) sinh ( α ) sinh ( α ) cosh ( α ) ] , \left[\begin{smallmatrix}\cosh(\alpha)&\sinh(\alpha)\\ \sinh(\alpha)&\cosh(\alpha)\end{smallmatrix}\right],
  3. η = diag ( 1 , , 1 p , - 1 , , - 1 q ) . \eta=\mathrm{diag}(\underbrace{1,\ldots,1}_{p},\underbrace{-1,\ldots,-1}_{q}).
  4. Q ( x 1 , , x n ) = x 1 2 + + x p 2 - x p + 1 2 - - x p + q 2 . Q(x_{1},\dots,x_{n})=x_{1}^{2}+\cdots+x_{p}^{2}-x_{p+1}^{2}-\cdots-x_{p+q}^{2}.
  5. Q ( M v ) = Q ( v ) Q(Mv)=Q(v)
  6. M T η M = η . M\text{T}\eta M=\eta.
  7. M - 1 = η - 1 M T η . M^{-1}=\eta^{-1}M\text{T}\eta.

Independence_of_irrelevant_alternatives.html

  1. R 1 R_{1}
  2. R n R_{n}
  3. R 1 R_{1}^{\prime}
  4. R n R_{n}^{\prime}
  5. R i R_{i}
  6. R i R_{i}^{\prime}
  7. L M \,L\prec M\,
  8. N \,N\,
  9. p ( 0 , 1 ] \,p\in(0,1]\,
  10. p L + ( 1 - p ) N p M + ( 1 - p ) N , \,pL+(1-p)N\prec pM+(1-p)N,
  11. L M \,L\prec M\,
  12. p = ( R 1 , , R n ) p=(R_{1},\ldots,R_{n})
  13. p = ( R 1 , , R n ) p^{\prime}=(R^{\prime}_{1},\ldots,R^{\prime}_{n})
  14. R i { x , y } 2 = R i { x , y } 2 R_{i}\cap\{x,y\}^{2}=R^{\prime}_{i}\cap\{x,y\}^{2}
  15. f ( p ) { x , y } 2 = f ( p ) { x , y } 2 f(p)\cap\{x,y\}^{2}=f(p^{\prime})\cap\{x,y\}^{2}

Index_of_a_subgroup.html

  1. | 𝐙 : n 𝐙 | = n |\mathbf{Z}:n\mathbf{Z}|=n
  2. | G : H | = | G | | H | . |G:H|=\frac{|G|}{|H|}.
  3. | G : K | = | G : H | | H : K | . |G:K|=|G:H|\,|H:K|.
  4. | G : H K | | G : H | | G : K | , |G:H\cap K|\leq|G:H|\,|G:K|,
  5. | H : H K | | G : K | , |H:H\cap K|\leq|G:K|,
  6. | G : ker φ | = | im φ | . |G:\operatorname{ker}\;\varphi|=|\operatorname{im}\;\varphi|.
  7. | G x | = | G : G x | . |Gx|=|G:G_{x}|.\!
  8. | G : Core ( H ) | | G : H | ! |G:\operatorname{Core}(H)|\leq|G:H|!
  9. A n A_{n}
  10. S n , S_{n},
  11. { ( x , y ) x is even } , { ( x , y ) y is even } , and { ( x , y ) x + y is even } \{(x,y)\mid x\,\text{ is even}\},\quad\{(x,y)\mid y\,\text{ is even}\},\quad\,% \text{and}\quad\{(x,y)\mid x+y\,\text{ is even}\}
  12. p k p^{k}
  13. [ G , G ] [G,G]
  14. p k p^{k}
  15. 𝐄 p ( G ) 𝐀 p ( G ) 𝐎 p ( G ) . \mathbf{E}^{p}(G)\supseteq\mathbf{A}^{p}(G)\supseteq\mathbf{O}^{p}(G).
  16. G / 𝐄 p ( G ) ( 𝐙 / p ) k G/\mathbf{E}^{p}(G)\cong(\mathbf{Z}/p)^{k}
  17. 𝐏 ( Hom ( G , 𝐙 / p ) ) . \mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)).
  18. Hom ( G , 𝐙 / p ) , \operatorname{Hom}(G,\mathbf{Z}/p),
  19. 𝐅 p = 𝐙 / p . \mathbf{F}_{p}=\mathbf{Z}/p.
  20. ( 𝐙 / p ) × (\mathbf{Z}/p)^{\times}
  21. 𝐏 ( Hom ( G , 𝐙 / p ) ) := ( Hom ( G , 𝐙 / p ) ) { 0 } ) / ( 𝐙 / p ) × \mathbf{P}(\operatorname{Hom}(G,\mathbf{Z}/p)):=(\operatorname{Hom}(G,\mathbf{% Z}/p))\setminus\{0\})/(\mathbf{Z}/p)^{\times}
  22. 𝐙 / p \mathbf{Z}/p
  23. 1 𝐙 / p , 1\in\mathbf{Z}/p,
  24. ( p k + 1 - 1 ) / ( p - 1 ) = 1 + p + + p k (p^{k+1}-1)/(p-1)=1+p+\cdots+p^{k}
  25. k = - 1 k=-1
  26. p + 1 p+1
  27. p = 2 , p=2,
  28. 0 , 1 , 3 , 7 , 15 , 0,1,3,7,15,\ldots

Index_of_coincidence.html

  1. 𝐈𝐂 = c × ( ( n a N × n a - 1 N - 1 ) + ( n b N × n b - 1 N - 1 ) + + ( n z N × n z - 1 N - 1 ) ) \mathbf{IC}=c\times\left({\left({\frac{n_{\mathrm{a}}}{N}\times\frac{n_{% \mathrm{a}}-1}{N-1}}\right)+\left({\frac{n_{\mathrm{b}}}{N}\times\frac{n_{% \mathrm{b}}-1}{N-1}}\right)+...+\left({\frac{n_{\mathrm{z}}}{N}\times\frac{n_{% \mathrm{z}}-1}{N-1}}\right)}\right)
  2. 𝐈𝐂 = i = 1 c n i ( n i - 1 ) N ( N - 1 ) / c \mathbf{IC}=\frac{\displaystyle\sum_{i=1}^{c}n_{i}(n_{i}-1)}{N(N-1)/c}
  3. n ( n 1 ) n(n−1)
  4. N ( N 1 ) N(N−1)
  5. 𝐈𝐂 expected = i = 1 c f i 2 1 / c . \mathbf{IC}_{\mathrm{expected}}=\frac{\displaystyle\sum_{i=1}^{c}{f_{i}}^{2}}{% 1/c}.
  6. c c
  7. 0.067 = 1.73 / 26 0.067=1.73/26
  8. 1 / c 1/c
  9. 0.0385 = 1 / 26 0.0385=1/26
  10. c c
  11. N N
  12. 𝐈𝐂 = j = 1 N [ a j = b j ] N / c , \mathbf{IC}=\frac{\displaystyle\sum_{j=1}^{N}[a_{j}=b_{j}]}{N/c},
  13. N N
  14. j j
  15. j j
  16. χ = i = 1 c n i f i \mathbf{\chi}=\sum_{i=1}^{c}n_{i}f_{i}
  17. n i n_{i}
  18. f i f_{i}

Indicator_function.html

  1. 𝟏 A : X { 0 , 1 } \mathbf{1}_{A}\colon X\to\{0,1\}\,
  2. 𝟏 A ( x ) := { 1 if x A , 0 if x A . \mathbf{1}_{A}(x):=\begin{cases}1&\,\text{if }x\in A,\\ 0&\,\text{if }x\notin A.\end{cases}
  3. [ x A ] [x\in A]
  4. 𝟏 A ( x ) \mathbf{1}_{A}(x)
  5. 𝟏 A \mathbf{1}_{A}
  6. I A I_{A}
  7. χ A \chi_{A}
  8. A A
  9. χ \chi
  10. 1 A 1_{A}
  11. χ A \chi_{A}
  12. \cap
  13. \cup
  14. A A
  15. B B
  16. X X
  17. 𝟏 A B = min { 𝟏 A , 𝟏 B } = 𝟏 A 𝟏 B , \mathbf{1}_{A\cap B}=\min\{\mathbf{1}_{A},\mathbf{1}_{B}\}=\mathbf{1}_{A}\cdot% \mathbf{1}_{B},
  18. 𝟏 A B = max { 𝟏 A , 𝟏 B } = 𝟏 A + 𝟏 B - 𝟏 A 𝟏 B , \mathbf{1}_{A\cup B}=\max\{{\mathbf{1}_{A},\mathbf{1}_{B}}\}=\mathbf{1}_{A}+% \mathbf{1}_{B}-\mathbf{1}_{A}\cdot\mathbf{1}_{B},
  19. A A
  20. A C A^{C}
  21. 𝟏 A = 1 - 𝟏 A \mathbf{1}_{A^{\complement}}=1-\mathbf{1}_{A}
  22. A 1 , , A n A_{1},\ldots,A_{n}
  23. k I ( 1 - 𝟏 A k ( x ) ) \prod_{k\in I}(1-\mathbf{1}_{A_{k}}(x))
  24. k I ( 1 - 𝟏 A k ) = 𝟏 X - k A k = 1 - 𝟏 k A k . \prod_{k\in I}(1-\mathbf{1}_{A_{k}})=\mathbf{1}_{X-\bigcup_{k}A_{k}}=1-\mathbf% {1}_{\bigcup_{k}A_{k}}.
  25. 𝟏 k A k = 1 - F { 1 , 2 , , n } ( - 1 ) | F | 𝟏 F A k = F { 1 , 2 , , n } ( - 1 ) | F | + 1 𝟏 F A k \mathbf{1}_{\bigcup_{k}A_{k}}=1-\sum_{F\subseteq\{1,2,\ldots,n\}}(-1)^{|F|}% \mathbf{1}_{\bigcap_{F}A_{k}}=\sum_{\emptyset\neq F\subseteq\{1,2,\ldots,n\}}(% -1)^{|F|+1}\mathbf{1}_{\bigcap_{F}A_{k}}
  26. X X
  27. \mathbb{P}
  28. A A
  29. 𝟏 A \mathbf{1}_{A}
  30. A A
  31. E ( 𝟏 A ) = X 𝟏 A ( x ) d = A d = P ( A ) \operatorname{E}(\mathbf{1}_{A})=\int_{X}\mathbf{1}_{A}(x)\,d\mathbb{P}=\int_{% A}d\mathbb{P}=\operatorname{P}(A)
  32. ( Ω , , ) \textstyle(\Omega,\mathcal{F},\mathbb{P})
  33. A A\in\mathcal{F}
  34. 𝟏 A : Ω \mathbf{1}_{A}\colon\Omega\rightarrow\mathbb{R}
  35. 𝟏 A ( ω ) = 1 \mathbf{1}_{A}(\omega)=1
  36. ω A , \omega\in A,
  37. 𝟏 A ( ω ) = 0. \mathbf{1}_{A}(\omega)=0.
  38. E ( 𝟏 A ( ω ) ) = P ( A ) \operatorname{E}(\mathbf{1}_{A}(\omega))=\operatorname{P}(A)
  39. Var ( 𝟏 A ( ω ) ) = P ( A ) ( 1 - P ( A ) ) \operatorname{Var}(\mathbf{1}_{A}(\omega))=\operatorname{P}(A)(1-\operatorname% {P}(A))
  40. Cov ( 𝟏 A ( ω ) , 𝟏 B ( ω ) ) = P ( A B ) - P ( A ) P ( B ) \operatorname{Cov}(\mathbf{1}_{A}(\omega),\mathbf{1}_{B}(\omega))=% \operatorname{P}(A\cap B)-\operatorname{P}(A)\operatorname{P}(B)
  41. δ ( x ) = d H ( x ) d x , \delta(x)=\tfrac{dH(x)}{dx},
  42. - f ( x ) δ ( x ) d x = f ( 0 ) . \int_{-\infty}^{\infty}f(x)\,\delta(x)dx=f(0).
  43. δ S ( 𝐱 ) = - 𝐧 x x 𝟏 𝐱 D \delta_{S}(\mathbf{x})=-\mathbf{n}_{x}\cdot\nabla_{x}\mathbf{1}_{\mathbf{x}\in D}
  44. - 𝐑 n f ( 𝐱 ) 𝐧 x x 𝟏 𝐱 D d n 𝐱 = S f ( β ) d n - 1 β . -\int_{\mathbf{R}^{n}}f(\mathbf{x})\,\mathbf{n}_{x}\cdot\nabla_{x}\mathbf{1}_{% \mathbf{x}\in D}\;d^{n}\mathbf{x}=\oint_{S}\,f(\mathbf{\beta})\;d^{n-1}\mathbf% {\beta}.

Inductance.html

  1. v ( t ) = L d i d t \displaystyle v(t)=L\frac{di}{dt}
  2. v m = n = 1 K L m , n d i n d t . \displaystyle v_{m}=\sum\limits_{n=1}^{K}L_{m,n}\frac{di_{n}}{dt}.
  3. N m Φ m = n = 1 K L m , n i n . \displaystyle N_{m}\Phi_{m}=\sum\limits_{n=1}^{K}L_{m,n}i_{n}.
  4. v m = N m d Φ m d t = n = 1 K L m , n d i n d t , \displaystyle v_{m}=N_{m}\frac{d\Phi_{m}}{dt}=\sum\limits_{n=1}^{K}L_{m,n}% \frac{di_{n}}{dt},
  5. m K i m v m d t = m , n = 1 K i m L m , n d i n = ! n = 1 K W ( i ) i n d i n . \displaystyle\sum\limits_{m}^{K}i_{m}v_{m}dt=\sum\limits_{m,n=1}^{K}i_{m}L_{m,% n}di_{n}\overset{!}{=}\sum\limits_{n=1}^{K}\frac{\partial W\left(i\right)}{% \partial i_{n}}di_{n}.
  6. 2 W i m i n = 2 W i n i m \displaystyle\frac{\partial^{2}W}{\partial i_{m}\partial i_{n}}=\frac{\partial% ^{2}W}{\partial i_{n}\partial i_{m}}
  7. W ( i ) = 1 2 m , n = 1 K i m L m , n i n . \displaystyle W\left(i\right)=\tfrac{1}{2}\sum\limits_{m,n=1}^{K}i_{m}L_{m,n}i% _{n}.
  8. M 21 = N 1 N 2 P 21 M_{21}=N_{1}N_{2}P_{21}\!
  9. M 21 M_{21}
  10. M = k L 1 L 2 M=k\sqrt{L_{1}L_{2}}\!
  11. v 1 = L 1 d i 1 d t - M d i 2 d t v_{1}=L_{1}\frac{di_{1}}{dt}-M\frac{di_{2}}{dt}
  12. [ 𝐳 ] = s [ L 1 M M L 2 ] [\mathbf{z}]=s\begin{bmatrix}L_{1}\ M\\ M\ L_{2}\end{bmatrix}
  13. A v = s M Z s 2 L 1 L 2 - s 2 M 2 + s L 1 Z A_{\mathrm{v}}=\frac{sMZ}{s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}
  14. A v = L 2 L 1 A_{\mathrm{v}}=\sqrt{L_{2}\over L_{1}}
  15. Z in = s 2 L 1 L 2 - s 2 M 2 + s L 1 Z s L 2 + Z Z_{\mathrm{in}}=\frac{s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}
  16. Z in = s L 1 Z s L 2 + Z Z_{\mathrm{in}}=\frac{sL_{1}Z}{sL_{2}+Z}
  17. | s L 2 | | Z | |sL_{2}|\gg|Z|
  18. Z in L 1 L 2 Z Z_{\mathrm{in}}\approx{L_{1}\over L_{2}}Z
  19. A i L 1 L 2 = 1 A v A_{\mathrm{i}}\approx\sqrt{L_{1}\over L_{2}}={1\over A_{\mathrm{v}}}
  20. V s = N s N p V p V\text{s}=\frac{N\text{s}}{N\text{p}}V\text{p}
  21. I s = N p N s I p I\text{s}=\frac{N\text{p}}{N\text{s}}I\text{p}
  22. L m , n = μ 0 4 π C m C n 𝐝𝐱 m 𝐝𝐱 n | 𝐱 m - 𝐱 n | L_{m,n}=\frac{\mu_{0}}{4\pi}\oint_{C_{m}}\oint_{C_{n}}\frac{\mathbf{dx}_{m}% \cdot\mathbf{dx}_{n}}{|\mathbf{x}_{m}-\mathbf{x}_{n}|}
  23. L = ( μ 0 4 π C C 𝐝𝐱 𝐝𝐱 | 𝐱 - 𝐱 | ) | 𝐱 - 𝐱 | > a / 2 + μ 0 4 π l Y + O ( μ 0 a ) . L=\left(\frac{\mu_{0}}{4\pi}\oint_{C}\oint_{C^{\prime}}\frac{\mathbf{dx}\cdot% \mathbf{dx}^{\prime}}{|\mathbf{x}-\mathbf{x}^{\prime}|}\right)_{|\mathbf{x}-% \mathbf{x}^{\prime}|>a/2}+\frac{\mu_{0}}{4\pi}lY+O\left(\mu_{0}a\right).
  24. L C = ε μ . \displaystyle L^{\prime}C^{\prime}={\varepsilon\mu}.
  25. μ 0 r 2 N 2 3 l [ - 8 w + 4 1 + m m ( K ( m 1 + m ) - ( 1 - m ) E ( m 1 + m ) ) ] \frac{\mu_{0}r^{2}N^{2}}{3l}\left[-8w+4\frac{\sqrt{1+m}}{m}\left(K\left(\sqrt{% \frac{m}{1+m}}\right)-\left(1-m\right)E\left(\sqrt{\frac{m}{1+m}}\right)\right% )\right]
  26. = μ 0 r 2 N 2 π l [ 1 - 8 w 3 π + n = 1 ( 2 n ) ! 2 n ! 4 ( n + 1 ) ( 2 n - 1 ) 2 2 n ( - 1 ) n + 1 w 2 n ] =\frac{\mu_{0}r^{2}N^{2}\pi}{l}\left[1-\frac{8w}{3\pi}+\sum_{n=1}^{\infty}% \frac{\left(2n\right)!^{2}}{n!^{4}\left(n+1\right)\left(2n-1\right)2^{2n}}% \left(-1\right)^{n+1}w^{2n}\right]
  27. = μ 0 r 2 N 2 π l ( 1 - 8 w 3 π + w 2 2 - w 4 4 + 5 w 6 16 - 35 w 8 64 + ) =\frac{\mu_{0}r^{2}N^{2}\pi}{l}\left(1-\frac{8w}{3\pi}+\frac{w^{2}}{2}-\frac{w% ^{4}}{4}+\frac{5w^{6}}{16}-\frac{35w^{8}}{64}+...\right)
  28. = μ 0 r N 2 [ ( 1 + 1 32 w 2 + O ( 1 w 4 ) ) ln ( 8 w ) - 1 / 2 + 1 128 w 2 + O ( 1 w 4 ) ] =\mu_{0}rN^{2}\left[\left(1+\frac{1}{32w^{2}}+O\left(\frac{1}{w^{4}}\right)% \right)\ln(8w)-1/2+\frac{1}{128w^{2}}+O\left(\frac{1}{w^{4}}\right)\right]
  29. μ 0 l 2 π ln ( a 1 a ) \frac{\mu_{0}l}{2\pi}\ln\left(\frac{a_{1}}{a}\right)
  30. μ 0 r ( ln ( 8 r a ) - 2 + Y 2 + O ( a 2 / r 2 ) ) \mu_{0}r\cdot\left(\ln\left(\frac{8r}{a}\right)-2+\frac{Y}{2}+O\left(a^{2}/r^{% 2}\right)\right)
  31. μ 0 π ( b ln ( 2 b a ) + d ln ( 2 d a ) - ( b + d ) ( 2 - Y 2 ) + 2 b 2 + d 2 ) \frac{\mu_{0}}{\pi}\left(b\ln\left(\frac{2b}{a}\right)+d\ln\left(\frac{2d}{a}% \right)-\left(b+d\right)\left(2-\frac{Y}{2}\right)+2\sqrt{b^{2}+d^{2}}\right)
  32. - μ 0 π ( b arsinh ( b d ) + d arsinh ( d b ) + O ( a ) ) \;\;-\frac{\mu_{0}}{\pi}\left(b\cdot\operatorname{arsinh}\left(\frac{b}{d}% \right)+d\cdot\operatorname{arsinh}\left(\frac{d}{b}\right)+O\left(a\right)\right)
  33. μ 0 l π ( ln ( d a ) + Y 2 ) \frac{\mu_{0}l}{\pi}\left(\ln\left(\frac{d}{a}\right)+\frac{Y}{2}\right)
  34. μ 0 l π arcosh ( d 2 a ) = μ 0 l π ln ( d 2 a + d 2 4 a 2 - 1 ) \frac{\mu_{0}l}{\pi}\operatorname{arcosh}\left(\frac{d}{2a}\right)=\frac{\mu_{% 0}l}{\pi}\ln\left(\frac{d}{2a}+\sqrt{\frac{d^{2}}{4a^{2}}-1}\right)
  35. μ 0 l 2 π ( ln ( 2 d a ) + Y 2 ) \frac{\mu_{0}l}{2\pi}\left(\ln\left(\frac{2d}{a}\right)+\frac{Y}{2}\right)
  36. μ 0 l 2 π arcosh ( d a ) = μ 0 l 2 π ln ( d a + d 2 a 2 - 1 ) \frac{\mu_{0}l}{2\pi}\operatorname{arcosh}\left(\frac{d}{a}\right)=\frac{\mu_{% 0}l}{2\pi}\ln\left(\frac{d}{a}+\sqrt{\frac{d^{2}}{a^{2}}-1}\right)
  37. B B
  38. B = μ 0 N i / l \displaystyle B=\mu_{0}Ni/l
  39. μ 0 \mu_{0}
  40. N N
  41. i i
  42. l l
  43. B B
  44. A A
  45. Φ = μ 0 N i A / l , \displaystyle\Phi=\mu_{0}NiA/l,
  46. L = N Φ / i \displaystyle L=N\Phi/i
  47. L = μ 0 N 2 A / l . \displaystyle L=\mu_{0}N^{2}A/l.
  48. B = μ 0 μ r N i / l \displaystyle B=\mu_{0}\mu_{r}Ni/l
  49. μ r \mu_{r}
  50. Φ = μ 0 μ r N i A / l , \displaystyle\Phi=\mu_{0}\mu_{r}NiA/l,
  51. L = μ 0 μ r N 2 A / l . \displaystyle L=\mu_{0}\mu_{r}N^{2}A/l.
  52. r i r_{i}
  53. μ i \mu_{i}
  54. μ d \mu_{d}
  55. r o 1 r_{o1}
  56. r o 2 r_{o2}
  57. μ o \mu_{o}
  58. I I
  59. r r
  60. 0 r r i : B ( r ) = μ i I r 2 π r i 2 0\leq r\leq r_{i}:B(r)=\frac{\mu_{i}Ir}{2\pi r_{i}^{2}}
  61. r i r r o 1 : B ( r ) = μ d I 2 π r r_{i}\leq r\leq r_{o1}:B(r)=\frac{\mu_{d}I}{2\pi r}
  62. r o 1 r r o 2 : B ( r ) = μ o I 2 π r ( r o 2 2 - r 2 r o 2 2 - r o 1 2 ) r_{o1}\leq r\leq r_{o2}:B(r)=\frac{\mu_{o}I}{2\pi r}\left(\frac{r_{o2}^{2}-r^{% 2}}{r_{o2}^{2}-r_{o1}^{2}}\right)
  63. l l
  64. d ϕ d d l = r i r o 1 B ( r ) d r = μ d I 2 π ln r o 1 r i \frac{d\phi_{d}}{dl}=\int_{r_{i}}^{r_{o1}}B(r)dr=\frac{\mu_{d}I}{2\pi}\ln\frac% {r_{o1}}{r_{i}}
  65. 1 2 L I 2 \frac{1}{2}LI^{2}
  66. 1 2 L I 2 = V B 2 2 μ d V \frac{1}{2}LI^{2}=\int_{V}\frac{B^{2}}{2\mu}dV
  67. l l
  68. 1 2 L I 2 = r 1 r 2 B 2 2 μ 2 π r d r \frac{1}{2}L^{\prime}I^{2}=\int_{r_{1}}^{r_{2}}\frac{B^{2}}{2\mu}2\pi r~{}dr
  69. L L^{\prime}
  70. μ i I 2 16 π \frac{\mu_{i}I^{2}}{16\pi}
  71. μ o I 2 4 π ( r o 2 2 r o 2 2 - r o 1 2 ) 2 ln r o 2 r o 1 - μ o I 2 8 π ( r o 2 2 r o 2 2 - r o 1 2 ) - μ o I 2 16 π \frac{\mu_{o}I^{2}}{4\pi}\left(\frac{r_{o2}^{2}}{r_{o2}^{2}-r_{o1}^{2}}\right)% ^{2}\ln\frac{r_{o2}}{r_{o1}}-\frac{\mu_{o}I^{2}}{8\pi}\left(\frac{r_{o2}^{2}}{% r_{o2}^{2}-r_{o1}^{2}}\right)-\frac{\mu_{o}I^{2}}{16\pi}
  72. L L^{\prime}
  73. L = μ i 8 π + μ d 2 π ln r o 1 r i + μ o 2 π ( r o 2 2 r o 2 2 - r o 1 2 ) 2 ln r o 2 r o 1 - μ o 4 π ( r o 2 2 r o 2 2 - r o 1 2 ) - μ o 8 π L^{\prime}=\frac{\mu_{i}}{8\pi}+\frac{\mu_{d}}{2\pi}\ln\frac{r_{o1}}{r_{i}}+% \frac{\mu_{o}}{2\pi}\left(\frac{r_{o2}^{2}}{r_{o2}^{2}-r_{o1}^{2}}\right)^{2}% \ln\frac{r_{o2}}{r_{o1}}-\frac{\mu_{o}}{4\pi}\left(\frac{r_{o2}^{2}}{r_{o2}^{2% }-r_{o1}^{2}}\right)-\frac{\mu_{o}}{8\pi}
  74. L = d L d l μ d 2 π ln r o 1 r i L^{\prime}=\frac{dL}{dl}\approx\frac{\mu_{d}}{2\pi}\ln\frac{r_{o1}}{r_{i}}
  75. Z L = V / I = j ω L Z_{L}=V/I=j\omega L\,
  76. L s ( i ) = def N Φ i = Λ i L_{s}(i)\ \overset{\underset{\mathrm{def}}{}}{=}\ \frac{N\Phi}{i}=\frac{% \Lambda}{i}
  77. L d ( i ) = def d ( N Φ ) d i = d Λ d i L_{d}(i)\ \overset{\underset{\mathrm{def}}{}}{=}\ \frac{d(N\Phi)}{di}=\frac{d% \Lambda}{di}
  78. v ( t ) = d Λ d t = d Λ d i d i d t = L d ( i ) d i d t v(t)=\frac{d\Lambda}{dt}=\frac{d\Lambda}{di}\frac{di}{dt}=L_{d}(i)\frac{di}{dt}

Induction_motor.html

  1. n s n_{s}
  2. n s n_{s}
  3. n s = 120 × f p n_{s}={120\times{f}\over{p}}
  4. f f
  5. p p
  6. p p
  7. n s n_{s}
  8. s s
  9. s = n s - n r n s s=\frac{n_{s}-n_{r}}{n_{s}}\,
  10. n s n_{s}
  11. n r n_{r}
  12. R r / s R_{r}^{{}^{\prime}}/s
  13. R r / s R_{r}^{{}^{\prime}}/s
  14. R s R_{s}
  15. X s X_{s}
  16. R r R_{r}
  17. X r X_{r}
  18. R r R_{r}^{{}^{\prime}}
  19. X r X_{r}^{{}^{\prime}}
  20. s s
  21. X m X_{m}
  22. f s f_{s}
  23. n r n_{r}
  24. n s n_{s}
  25. I s I_{s}
  26. I r I_{r}^{{}^{\prime}}
  27. I m I_{m}
  28. j = - 1 j=\sqrt{-1}
  29. K T E K_{TE}
  30. = X m / ( X s + X m ) =X_{m}/(X_{s}+X_{m})
  31. m m
  32. p p
  33. P e m P_{em}
  34. P g a p P_{gap}
  35. P r P_{r}
  36. P o P_{o}
  37. P h P_{h}
  38. P f P_{f}
  39. P r l P_{rl}
  40. P s l P_{sl}
  41. R s , X s R_{s},X_{s}
  42. R r , X r R_{r}^{{}^{\prime}},X_{r}^{{}^{\prime}}
  43. R o , X o R_{o},X_{o}
  44. R T E , X T E R_{TE},X_{TE}
  45. R s , X s R_{s},X_{s}
  46. X m X_{m}
  47. s s
  48. T e m T_{em}
  49. T m a x T_{max}
  50. V s V_{s}
  51. X m X_{m}
  52. X X
  53. X s + X r X_{s}+X_{r}^{{}^{\prime}}
  54. Z s Z_{s}
  55. Z r Z_{r}^{{}^{\prime}}
  56. Z o Z_{o}
  57. Z Z
  58. Z T E Z_{TE}
  59. R T E + X T E R_{TE}+X_{TE}
  60. ω r \omega_{r}
  61. ω s \omega_{s}
  62. Y Y
  63. = G - j B = 1 Z = 1 R + j X = R Z 2 - j X Z 2 =G-jB=\frac{1}{Z}=\frac{1}{R+jX}=\frac{R}{Z^{2}}-\frac{jX}{Z^{2}}
  64. | Z | \left|Z\right|
  65. R 2 + X 2 \sqrt{R^{2}+X^{2}}
  66. V s / X {V_{s}}/X
  67. T m a x T_{max}
  68. s R r / X s\approx{R_{r}^{{}^{\prime}}/X}
  69. I s 0.7 L R C I_{s}\approx{0.7}LRC
  70. T m a x K * V s 2 / ( 2 X ) T_{max}\approx{K*V_{s}^{2}}/(2X)
  71. X s X r 0.4 0.6 \frac{X_{s}}{X_{r}^{{}^{\prime}}}\approx\frac{0.4}{0.6}
  72. T e m 2 T m a x s s m a x + s m a x s T_{em}\approx\frac{2T_{max}}{\frac{s}{s_{max}}+\frac{s_{max}}{s}}
  73. s m a x s_{max}
  74. T m a x T_{max}
  75. ω s = 2 π n s 60 = 4 π f s p \omega_{s}=\frac{2{\pi}n_{s}}{60}=\frac{4{\pi}f_{s}}{p}
  76. Z m = R s + j X s + ( R r s + j X r ) ( j X m ) R r s + j ( X r + X m ) Z_{m}=R_{s}+jX_{s}+\frac{(\frac{R_{r}^{{}^{\prime}}}{s}+jX_{r}^{{}^{\prime}})(% jX_{m})}{\frac{R_{r}^{{}^{\prime}}}{s}+j(X_{r}^{{}^{\prime}}+X_{m})}
  77. I s = V s / Z m = V s / ( R s + j X s + ( R r s + j X r ) ( j X m ) R r s + j ( X r + X m ) ) I_{s}=V_{s}/Z_{m}=V_{s}/(R_{s}+jX_{s}+\frac{(\frac{R_{r}^{{}^{\prime}}}{s}+jX_% {r}^{{}^{\prime}})(jX_{m})}{\frac{R_{r}^{{}^{\prime}}}{s}+j(X_{r}^{{}^{\prime}% }+X_{m})})
  78. I r = j X m R r s + j ( X r + X m ) I s I_{r}^{{}^{\prime}}=\frac{jX_{m}}{\frac{R_{r}^{{}^{\prime}}}{s}+j(X_{r}^{{}^{% \prime}}+X_{m})}I_{s}
  79. R r s = R r ( 1 - s ) s + R r \frac{R_{r}^{{}^{\prime}}}{s}=\frac{R_{r}^{{}^{\prime}}(1-s)}{s}+R_{r}^{{}^{% \prime}}
  80. P g a p = P e m + P r P_{gap}=P_{em}+P_{r}
  81. P r = 3 R r I r 2 P_{r}=3R_{r}^{{}^{\prime}}I_{r}^{{}^{\prime}2}
  82. P g a p = 3 R r I r 2 s P_{gap}=\frac{3R_{r}^{{}^{\prime}}I_{r}^{{}^{\prime}2}}{s}
  83. P e m = 3 R r I r 2 ( 1 - s ) s P_{em}=\frac{3R_{r}^{{}^{\prime}}I_{r}^{{}^{\prime}2}(1-s)}{s}
  84. P e m = P g a p ( 1 - s ) P_{em}=P_{gap}(1-s)
  85. P e m = 3 R r I r 2 n r s n s P_{em}=\frac{3R_{r}^{{}^{\prime}}I_{r}^{{}^{\prime}2}n_{r}}{sn_{s}}
  86. P e m = 3 R r I r 2 n r 746 s n s P_{em}=\frac{3R_{r}^{{}^{\prime}}I_{r}^{{}^{\prime}2}n_{r}}{746sn_{s}}
  87. T e m T_{em}
  88. P e m = T e m n r 5252 P_{em}=\frac{T_{em}n_{r}}{5252}
  89. T e m = P e m ω r = P r s ω s = 3 I r 2 R r ω s s T_{em}=\frac{P_{em}}{\omega_{r}}=\frac{\frac{P_{r}}{s}}{\omega_{s}}=\frac{3I_{% r}^{{}^{\prime}2}R_{r}^{{}^{\prime}}}{\omega_{s}s}
  90. T e m T_{em}
  91. s s
  92. R s , X s R_{s},X_{s}
  93. X m X_{m}
  94. V T E = X m R s 2 + ( X s + X m ) 2 V s V_{TE}=\frac{X_{m}}{\sqrt{R_{s}^{2}+(X_{s}+X_{m})^{2}}}V_{s}
  95. Z T E = R T E + j X T E = j X m ( R s + j X s ) R s + j ( X s + X m ) Z_{TE}=R_{TE}+jX_{TE}=\frac{jX_{m}(R_{s}+jX_{s})}{R_{s}+j(X_{s}+X_{m})}
  96. R s 2 ( X s + X m ) 2 R_{s}^{2}\gg{(X_{s}+X_{m})^{2}}
  97. X s X m X_{s}\ll{X_{m}}
  98. K T E = X m X s + X m K_{TE}=\frac{X_{m}}{X_{s}+X_{m}}
  99. V T E Z T E V s V_{TE}\approx{Z_{TE}V_{s}}
  100. Z T E K T E 2 R s + j X s Z_{TE}\approx{K_{TE}^{2}R_{s}+jX_{s}}
  101. T e m = 3 V T E 2 ( R T E + R r s ) 2 + ( R T E + X r ) 2 . R r s . 1 ω s ( N . m ) T_{em}=\frac{3V_{TE}^{2}}{(R_{TE}+\frac{R_{r}^{{}^{\prime}}}{s})^{2}+(R_{TE}+X% _{r}^{{}^{\prime}})^{2}}.\frac{R_{r}^{{}^{\prime}}}{s}.\frac{1}{\omega_{s}}(N.m)
  102. R T E + R r R T E + X r R_{TE}+R_{r}^{{}^{\prime}}\gg{R_{TE}+X_{r}^{{}^{\prime}}}
  103. R r R T E R_{r}^{{}^{\prime}}\gg{R_{TE}}
  104. T e m 1 ω s . 3 V T E 2 R r . s T_{em}\approx\frac{1}{\omega_{s}}.\frac{3V_{TE}^{2}}{R_{r}^{{}^{\prime}}}.s
  105. R T E + R r R T E + X r R_{TE}+R_{r}^{{}^{\prime}}\ll{R_{TE}+X_{r}^{{}^{\prime}}}
  106. T e m 1 ω s . 3 V T E 2 ( X s + X r ) 2 . R r 2 s T_{em}\approx\frac{1}{\omega_{s}}.\frac{3V_{TE}^{2}}{(X_{s}+X_{r}^{{}^{\prime}% })^{2}}.\frac{R_{r}^{{}^{\prime}2}}{s}
  107. T m a x = 1 2 ω s . 3 V T E 2 R T E + R T E 2 + ( X T E + X r ) 2 T_{max}=\frac{1}{2\omega_{s}}.\frac{3V_{TE}^{2}}{R_{TE}+\sqrt{R_{TE}^{2}+(X_{% TE}+X_{r}^{{}^{\prime}})^{2}}}
  108. s = R r R T E 2 + ( X T E + X r ) 2 s=\frac{R_{r}^{{}^{\prime}}}{\sqrt{R_{TE}^{2}+(X_{TE}+X_{r}^{{}^{\prime}})^{2}}}
  109. T e m = 21.21 I r 2 R r n r s T_{em}=\frac{21.21I_{r}^{{}^{\prime}2}R_{r}^{{}^{\prime}}}{n_{r}s}
  110. T e m = 7.04 P g a p n s T_{em}=\frac{7.04P_{gap}}{n_{s}}

Ineffable_cardinal.html

  1. κ \kappa
  2. f : κ 𝒫 ( κ ) f:\kappa\to\mathcal{P}(\kappa)
  3. 𝒫 ( κ ) \mathcal{P}(\kappa)
  4. κ \kappa
  5. f ( δ ) f(\delta)
  6. δ \delta
  7. δ < κ \delta<\kappa
  8. S S
  9. κ \kappa
  10. κ \kappa
  11. f f
  12. δ 1 < δ 2 \delta_{1}<\delta_{2}
  13. S S
  14. f ( δ 1 ) = f ( δ 2 ) δ 1 f(\delta_{1})=f(\delta_{2})\cap\delta_{1}
  15. κ \kappa
  16. f : 𝒫 = 2 ( κ ) { 0 , 1 } f:\mathcal{P}_{=2}(\kappa)\to\{0,1\}
  17. κ \kappa
  18. f f
  19. f f
  20. κ \kappa
  21. n n
  22. n n
  23. f : 𝒫 = n ( κ ) { 0 , 1 } f:\mathcal{P}_{=n}(\kappa)\to\{0,1\}
  24. κ \kappa
  25. f f
  26. n n
  27. n n
  28. n n
  29. 2 n < 0 2\leq n<\aleph_{0}
  30. κ \kappa
  31. ( n + 1 ) (n+1)
  32. n n
  33. κ \kappa
  34. κ \kappa

Inequality_(mathematics).html

  1. H = n 1 / a 1 + 1 / a 2 + + 1 / a n H=\frac{n}{1/a_{1}+1/a_{2}+\cdots+1/a_{n}}
  2. G = a 1 a 2 a n n G=\sqrt[n]{a_{1}\cdot a_{2}\cdots a_{n}}
  3. A = a 1 + a 2 + + a n n A=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}
  4. Q = a 1 2 + a 2 2 + + a n 2 n Q=\sqrt{\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}}
  5. e x 1 + x . e^{x}\geq 1+x.\,
  6. x x ( 1 e ) 1 / e . x^{x}\geq\left(\frac{1}{e}\right)^{1/e}.\,
  7. x x x x . x^{x^{x}}\geq x.\,
  8. ( x + y ) z + ( x + z ) y + ( y + z ) x > 2. (x+y)^{z}+(x+z)^{y}+(y+z)^{x}>2.\,
  9. e b - e a b - a > e ( a + b ) / 2 . \frac{e^{b}-e^{a}}{b-a}>e^{(a+b)/2}.
  10. ( x + y ) p < x p + y p . (x+y)^{p}<x^{p}+y^{p}.\,
  11. x x y y z z ( x y z ) ( x + y + z ) / 3 . x^{x}y^{y}z^{z}\geq(xyz)^{(x+y+z)/3}.\,
  12. a a + b b a b + b a . a^{a}+b^{b}\geq a^{b}+b^{a}.\,
  13. a e a + b e b a e b + b e a . a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.\,
  14. a 2 a + b 2 b + c 2 c a 2 b + b 2 c + c 2 a . a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.\,
  15. a b + b a > 1. a^{b}+b^{a}>1.\,
  16. a 1 a 2 + a 2 a 3 + + a n a 1 > 1 a_{1}^{a_{2}}+a_{2}^{a_{3}}+\cdots+a_{n}^{a_{1}}>1
  17. \mathbb{C}
  18. ( , + , × , ) (\mathbb{C},+,\times,\leq)
  19. ( , + , × , ) (\mathbb{C},+,\times,\leq)
  20. - a -a
  21. i 2 > 0 i^{2}>0
  22. 1 2 > 0 1^{2}>0
  23. - 1 > 0 -1>0
  24. 1 > 0 1>0
  25. ( - 1 + 1 ) > 0 (-1+1)>0
  26. R e ( a ) Re(a)
  27. R e ( a ) = R e ( b ) Re(a)=Re(b)
  28. I m ( a ) Im(a)
  29. I m ( b ) Im(b)
  30. x , y n x,y\in\mathbb{R}^{n}
  31. x = ( x 1 , x 2 , , x n ) 𝖳 x=\left(x_{1},x_{2},\ldots,x_{n}\right)^{\mathsf{T}}
  32. y = ( y 1 , y 2 , , y n ) 𝖳 y=\left(y_{1},y_{2},\ldots,y_{n}\right)^{\mathsf{T}}
  33. x i x_{i}
  34. y i y_{i}
  35. i = 1 , , n i=1,\ldots,n
  36. x = y x=y
  37. x i = y i x_{i}=y_{i}
  38. i = 1 , , n i=1,\ldots,n
  39. x < y x<y
  40. x i < y i x_{i}<y_{i}
  41. i = 1 , , n i=1,\ldots,n
  42. x y x\leq y
  43. x i y i x_{i}\leq y_{i}
  44. i = 1 , , n i=1,\ldots,n
  45. x y x\neq y
  46. x y x\leqq y
  47. x i y i x_{i}\leq y_{i}
  48. i = 1 , , n i=1,\ldots,n
  49. x > y x>y
  50. x y x\geq y
  51. x y x\geqq y
  52. x = [ 2 , 5 ] 𝖳 x=\left[2,5\right]^{\mathsf{T}}
  53. y = [ 3 , 4 ] 𝖳 y=\left[3,4\right]^{\mathsf{T}}
  54. X T A X 0 X^{T}AX\geq 0
  55. X = ( x , y , z , . , 1 ) T X=(x,y,z,....,1)^{T}

Infinitesimal.html

  1. sin θ \sin\theta
  2. cos θ \cos\theta
  3. θ \theta
  4. e ln ln x + ln ln x + j = 0 e x x - j , e^{\sqrt{\ln\ln x}}+\ln\ln x+\sum_{j=0}^{\infty}e^{x}x^{-j},
  5. \mathbb{N}
  6. * {}^{*}\mathbb{N}
  7. n , sin n π = 0 \forall n\in\mathbb{N},\sin n\pi=0
  8. n * , sin * n π = 0 \forall n\in{}^{*}\mathbb{N},{}^{*}\!\!\sin n\pi=0
  9. α \alpha
  10. δ α \delta_{\alpha}
  11. F ( x ) δ α ( x ) = F ( 0 ) \int F(x)\delta_{\alpha}(x)=F(0)

Infinitesimal_strain_theory.html

  1. 𝐮 1 \|\mathbf{u}\|\ll 1\,\!
  2. 𝐮 1 \|\nabla\mathbf{u}\|\ll 1\,\!
  3. 𝐄 \mathbf{E}\,\!
  4. 𝐞 \mathbf{e}\,\!
  5. 𝐄 = 1 2 ( 𝐗 𝐮 + ( 𝐗 𝐮 ) T + 𝐗 𝐮 ( 𝐗 𝐮 ) T ) 1 2 ( 𝐗 𝐮 + ( 𝐗 𝐮 ) T ) \mathbf{E}=\frac{1}{2}\left(\nabla_{\mathbf{X}}\mathbf{u}+(\nabla_{\mathbf{X}}% \mathbf{u})^{T}+\nabla_{\mathbf{X}}\mathbf{u}(\nabla_{\mathbf{X}}\mathbf{u})^{% T}\right)\approx\frac{1}{2}\left(\nabla_{\mathbf{X}}\mathbf{u}+(\nabla_{% \mathbf{X}}\mathbf{u})^{T}\right)\,\!
  6. E K L = 1 2 ( U K X L + U L X K + U M X K U M X L ) 1 2 ( U K X L + U L X K ) E_{KL}=\frac{1}{2}\left(\frac{\partial U_{K}}{\partial X_{L}}+\frac{\partial U% _{L}}{\partial X_{K}}+\frac{\partial U_{M}}{\partial X_{K}}\frac{\partial U_{M% }}{\partial X_{L}}\right)\approx\frac{1}{2}\left(\frac{\partial U_{K}}{% \partial X_{L}}+\frac{\partial U_{L}}{\partial X_{K}}\right)\,\!
  7. 𝐞 = 1 2 ( 𝐱 𝐮 + ( 𝐱 𝐮 ) T - 𝐱 𝐮 ( 𝐱 𝐮 ) T ) 1 2 ( 𝐱 𝐮 + ( 𝐱 𝐮 ) T ) \mathbf{e}=\frac{1}{2}\left(\nabla_{\mathbf{x}}\mathbf{u}+(\nabla_{\mathbf{x}}% \mathbf{u})^{T}-\nabla_{\mathbf{x}}\mathbf{u}(\nabla_{\mathbf{x}}\mathbf{u})^{% T}\right)\approx\frac{1}{2}\left(\nabla_{\mathbf{x}}\mathbf{u}+(\nabla_{% \mathbf{x}}\mathbf{u})^{T}\right)\,\!
  8. e r s = 1 2 ( u r x s + u s x r - u k x r u k x s ) 1 2 ( u r x s + u s x r ) e_{rs}=\frac{1}{2}\left(\frac{\partial u_{r}}{\partial x_{s}}+\frac{\partial u% _{s}}{\partial x_{r}}-\frac{\partial u_{k}}{\partial x_{r}}\frac{\partial u_{k% }}{\partial x_{s}}\right)\approx\frac{1}{2}\left(\frac{\partial u_{r}}{% \partial x_{s}}+\frac{\partial u_{s}}{\partial x_{r}}\right)\,\!
  9. 𝐄 𝐞 s y m b o l ε = 1 2 ( ( 𝐮 ) T + 𝐮 ) \mathbf{E}\approx\mathbf{e}\approx symbol\varepsilon=\frac{1}{2}\left((\nabla% \mathbf{u})^{T}+\nabla\mathbf{u}\right)\qquad
  10. E K L e r s ε i j = 1 2 ( u i , j + u j , i ) \qquad E_{KL}\approx e_{rs}\approx\varepsilon_{ij}=\frac{1}{2}\left(u_{i,j}+u_% {j,i}\right)\,\!
  11. ε i j \varepsilon_{ij}\,\!
  12. s y m b o l ε symbol\varepsilon\,\!
  13. ε i j = 1 2 ( u i , j + u j , i ) = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] = [ u 1 x 1 1 2 ( u 1 x 2 + u 2 x 1 ) 1 2 ( u 1 x 3 + u 3 x 1 ) 1 2 ( u 2 x 1 + u 1 x 2 ) u 2 x 2 1 2 ( u 2 x 3 + u 3 x 2 ) 1 2 ( u 3 x 1 + u 1 x 3 ) 1 2 ( u 3 x 2 + u 2 x 3 ) u 3 x 3 ] \begin{aligned}\displaystyle\varepsilon_{ij}&\displaystyle=\frac{1}{2}\left(u_% {i,j}+u_{j,i}\right)\\ &\displaystyle=\left[\begin{matrix}\varepsilon_{11}&\varepsilon_{12}&% \varepsilon_{13}\\ \varepsilon_{21}&\varepsilon_{22}&\varepsilon_{23}\\ \varepsilon_{31}&\varepsilon_{32}&\varepsilon_{33}\\ \end{matrix}\right]\\ &\displaystyle=\left[\begin{matrix}\frac{\partial u_{1}}{\partial x_{1}}&\frac% {1}{2}\left(\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{% \partial x_{1}}\right)&\frac{1}{2}\left(\frac{\partial u_{1}}{\partial x_{3}}+% \frac{\partial u_{3}}{\partial x_{1}}\right)\\ \frac{1}{2}\left(\frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial u_{1}}{% \partial x_{2}}\right)&\frac{\partial u_{2}}{\partial x_{2}}&\frac{1}{2}\left(% \frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}% \right)\\ \frac{1}{2}\left(\frac{\partial u_{3}}{\partial x_{1}}+\frac{\partial u_{1}}{% \partial x_{3}}\right)&\frac{1}{2}\left(\frac{\partial u_{3}}{\partial x_{2}}+% \frac{\partial u_{2}}{\partial x_{3}}\right)&\frac{\partial u_{3}}{\partial x_% {3}}\\ \end{matrix}\right]\end{aligned}
  14. [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ u x x 1 2 ( u x y + u y x ) 1 2 ( u x z + u z x ) 1 2 ( u y x + u x y ) u y y 1 2 ( u y z + u z y ) 1 2 ( u z x + u x z ) 1 2 ( u z y + u y z ) u z z ] \left[\begin{matrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz}\\ \end{matrix}\right]=\left[\begin{matrix}\frac{\partial u_{x}}{\partial x}&% \frac{1}{2}\left(\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{% \partial x}\right)&\frac{1}{2}\left(\frac{\partial u_{x}}{\partial z}+\frac{% \partial u_{z}}{\partial x}\right)\\ \frac{1}{2}\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{% \partial y}\right)&\frac{\partial u_{y}}{\partial y}&\frac{1}{2}\left(\frac{% \partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\right)\\ \frac{1}{2}\left(\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{% \partial z}\right)&\frac{1}{2}\left(\frac{\partial u_{z}}{\partial y}+\frac{% \partial u_{y}}{\partial z}\right)&\frac{\partial u_{z}}{\partial z}\\ \end{matrix}\right]\,\!
  15. s y m b o l F = s y m b o l 𝐮 + s y m b o l I symbol{F}=symbol{\nabla}\mathbf{u}+symbol{I}
  16. s y m b o l I symbol{I}
  17. s y m b o l ε = 1 2 ( s y m b o l F T + s y m b o l F ) - s y m b o l I symbol\varepsilon=\frac{1}{2}\left(symbol{F}^{T}+symbol{F}\right)-symbol{I}\,\!
  18. 𝐄 ( m ) = 1 2 m ( 𝐔 2 m - s y m b o l I ) = 1 2 m [ ( s y m b o l F T s y m b o l F ) m - s y m b o l I ] 1 2 m [ { s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T + s y m b o l I } m - s y m b o l I ] s y m b o l ε 𝐞 ( m ) = 1 2 m ( 𝐕 2 m - s y m b o l I ) = 1 2 m [ ( s y m b o l F s y m b o l F T ) m - s y m b o l I ] s y m b o l ε \begin{aligned}\displaystyle\mathbf{E}_{(m)}&\displaystyle=\frac{1}{2m}(% \mathbf{U}^{2m}-symbol{I})=\frac{1}{2m}[(symbol{F}^{T}symbol{F})^{m}-symbol{I}% ]\approx\frac{1}{2m}[\{symbol{\nabla}\mathbf{u}+(symbol{\nabla}\mathbf{u})^{T}% +symbol{I}\}^{m}-symbol{I}]\approx symbol{\varepsilon}\\ \displaystyle\mathbf{e}_{(m)}&\displaystyle=\frac{1}{2m}(\mathbf{V}^{2m}-% symbol{I})=\frac{1}{2m}[(symbol{F}symbol{F}^{T})^{m}-symbol{I}]\approx symbol{% \varepsilon}\end{aligned}
  19. d x dx\,\!
  20. d y dy\,\!
  21. a b ¯ = ( d x + u x x d x ) 2 + ( u y x d x ) 2 = d x 1 + 2 u x x + ( u x x ) 2 + ( u y x ) 2 \begin{aligned}\displaystyle\overline{ab}&\displaystyle=\sqrt{\left(dx+\frac{% \partial u_{x}}{\partial x}dx\right)^{2}+\left(\frac{\partial u_{y}}{\partial x% }dx\right)^{2}}\\ &\displaystyle=dx\sqrt{1+2\frac{\partial u_{x}}{\partial x}+\left(\frac{% \partial u_{x}}{\partial x}\right)^{2}+\left(\frac{\partial u_{y}}{\partial x}% \right)^{2}}\\ \end{aligned}\,\!
  22. 𝐮 1 \|\nabla\mathbf{u}\|\ll 1\,\!
  23. a b ¯ d x + u x x d x \overline{ab}\approx dx+\frac{\partial u_{x}}{\partial x}dx\,\!
  24. x x\,\!
  25. ε x = a b ¯ - A B ¯ A B ¯ \varepsilon_{x}=\frac{\overline{ab}-\overline{AB}}{\overline{AB}}\,\!
  26. A B ¯ = d x \overline{AB}=dx\,\!
  27. ε x = u x x \varepsilon_{x}=\frac{\partial u_{x}}{\partial x}\,\!
  28. y y\,\!
  29. z z\,\!
  30. ε y = u y y , ε z = u z z \varepsilon_{y}=\frac{\partial u_{y}}{\partial y}\quad,\qquad\varepsilon_{z}=% \frac{\partial u_{z}}{\partial z}\,\!
  31. A C ¯ \overline{AC}\,\!
  32. A B ¯ \overline{AB}\,\!
  33. γ x y = α + β \gamma_{xy}=\alpha+\beta\,\!
  34. tan α = u y x d x d x + u x x d x = u y x 1 + u x x , tan β = u x y d y d y + u y y d y = u x y 1 + u y y \tan\alpha=\frac{\dfrac{\partial u_{y}}{\partial x}dx}{dx+\dfrac{\partial u_{x% }}{\partial x}dx}=\frac{\dfrac{\partial u_{y}}{\partial x}}{1+\dfrac{\partial u% _{x}}{\partial x}}\quad,\qquad\tan\beta=\frac{\dfrac{\partial u_{x}}{\partial y% }dy}{dy+\dfrac{\partial u_{y}}{\partial y}dy}=\frac{\dfrac{\partial u_{x}}{% \partial y}}{1+\dfrac{\partial u_{y}}{\partial y}}\,\!
  35. α \alpha\,\!
  36. β \beta\,\!
  37. 1 \ll 1\,\!
  38. tan α α , tan β β \tan\alpha\approx\alpha\quad,\qquad\tan\beta\approx\beta\,\!
  39. α = u y x , β = u x y \alpha=\frac{\partial u_{y}}{\partial x}\quad,\qquad\beta=\frac{\partial u_{x}% }{\partial y}\,\!
  40. γ x y = α + β = u y x + u x y \gamma_{xy}=\alpha+\beta=\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x% }}{\partial y}\,\!
  41. x x\,\!
  42. y y\,\!
  43. u x u_{x}\,\!
  44. u y u_{y}\,\!
  45. γ x y = γ y x \gamma_{xy}=\gamma_{yx}\,\!
  46. y y\,\!
  47. z z\,\!
  48. x x\,\!
  49. z z\,\!
  50. γ y z = γ z y = u y z + u z y , γ z x = γ x z = u z x + u x z \gamma_{yz}=\gamma_{zy}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}% }{\partial y}\quad,\qquad\gamma_{zx}=\gamma_{xz}=\frac{\partial u_{z}}{% \partial x}+\frac{\partial u_{x}}{\partial z}\,\!
  51. γ \gamma\,\!
  52. [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ ε x x γ x y / 2 γ x z / 2 γ y x / 2 ε y y γ y z / 2 γ z x / 2 γ z y / 2 ε z z ] \left[\begin{matrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz}\\ \end{matrix}\right]=\left[\begin{matrix}\varepsilon_{xx}&\gamma_{xy}/2&\gamma_% {xz}/2\\ \gamma_{yx}/2&\varepsilon_{yy}&\gamma_{yz}/2\\ \gamma_{zx}/2&\gamma_{zy}/2&\varepsilon_{zz}\\ \end{matrix}\right]\,\!
  53. d 𝐱 2 - d 𝐗 2 = d 𝐗 2 𝐄 d 𝐗 or ( d x ) 2 - ( d X ) 2 = 2 E K L d X K d X L d\mathbf{x}^{2}-d\mathbf{X}^{2}=d\mathbf{X}\cdot 2\mathbf{E}\cdot d\mathbf{X}% \quad\,\text{or}\quad(dx)^{2}-(dX)^{2}=2E_{KL}\,dX_{K}\,dX_{L}\,\!
  54. d 𝐱 2 - d 𝐗 2 = d 𝐗 2 𝐬𝐲𝐦𝐛𝐨𝐥 ε d 𝐗 or ( d x ) 2 - ( d X ) 2 = 2 ε K L d X K d X L d\mathbf{x}^{2}-d\mathbf{X}^{2}=d\mathbf{X}\cdot 2\mathbf{symbol\varepsilon}% \cdot d\mathbf{X}\quad\,\text{or}\quad(dx)^{2}-(dX)^{2}=2\varepsilon_{KL}\,dX_% {K}\,dX_{L}\,\!
  55. ( d X ) 2 (dX)^{2}\,\!
  56. d x - d X d X d x + d X d X = 2 ε i j d X i d X d X j d X \frac{dx-dX}{dX}\frac{dx+dX}{dX}=2\varepsilon_{ij}\frac{dX_{i}}{dX}\frac{dX_{j% }}{dX}\,\!
  57. d x d X dx\approx dX\,\!
  58. d x + d X d X 2 \frac{dx+dX}{dX}\approx 2\,\!
  59. d x - d X d X = ε i j N i N j = 𝐍 s y m b o l ε 𝐍 \frac{dx-dX}{dX}=\varepsilon_{ij}N_{i}N_{j}=\mathbf{N}\cdot symbol\varepsilon% \cdot\mathbf{N}\,\!
  60. N i = d X i d X N_{i}=\frac{dX_{i}}{dX}\,\!
  61. d 𝐗 d\mathbf{X}\,\!
  62. e ( 𝐍 ) e_{(\mathbf{N})}\,\!
  63. 𝐍 \mathbf{N}\,\!
  64. 𝐍 \mathbf{N}\,\!
  65. X 1 X_{1}\,\!
  66. 𝐍 = 𝐈 1 \mathbf{N}=\mathbf{I}_{1}\,\!
  67. e ( 𝐈 1 ) = 𝐈 1 s y m b o l ε 𝐈 1 = ε 11 e_{(\mathbf{I}_{1})}=\mathbf{I}_{1}\cdot symbol\varepsilon\cdot\mathbf{I}_{1}=% \varepsilon_{11}\,\!
  68. 𝐍 = 𝐈 2 \mathbf{N}=\mathbf{I}_{2}\,\!
  69. 𝐍 = 𝐈 3 \mathbf{N}=\mathbf{I}_{3}\,\!
  70. ε 22 \varepsilon_{22}\,\!
  71. ε 33 \varepsilon_{33}\,\!
  72. 𝐞 1 , 𝐞 2 , 𝐞 3 \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}
  73. s y m b o l ε = i = 1 3 j = 1 3 ε i j 𝐞 i 𝐞 j symbol{\varepsilon}=\sum_{i=1}^{3}\sum_{j=1}^{3}\varepsilon_{ij}\mathbf{e}_{i}% \otimes\mathbf{e}_{j}
  74. s y m b o l ε ¯ ¯ = [ ε 11 ε 12 ε 13 ε 12 ε 22 ε 23 ε 13 ε 23 ε 33 ] \underline{\underline{symbol{\varepsilon}}}=\begin{bmatrix}\varepsilon_{11}&% \varepsilon_{12}&\varepsilon_{13}\\ \varepsilon_{12}&\varepsilon_{22}&\varepsilon_{23}\\ \varepsilon_{13}&\varepsilon_{23}&\varepsilon_{33}\end{bmatrix}
  75. 𝐞 ^ 1 , 𝐞 ^ 2 , 𝐞 ^ 3 \hat{\mathbf{e}}_{1},\hat{\mathbf{e}}_{2},\hat{\mathbf{e}}_{3}
  76. s y m b o l ε = i = 1 3 j = 1 3 ε ^ i j 𝐞 ^ i 𝐞 ^ j s y m b o l ε ¯ ¯ = [ ε ^ 11 ε ^ 12 ε ^ 13 ε ^ 12 ε ^ 22 ε ^ 23 ε ^ 13 ε ^ 23 ε ^ 33 ] symbol{\varepsilon}=\sum_{i=1}^{3}\sum_{j=1}^{3}\hat{\varepsilon}_{ij}\hat{% \mathbf{e}}_{i}\otimes\hat{\mathbf{e}}_{j}\quad\implies\quad\underline{% \underline{symbol{\varepsilon}}}=\begin{bmatrix}\hat{\varepsilon}_{11}&\hat{% \varepsilon}_{12}&\hat{\varepsilon}_{13}\\ \hat{\varepsilon}_{12}&\hat{\varepsilon}_{22}&\hat{\varepsilon}_{23}\\ \hat{\varepsilon}_{13}&\hat{\varepsilon}_{23}&\hat{\varepsilon}_{33}\end{bmatrix}
  77. ε ^ i j = i p j q ε p q \hat{\varepsilon}_{ij}=\ell_{ip}~{}\ell_{jq}~{}\varepsilon_{pq}
  78. i j = 𝐞 ^ i 𝐞 j \ell_{ij}=\hat{\mathbf{e}}_{i}\cdot\mathbf{e}_{j}
  79. s y m b o l ε ¯ ^ ¯ = 𝐋 ¯ ¯ s y m b o l ε ¯ ¯ 𝐋 ¯ ¯ T \underline{\underline{\hat{symbol{\varepsilon}}}}=\underline{\underline{% \mathbf{L}}}~{}\underline{\underline{symbol{\varepsilon}}}~{}\underline{% \underline{\mathbf{L}}}^{T}
  80. [ ε ^ 11 ε ^ 12 ε ^ 13 ε ^ 21 ε ^ 22 ε ^ 23 ε ^ 31 ε ^ 32 ε ^ 33 ] = [ 11 12 13 21 22 23 31 32 33 ] [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] [ 11 12 13 21 22 23 31 32 33 ] T \begin{bmatrix}\hat{\varepsilon}_{11}&\hat{\varepsilon}_{12}&\hat{\varepsilon}% _{13}\\ \hat{\varepsilon}_{21}&\hat{\varepsilon}_{22}&\hat{\varepsilon}_{23}\\ \hat{\varepsilon}_{31}&\hat{\varepsilon}_{32}&\hat{\varepsilon}_{33}\end{% bmatrix}=\begin{bmatrix}\ell_{11}&\ell_{12}&\ell_{13}\\ \ell_{21}&\ell_{22}&\ell_{23}\\ \ell_{31}&\ell_{32}&\ell_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}&% \varepsilon_{12}&\varepsilon_{13}\\ \varepsilon_{21}&\varepsilon_{22}&\varepsilon_{23}\\ \varepsilon_{31}&\varepsilon_{32}&\varepsilon_{33}\end{bmatrix}\begin{bmatrix}% \ell_{11}&\ell_{12}&\ell_{13}\\ \ell_{21}&\ell_{22}&\ell_{23}\\ \ell_{31}&\ell_{32}&\ell_{33}\end{bmatrix}^{T}
  81. I 1 = tr ( s y m b o l ε ) I 2 = 1 2 { tr ( s y m b o l ε 2 ) - [ tr ( s y m b o l ε ) ] 2 } I 3 = det ( s y m b o l ε ) \begin{aligned}\displaystyle I_{1}&\displaystyle=\mathrm{tr}(symbol{% \varepsilon})\\ \displaystyle I_{2}&\displaystyle=\tfrac{1}{2}\{\mathrm{tr}(symbol{\varepsilon% }^{2})-[\mathrm{tr}(symbol{\varepsilon})]^{2}\}\\ \displaystyle I_{3}&\displaystyle=\det(symbol{\varepsilon})\end{aligned}
  82. I 1 = ε 11 + ε 22 + ε 33 I 2 = ε 12 2 + ε 23 2 + ε 31 2 - ε 11 ε 22 - ε 22 ε 33 - ε 33 ε 11 I 3 = ε 11 ( ε 22 ε 33 - ε 23 2 ) - ε 12 ( ε 12 ε 33 - ε 23 ε 31 ) + ε 13 ( ε 12 ε 23 - ε 22 ε 31 ) \begin{aligned}\displaystyle I_{1}&\displaystyle=\varepsilon_{11}+\varepsilon_% {22}+\varepsilon_{33}\\ \displaystyle I_{2}&\displaystyle=\varepsilon_{12}^{2}+\varepsilon_{23}^{2}+% \varepsilon_{31}^{2}-\varepsilon_{11}\varepsilon_{22}-\varepsilon_{22}% \varepsilon_{33}-\varepsilon_{33}\varepsilon_{11}\\ \displaystyle I_{3}&\displaystyle=\varepsilon_{11}(\varepsilon_{22}\varepsilon% _{33}-\varepsilon_{23}^{2})-\varepsilon_{12}(\varepsilon_{12}\varepsilon_{33}-% \varepsilon_{23}\varepsilon_{31})+\varepsilon_{13}(\varepsilon_{12}\varepsilon% _{23}-\varepsilon_{22}\varepsilon_{31})\end{aligned}
  83. 𝐧 1 , 𝐧 2 , 𝐧 3 \mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}
  84. s y m b o l ε ¯ ¯ = [ ε 1 0 0 0 ε 2 0 0 0 ε 3 ] s y m b o l ε = ε 1 𝐧 1 𝐧 1 + ε 2 𝐧 2 𝐧 2 + ε 3 𝐧 3 𝐧 3 \underline{\underline{symbol{\varepsilon}}}=\begin{bmatrix}\varepsilon_{1}&0&0% \\ 0&\varepsilon_{2}&0\\ 0&0&\varepsilon_{3}\end{bmatrix}\quad\implies\quad symbol{\varepsilon}=% \varepsilon_{1}\mathbf{n}_{1}\otimes\mathbf{n}_{1}+\varepsilon_{2}\mathbf{n}_{% 2}\otimes\mathbf{n}_{2}+\varepsilon_{3}\mathbf{n}_{3}\otimes\mathbf{n}_{3}
  85. 𝐧 1 , 𝐧 2 , 𝐧 3 \mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}
  86. 𝐧 i \mathbf{n}_{i}
  87. ( s y m b o l ε ¯ ¯ - ε i 𝐈 ¯ ¯ ) 𝐧 i = 𝟎 ¯ ¯ (\underline{\underline{symbol{\varepsilon}}}-\varepsilon_{i}~{}\underline{% \underline{\mathbf{I}}})~{}\mathbf{n}_{i}=\underline{\underline{\mathbf{0}}}
  88. 𝐧 i \mathbf{n}_{i}
  89. δ = Δ V V 0 = ε 11 + ε 22 + ε 33 \delta=\frac{\Delta V}{V_{0}}=\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{3% 3}\,\!
  90. a ( 1 + ε 11 ) × a ( 1 + ε 22 ) × a ( 1 + ε 33 ) a\cdot(1+\varepsilon_{11})\times a\cdot(1+\varepsilon_{22})\times a\cdot(1+% \varepsilon_{33})\,\!
  91. Δ V V 0 = ( 1 + ε 11 + ε 22 + ε 33 + ε 11 ε 22 + ε 11 ε 33 + ε 22 ε 33 + ε 11 ε 22 ε 33 ) a 3 - a 3 a 3 \frac{\Delta V}{V_{0}}=\frac{\left(1+\varepsilon_{11}+\varepsilon_{22}+% \varepsilon_{33}+\varepsilon_{11}\cdot\varepsilon_{22}+\varepsilon_{11}\cdot% \varepsilon_{33}+\varepsilon_{22}\cdot\varepsilon_{33}+\varepsilon_{11}\cdot% \varepsilon_{22}\cdot\varepsilon_{33}\right)\cdot a^{3}-a^{3}}{a^{3}}\,\!
  92. 1 ε i i ε i i ε j j ε 11 ε 22 ε 33 1\gg\varepsilon_{ii}\gg\varepsilon_{ii}\cdot\varepsilon_{jj}\gg\varepsilon_{11% }\cdot\varepsilon_{22}\cdot\varepsilon_{33}\,\!
  93. ε i j \varepsilon_{ij}\,\!
  94. ε M δ i j \varepsilon_{M}\delta_{ij}\,\!
  95. ε i j \varepsilon^{\prime}_{ij}\,\!
  96. ε i j = ε i j + ε M δ i j \varepsilon_{ij}=\varepsilon^{\prime}_{ij}+\varepsilon_{M}\delta_{ij}\,\!
  97. ε M \varepsilon_{M}\,\!
  98. ε M = ε k k 3 = ε 11 + ε 22 + ε 33 3 = 1 3 I 1 e \varepsilon_{M}=\frac{\varepsilon_{kk}}{3}=\frac{\varepsilon_{11}+\varepsilon_% {22}+\varepsilon_{33}}{3}=\tfrac{1}{3}I^{e}_{1}\,\!
  99. ε i j = ε i j - ε k k 3 δ i j [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] = [ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] - [ ε M 0 0 0 ε M 0 0 0 ε M ] = [ ε 11 - ε M ε 12 ε 13 ε 21 ε 22 - ε M ε 23 ε 31 ε 32 ε 33 - ε M ] \begin{aligned}\displaystyle\ \varepsilon^{\prime}_{ij}&\displaystyle=% \varepsilon_{ij}-\frac{\varepsilon_{kk}}{3}\delta_{ij}\\ \displaystyle\left[{\begin{matrix}\varepsilon^{\prime}_{11}&\varepsilon^{% \prime}_{12}&\varepsilon^{\prime}_{13}\\ \varepsilon^{\prime}_{21}&\varepsilon^{\prime}_{22}&\varepsilon^{\prime}_{23}% \\ \varepsilon^{\prime}_{31}&\varepsilon^{\prime}_{32}&\varepsilon^{\prime}_{33}% \\ \end{matrix}}\right]&\displaystyle=\left[{\begin{matrix}\varepsilon_{11}&% \varepsilon_{12}&\varepsilon_{13}\\ \varepsilon_{21}&\varepsilon_{22}&\varepsilon_{23}\\ \varepsilon_{31}&\varepsilon_{32}&\varepsilon_{33}\\ \end{matrix}}\right]-\left[{\begin{matrix}\varepsilon_{M}&0&0\\ 0&\varepsilon_{M}&0\\ 0&0&\varepsilon_{M}\\ \end{matrix}}\right]\\ &\displaystyle=\left[{\begin{matrix}\varepsilon_{11}-\varepsilon_{M}&% \varepsilon_{12}&\varepsilon_{13}\\ \varepsilon_{21}&\varepsilon_{22}-\varepsilon_{M}&\varepsilon_{23}\\ \varepsilon_{31}&\varepsilon_{32}&\varepsilon_{33}-\varepsilon_{M}\\ \end{matrix}}\right]\\ \end{aligned}\,\!
  100. 𝐧 1 , 𝐧 2 , 𝐧 3 \mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}
  101. γ oct = 2 3 ( ε 1 - ε 2 ) 2 + ( ε 2 - ε 3 ) 2 + ( ε 3 - ε 1 ) 2 \gamma_{\mathrm{oct}}=\tfrac{2}{3}\sqrt{(\varepsilon_{1}-\varepsilon_{2})^{2}+% (\varepsilon_{2}-\varepsilon_{3})^{2}+(\varepsilon_{3}-\varepsilon_{1})^{2}}
  102. ε 1 , ε 2 , ε 3 \varepsilon_{1},\varepsilon_{2},\varepsilon_{3}
  103. ε oct = 1 3 ( ε 1 + ε 2 + ε 3 ) \varepsilon_{\mathrm{oct}}=\tfrac{1}{3}(\varepsilon_{1}+\varepsilon_{2}+% \varepsilon_{3})
  104. ε eq = 2 3 s y m b o l ε dev : s y m b o l ε dev = 2 3 ε i j dev ε i j dev ; s y m b o l ε dev = s y m b o l ε - 1 3 tr ( s y m b o l ε ) s y m b o l 1 \varepsilon_{\mathrm{eq}}=\sqrt{\tfrac{2}{3}symbol{\varepsilon}^{\mathrm{dev}}% :symbol{\varepsilon}^{\mathrm{dev}}}=\sqrt{\tfrac{2}{3}\varepsilon_{ij}^{% \mathrm{dev}}\varepsilon_{ij}^{\mathrm{dev}}}~{};~{}~{}symbol{\varepsilon}^{% \mathrm{dev}}=symbol{\varepsilon}-\tfrac{1}{3}\mathrm{tr}(symbol{\varepsilon})% ~{}symbol{1}
  105. σ eq = 3 2 s y m b o l σ dev : s y m b o l σ dev \sigma_{\mathrm{eq}}=\sqrt{\tfrac{3}{2}symbol{\sigma}^{\mathrm{dev}}:symbol{% \sigma}^{\mathrm{dev}}}
  106. ε i j \varepsilon_{ij}\,\!
  107. u i , j + u j , i = 2 ε i j u_{i,j}+u_{j,i}=2\varepsilon_{ij}\,\!
  108. u i u_{i}\,\!
  109. u i u_{i}\,\!
  110. ε i j , k m + ε k m , i j - ε i k , j m - ε j m , i k = 0 \varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik% }=0\,\!
  111. 2 ϵ x y 2 + 2 ϵ y x 2 = 2 2 ϵ x y x y \frac{\partial^{2}\epsilon_{x}}{\partial y^{2}}+\frac{\partial^{2}\epsilon_{y}% }{\partial x^{2}}=2\frac{\partial^{2}\epsilon_{xy}}{\partial x\partial y}\,\!
  112. 2 ϵ y z 2 + 2 ϵ z y 2 = 2 2 ϵ y z y z \frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!
  113. 2 ϵ x z 2 + 2 ϵ z x 2 = 2 2 ϵ z x z x \frac{\partial^{2}\epsilon_{x}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial x^{2}}=2\frac{\partial^{2}\epsilon_{zx}}{\partial z\partial x}\,\!
  114. 2 ϵ x y z = x ( - ϵ y z x + ϵ z x y + ϵ x y z ) \frac{\partial^{2}\epsilon_{x}}{\partial y\partial z}=\frac{\partial}{\partial x% }\left(-\frac{\partial\epsilon_{yz}}{\partial x}+\frac{\partial\epsilon_{zx}}{% \partial y}+\frac{\partial\epsilon_{xy}}{\partial z}\right)\,\!
  115. 2 ϵ y z x = y ( ϵ y z x - ϵ z x y + ϵ x y z ) \frac{\partial^{2}\epsilon_{y}}{\partial z\partial x}=\frac{\partial}{\partial y% }\left(\frac{\partial\epsilon_{yz}}{\partial x}-\frac{\partial\epsilon_{zx}}{% \partial y}+\frac{\partial\epsilon_{xy}}{\partial z}\right)\,\!
  116. 2 ϵ z x y = z ( ϵ y z x + ϵ z x y - ϵ x y z ) \frac{\partial^{2}\epsilon_{z}}{\partial x\partial y}=\frac{\partial}{\partial z% }\left(\frac{\partial\epsilon_{yz}}{\partial x}+\frac{\partial\epsilon_{zx}}{% \partial y}-\frac{\partial\epsilon_{xy}}{\partial z}\right)\,\!
  117. ε 33 \varepsilon_{33}\,\!
  118. ε 13 \varepsilon_{13}\,\!
  119. ε 23 \varepsilon_{23}\,\!
  120. s y m b o l ε ¯ ¯ = [ ε 11 ε 12 0 ε 21 ε 22 0 0 0 0 ] \underline{\underline{symbol{\varepsilon}}}=\begin{bmatrix}\varepsilon_{11}&% \varepsilon_{12}&0\\ \varepsilon_{21}&\varepsilon_{22}&0\\ 0&0&0\end{bmatrix}\,\!
  121. s y m b o l σ ¯ ¯ = [ σ 11 σ 12 0 σ 21 σ 22 0 0 0 σ 33 ] \underline{\underline{symbol{\sigma}}}=\begin{bmatrix}\sigma_{11}&\sigma_{12}&% 0\\ \sigma_{21}&\sigma_{22}&0\\ 0&0&\sigma_{33}\end{bmatrix}\,\!
  122. σ 33 \sigma_{33}\,\!
  123. ϵ 33 = 0 \epsilon_{33}=0\,\!
  124. s y m b o l ε ¯ ¯ = [ 0 0 ε 13 0 0 ε 23 ε 13 ε 23 0 ] \underline{\underline{symbol{\varepsilon}}}=\begin{bmatrix}0&0&\varepsilon_{13% }\\ 0&0&\varepsilon_{23}\\ \varepsilon_{13}&\varepsilon_{23}&0\end{bmatrix}\,\!
  125. s y m b o l ε = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] symbol{\varepsilon}=\frac{1}{2}[symbol{\nabla}\mathbf{u}+(symbol{\nabla}% \mathbf{u})^{T}]
  126. s y m b o l 𝐮 = s y m b o l ε + s y m b o l ω symbol{\nabla}\mathbf{u}=symbol{\varepsilon}+symbol{\omega}
  127. s y m b o l ω := 1 2 [ s y m b o l 𝐮 - ( s y m b o l 𝐮 ) T ] symbol{\omega}:=\frac{1}{2}[symbol{\nabla}\mathbf{u}-(symbol{\nabla}\mathbf{u}% )^{T}]
  128. s y m b o l ω symbol{\omega}
  129. s y m b o l ω symbol{\omega}
  130. | ω i j | 1 |\omega_{ij}|\ll 1
  131. 𝐰 \mathbf{w}
  132. ω i j = - ϵ i j k w k ; w i = - 1 2 ϵ i j k ω j k \omega_{ij}=-\epsilon_{ijk}~{}w_{k}~{};~{}~{}w_{i}=-\tfrac{1}{2}~{}\epsilon_{% ijk}~{}\omega_{jk}
  133. ϵ i j k \epsilon_{ijk}
  134. s y m b o l ω ¯ ¯ = [ 0 - w 3 w 2 w 3 0 - w 1 - w 2 w 1 0 ] ; 𝐰 ¯ = [ w 1 w 2 w 3 ] \underline{\underline{symbol{\omega}}}=\begin{bmatrix}0&-w_{3}&w_{2}\\ w_{3}&0&-w_{1}\\ -w_{2}&w_{1}&0\end{bmatrix}~{};~{}~{}\underline{\mathbf{w}}=\begin{bmatrix}w_{% 1}\\ w_{2}\\ w_{3}\end{bmatrix}
  135. 𝐰 = 1 2 s y m b o l × 𝐮 \mathbf{w}=\tfrac{1}{2}~{}symbol{\nabla}\times\mathbf{u}
  136. w i = 1 2 ϵ i j k u k , j w_{i}=\tfrac{1}{2}~{}\epsilon_{ijk}~{}u_{k,j}
  137. \lVertsymbol ω 1 \lVertsymbol{\omega}\rVert\ll 1
  138. s y m b o l ε = s y m b o l 0 symbol{\varepsilon}=symbol{0}
  139. | 𝐰 | |\mathbf{w}|
  140. 𝐰 \mathbf{w}
  141. 𝐮 \mathbf{u}
  142. s y m b o l ε symbol{\varepsilon}
  143. s y m b o l \timessymbol ε = e i j k ε l j , i 𝐞 k 𝐞 l = 1 2 e i j k [ u l , j i + u j , l i ] 𝐞 k 𝐞 l symbol{\nabla}\timessymbol{\varepsilon}=e_{ijk}~{}\varepsilon_{lj,i}~{}\mathbf% {e}_{k}\otimes\mathbf{e}_{l}=\tfrac{1}{2}~{}e_{ijk}~{}[u_{l,ji}+u_{j,li}]~{}% \mathbf{e}_{k}\otimes\mathbf{e}_{l}
  144. u l , j i = u l , i j u_{l,ji}=u_{l,ij}\,
  145. e i j k u l , j i = ( e 12 k + e 21 k ) u l , 12 + ( e 13 k + e 31 k ) u l , 13 + ( e 23 k + e 32 k ) u l , 32 = 0 \,e_{ijk}u_{l,ji}=(e_{12k}+e_{21k})u_{l,12}+(e_{13k}+e_{31k})u_{l,13}+(e_{23k}% +e_{32k})u_{l,32}=0
  146. 1 2 e i j k u j , l i = ( 1 2 e i j k u j , i ) , l = ( 1 2 e k i j u j , i ) , l = w k , l \tfrac{1}{2}~{}e_{ijk}~{}u_{j,li}=\left(\tfrac{1}{2}~{}e_{ijk}~{}u_{j,i}\right% )_{,l}=\left(\tfrac{1}{2}~{}e_{kij}~{}u_{j,i}\right)_{,l}=w_{k,l}
  147. s y m b o l \timessymbol ε = w k , l 𝐞 k 𝐞 l = s y m b o l 𝐰 symbol{\nabla}\timessymbol{\varepsilon}=w_{k,l}~{}\mathbf{e}_{k}\otimes\mathbf% {e}_{l}=symbol{\nabla}\mathbf{w}
  148. 𝐮 \mathbf{u}
  149. s y m b o l × ( s y m b o l 𝐮 ) = s y m b o l 0. symbol{\nabla}\times(symbol{\nabla}\mathbf{u})=symbol{0}.
  150. s y m b o l 𝐮 = s y m b o l ε + s y m b o l ω symbol{\nabla}\mathbf{u}=symbol{\varepsilon}+symbol{\omega}
  151. s y m b o l \timessymbol ω = - s y m b o l \timessymbol ε = - s y m b o l 𝐰 . symbol{\nabla}\timessymbol{\omega}=-symbol{\nabla}\timessymbol{\varepsilon}=-% symbol{\nabla}\mathbf{w}.
  152. r , θ , z r,\theta,z
  153. 𝐮 = u r 𝐞 r + u θ 𝐞 θ + u z 𝐞 z \mathbf{u}=u_{r}~{}\mathbf{e}_{r}+u_{\theta}~{}\mathbf{e}_{\theta}+u_{z}~{}% \mathbf{e}_{z}
  154. ε r r = u r r ε θ θ = 1 r ( u θ θ + u r ) ε z z = u z z ε r θ = 1 2 ( 1 r u r θ + u θ r - u θ r ) ε θ z = 1 2 ( u θ z + 1 r u z θ ) ε z r = 1 2 ( u r z + u z r ) \begin{aligned}\displaystyle\varepsilon_{rr}&\displaystyle=\cfrac{\partial u_{% r}}{\partial r}\\ \displaystyle\varepsilon_{\theta\theta}&\displaystyle=\cfrac{1}{r}\left(\cfrac% {\partial u_{\theta}}{\partial\theta}+u_{r}\right)\\ \displaystyle\varepsilon_{zz}&\displaystyle=\cfrac{\partial u_{z}}{\partial z}% \\ \displaystyle\varepsilon_{r\theta}&\displaystyle=\cfrac{1}{2}\left(\cfrac{1}{r% }\cfrac{\partial u_{r}}{\partial\theta}+\cfrac{\partial u_{\theta}}{\partial r% }-\cfrac{u_{\theta}}{r}\right)\\ \displaystyle\varepsilon_{\theta z}&\displaystyle=\cfrac{1}{2}\left(\cfrac{% \partial u_{\theta}}{\partial z}+\cfrac{1}{r}\cfrac{\partial u_{z}}{\partial% \theta}\right)\\ \displaystyle\varepsilon_{zr}&\displaystyle=\cfrac{1}{2}\left(\cfrac{\partial u% _{r}}{\partial z}+\cfrac{\partial u_{z}}{\partial r}\right)\end{aligned}
  155. r , θ , ϕ r,\theta,\phi
  156. 𝐮 = u r 𝐞 r + u θ 𝐞 θ + u ϕ 𝐞 ϕ \mathbf{u}=u_{r}~{}\mathbf{e}_{r}+u_{\theta}~{}\mathbf{e}_{\theta}+u_{\phi}~{}% \mathbf{e}_{\phi}
  157. ε r r = u r r ε θ θ = 1 r ( u θ θ + u r ) ε ϕ ϕ = 1 r sin θ ( u ϕ ϕ + u r sin θ + u θ cos θ ) ε r θ = 1 2 ( 1 r u r θ + u θ r - u θ r ) ε θ ϕ = 1 2 r ( 1 sin θ u θ ϕ + u ϕ θ - u ϕ cot θ ) ε ϕ r = 1 2 ( 1 r sin θ u r ϕ + u ϕ r - u ϕ r ) \begin{aligned}\displaystyle\varepsilon_{rr}&\displaystyle=\cfrac{\partial u_{% r}}{\partial r}\\ \displaystyle\varepsilon_{\theta\theta}&\displaystyle=\cfrac{1}{r}\left(\cfrac% {\partial u_{\theta}}{\partial\theta}+u_{r}\right)\\ \displaystyle\varepsilon_{\phi\phi}&\displaystyle=\cfrac{1}{r\sin\theta}\left(% \cfrac{\partial u_{\phi}}{\partial\phi}+u_{r}\sin\theta+u_{\theta}\cos\theta% \right)\\ \displaystyle\varepsilon_{r\theta}&\displaystyle=\cfrac{1}{2}\left(\cfrac{1}{r% }\cfrac{\partial u_{r}}{\partial\theta}+\cfrac{\partial u_{\theta}}{\partial r% }-\cfrac{u_{\theta}}{r}\right)\\ \displaystyle\varepsilon_{\theta\phi}&\displaystyle=\cfrac{1}{2r}\left(\cfrac{% 1}{\sin\theta}\cfrac{\partial u_{\theta}}{\partial\phi}+\cfrac{\partial u_{% \phi}}{\partial\theta}-u_{\phi}\cot\theta\right)\\ \displaystyle\varepsilon_{\phi r}&\displaystyle=\cfrac{1}{2}\left(\cfrac{1}{r% \sin\theta}\cfrac{\partial u_{r}}{\partial\phi}+\cfrac{\partial u_{\phi}}{% \partial r}-\cfrac{u_{\phi}}{r}\right)\end{aligned}

Initialized_fractional_calculus.html

  1. 𝔻 q 𝔻 - q = 𝕀 \mathbb{D}^{q}\mathbb{D}^{-q}=\mathbb{I}
  2. 𝔻 - q 𝔻 q 𝕀 \mathbb{D}^{-q}\mathbb{D}^{q}\neq\mathbb{I}
  3. d d x [ ( 3 x 2 + 1 ) d x ] = d d x [ x 3 + x + c ] = 3 x 2 + 1 , \frac{d}{dx}\left[\int(3x^{2}+1)dx\right]=\frac{d}{dx}[x^{3}+x+c]=3x^{2}+1\,,
  4. [ d d x ( 3 x 2 + 1 ) ] = 6 x d x = 3 x 2 + c , \int\left[\frac{d}{dx}(3x^{2}+1)\right]=\int 6x\,dx=3x^{2}+c\,,
  5. Ψ \Psi
  6. 𝔻 t q f ( t ) = 1 Γ ( n - q ) d n d t n 0 t ( t - τ ) n - q - 1 f ( τ ) d τ + Ψ ( x ) \mathbb{D}^{q}_{t}f(t)=\frac{1}{\Gamma(n-q)}\frac{d^{n}}{dt^{n}}\int_{0}^{t}(t% -\tau)^{n-q-1}f(\tau)\,d\tau+\Psi(x)

Instanton.html

  1. \hbar
  2. d 2 ψ d x 2 = 2 m ( V ( x ) - E ) 2 ψ . \frac{d^{2}\psi}{dx^{2}}=\frac{2m(V(x)-E)}{\hbar^{2}}\psi.
  3. ψ = exp ( - i k x ) \psi=\exp(-\mathrm{i}kx)\,
  4. k = 2 m ( E - V ) . k=\frac{\sqrt{2m(E-V)}}{\hbar}.
  5. e - 1 a b 2 m ( V ( x ) - E ) d x , e^{-\frac{1}{\hbar}\int_{a}^{b}\sqrt{2m(V(x)-E)}\,dx},
  6. K ( a , b ; t ) = x = a | e - i t | x = b = d [ x ( t ) ] e i S [ x ( t ) ] . K(a,b;t)=\langle x=a|e^{-\frac{i\mathbb{H}t}{\hbar}}|x=b\rangle=\int d[x(t)]e^% {\frac{iS[x(t)]}{\hbar}}.
  7. i t τ it\rightarrow\tau
  8. K E ( a , b ; τ ) = x = a | e - τ | x = b = d [ x ( τ ) ] e - S E [ x ( τ ) ] , K_{E}(a,b;\tau)=\langle x=a|e^{-\frac{\mathbb{H}\tau}{\hbar}}|x=b\rangle=\int d% [x(\tau)]e^{-\frac{S_{E}[x(\tau)]}{\hbar}},
  9. S E = τ a τ b ( 1 2 m ( d x d τ ) 2 + V ( x ) ) d τ . S_{E}=\int_{\tau_{a}}^{\tau_{b}}\left(\frac{1}{2}m\left(\frac{dx}{d\tau}\right% )^{2}+V(x)\right)d\tau.
  10. V ( x ) - V ( x ) V(x)\rightarrow-V(x)
  11. V ( x ) V(x)
  12. V ( x ) V(x)
  13. S 3 S^{3}
  14. S 3 S^{3}
  15. S 3 S^{3}
  16. π 3 \pi_{3}
  17. ( S 3 ) = (S^{3})=
  18. \mathbb{Z}\,
  19. | N |N\rangle
  20. N N
  21. Q Q
  22. Q Q
  23. | N |N\rangle
  24. | N + Q |N+Q\rangle
  25. | θ = N = - N = + e i θ N | N . |\theta\rangle=\sum_{N=-\infty}^{N=+\infty}e^{i\theta N}|N\rangle.
  26. S Y M = M | F | 2 d vol M , S_{YM}=\int_{M}\left|F\right|^{2}d\mathrm{vol}_{M},
  27. d vol M d\mathrm{vol}_{M}
  28. M M
  29. 𝔤 \mathfrak{g}
  30. G G
  31. F F
  32. 𝔤 \mathfrak{g}
  33. M Tr ( F * F ) \int_{M}\mathrm{Tr}(F\wedge*F)
  34. F * F = F , F d vol M . F\wedge*F=\langle F,F\rangle d\mathrm{vol}_{M}.
  35. d F = 0 , d * F = 0. \mathrm{d}F=0,\quad\mathrm{d}{*F}=0.
  36. 𝐉 \mathbf{J}
  37. * F = ± F {*F}=\pm F\,
  38. D F = 0 DF=0
  39. D * F = 0 D*F=0
  40. D F = d F + A F - F A = d ( d A + A A ) + A ( d A + A A ) - ( d A + A A ) A = 0 DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge(dA+A\wedge A)-(dA+A\wedge A)% \wedge A=0
  41. F = d A + A A {F}=d{A}+{A}\wedge{A}
  42. 4 \mathbb{R}^{4}
  43. 1 2 4 Tr [ * F F ] \frac{1}{2}\int_{\mathbb{R}^{4}}\operatorname{Tr}[*{F}\wedge{F}]
  44. 4 Tr [ F F ] . \int_{\mathbb{R}^{4}}\operatorname{Tr}[{F}\wedge{F}].
  45. 0 1 2 4 Tr [ ( * F + e - i θ F ) ( F + e i θ * F ) ] = 4 Tr [ * F F + cos θ F F ] 0\leq\frac{1}{2}\int_{\mathbb{R}^{4}}\operatorname{Tr}[(*{F}+e^{-i\theta}{F})% \wedge({F}+e^{i\theta}*{F})]=\int_{\mathbb{R}^{4}}\operatorname{Tr}[*{F}\wedge% {F}+\cos\theta{F}\wedge{F}]
  46. 1 2 4 Tr [ * F F ] 1 2 | 4 Tr [ F F ] | . \frac{1}{2}\int_{\mathbb{R}^{4}}\operatorname{Tr}[*{F}\wedge{F}]\geq\frac{1}{2% }\left|\int_{\mathbb{R}^{4}}\operatorname{Tr}[{F}\wedge{F}]\right|.

Integration_by_parts.html

  1. u u
  2. = =
  3. u ( x ) u(x)
  4. d u du
  5. = =
  6. u ( x ) d x u′(x)dx
  7. v v
  8. = =
  9. v ( x ) v(x)
  10. d v dv
  11. = =
  12. v ( x ) d x v′(x)dx
  13. u ( x ) v ( x ) d x = u ( x ) v ( x ) - v ( x ) u ( x ) d x \int u(x)v^{\prime}(x)\,dx=u(x)v(x)-\int v(x)\,u^{\prime}(x)dx
  14. u d v = u v - v d u . \int u\,dv=uv-\int v\,du.\!
  15. d d x ( u ( x ) v ( x ) ) = v ( x ) d d x ( u ( x ) ) + u ( x ) d d x ( v ( x ) ) . \frac{d}{dx}\left(u(x)v(x)\right)=v(x)\frac{d}{dx}\left(u(x)\right)+u(x)\frac{% d}{dx}\left(v(x)\right).\!
  16. d d x ( u ( x ) v ( x ) ) d x = u ( x ) v ( x ) d x + u ( x ) v ( x ) d x \int\frac{d}{dx}\left(u(x)v(x)\right)\,dx=\int u^{\prime}(x)v(x)\,dx+\int u(x)% v^{\prime}(x)\,dx
  17. u ( x ) v ( x ) = u ( x ) v ( x ) d x + u ( x ) v ( x ) d x u(x)v(x)=\int u^{\prime}(x)v(x)\,dx+\int u(x)v^{\prime}(x)\,dx
  18. u ( x ) v ( x ) d x = u ( x ) v ( x ) - u ( x ) v ( x ) d x \int u(x)v^{\prime}(x)\,dx=u(x)v(x)-\int u^{\prime}(x)v(x)\,dx
  19. d u = u ( x ) d x d v = v ( x ) d x du=u^{\prime}(x)dx\quad dv=v^{\prime}(x)dx
  20. u ( x ) d v = u ( x ) v ( x ) - v ( x ) d u \int u(x)\,dv=u(x)v(x)-\int v(x)\,du
  21. a b u v d w = [ u v w ] a b - a b u w d v - a b v w d u . \int_{a}^{b}uv\,dw=[uvw]^{b}_{a}-\int_{a}^{b}uw\,dv-\int_{a}^{b}vw\,du.
  22. d d x ( i = 1 n u i ( x ) ) = j = 1 n i j n u i ( x ) d u j ( x ) d x , \frac{d}{dx}\left(\prod_{i=1}^{n}u_{i}(x)\right)=\sum_{j=1}^{n}\prod_{i\neq j}% ^{n}u_{i}(x)\frac{du_{j}(x)}{dx},
  23. [ i = 1 n u i ( x ) ] a b = j = 1 n a b i j n u i ( x ) d u j ( x ) , \Bigl[\prod_{i=1}^{n}u_{i}(x)\Bigr]_{a}^{b}=\sum_{j=1}^{n}\int_{a}^{b}\prod_{i% \neq j}^{n}u_{i}(x)\,du_{j}(x),
  24. x ( y ) = f ( g - 1 ( y ) ) x(y)=f(g^{-1}(y))
  25. y ( x ) = g ( f - 1 ( x ) ) y(x)=g(f^{-1}(x))
  26. A 1 = y 1 y 2 x ( y ) d y A_{1}=\int_{y_{1}}^{y_{2}}x(y)dy
  27. A 2 = x 1 x 2 y ( x ) d x A_{2}=\int_{x_{1}}^{x_{2}}y(x)dx
  28. y 1 y 2 x ( y ) d y A 1 + x 1 x 2 y ( x ) d x A 2 = . x . y ( x ) | x 1 x 2 = . y . x ( y ) | y 1 y 2 \overbrace{\int_{y_{1}}^{y_{2}}x(y)dy}^{A_{1}}+\overbrace{\int_{x_{1}}^{x_{2}}% y(x)dx}^{A_{2}}=\biggl.x.y(x)\biggl|_{x1}^{x2}=\biggl.y.x(y)\biggl|_{y1}^{y2}
  29. x d y + y d x = x y \int xdy+\int ydx=xy
  30. x d y = x y - y d x \int xdy=xy-\int ydx
  31. u v d x = u v d x - ( u v d x ) d x . \int uv\,dx=u\int v\,dx-\int\left(u^{\prime}\int v\,dx\right)\,dx.\!
  32. ln x x 2 d x . \int\frac{\ln x}{x^{2}}\,dx.\!
  33. ln x x 2 d x = - ln x x - ( 1 x ) ( - 1 x ) d x . \int\frac{\ln x}{x^{2}}\,dx=-\frac{\ln x}{x}-\int\biggl(\frac{1}{x}\biggr)% \biggl(-\frac{1}{x}\biggr)\,dx.\!
  34. sec 2 x ln | sin x | d x . \int\sec^{2}x\ln|\sin x|dx.
  35. sec 2 x ln | sin x | d x = tan x ln | sin x | - tan x 1 tan x d x . \int\sec^{2}x\ln|\sin x|dx=\tan x\ln|\sin x|-\int\tan x\frac{1}{\tan x}dx.
  36. I = x cos ( x ) d x , I=\int x\cos(x)\,dx\,,
  37. u = x d u = d x u=x\Rightarrow du=dx
  38. d v = cos ( x ) d x v = cos ( x ) d x = sin ( x ) dv=\cos(x)\,dx\Rightarrow v=\int\cos(x)\,dx=\sin(x)
  39. x cos ( x ) d x = u d v = u v - v d u = x sin ( x ) - sin ( x ) d x = x sin ( x ) + cos ( x ) + C , \begin{aligned}\displaystyle\int x\cos(x)\,dx&\displaystyle=\int u\,dv\\ &\displaystyle=uv-\int v\,du\\ &\displaystyle=x\sin(x)-\int\sin(x)\,dx\\ &\displaystyle=x\sin(x)+\cos(x)+C,\end{aligned}\!
  40. x n e x d x , x n sin ( x ) d x , x n cos ( x ) d x , \int x^{n}e^{x}\,dx,\,\int x^{n}\sin(x)\,dx,\,\int x^{n}\cos(x)\,dx\,,
  41. I = e x cos ( x ) d x . I=\int e^{x}\cos(x)\,dx.
  42. u = cos ( x ) d u = - sin ( x ) d x u=\cos(x)\Rightarrow du=-\sin(x)\,dx
  43. d v = e x d x v = e x d x = e x dv=e^{x}\,dx\Rightarrow v=\int e^{x}\,dx=e^{x}
  44. e x cos ( x ) d x = e x cos ( x ) + e x sin ( x ) d x . \int e^{x}\cos(x)\,dx=e^{x}\cos(x)+\int e^{x}\sin(x)\,dx.\!
  45. u = sin ( x ) d u = cos ( x ) d x u=\sin(x)\Rightarrow du=\cos(x)\,dx
  46. d v = e x d x v = e x d x = e x . dv=e^{x}\,dx\Rightarrow v=\int e^{x}\,dx=e^{x}.
  47. e x sin ( x ) d x = e x sin ( x ) - e x cos ( x ) d x . \int e^{x}\sin(x)\,dx=e^{x}\sin(x)-\int e^{x}\cos(x)\,dx.
  48. e x cos ( x ) d x = e x cos ( x ) + e x sin ( x ) - e x cos ( x ) d x . \int e^{x}\cos(x)\,dx=e^{x}\cos(x)+e^{x}\sin(x)-\int e^{x}\cos(x)\,dx.
  49. 2 e x cos ( x ) d x = e x ( sin ( x ) + cos ( x ) ) + C 2\int e^{x}\cos(x)\,dx=e^{x}(\sin(x)+\cos(x))+C\!
  50. e x cos ( x ) d x = e x ( sin ( x ) + cos ( x ) ) 2 + C \int e^{x}\cos(x)\,dx={e^{x}(\sin(x)+\cos(x))\over 2}+C^{\prime}\!
  51. I = ln ( x ) 1 d x . I=\int\ln(x)\cdot 1\,dx.\!
  52. u = ln ( x ) d u = d x x u=\ln(x)\Rightarrow du=\frac{dx}{x}
  53. d v = d x v = x dv=dx\Rightarrow v=x\,
  54. ln ( x ) d x \displaystyle\int\ln(x)\,dx
  55. I = arctan ( x ) d x . I=\int\arctan(x)\,dx.
  56. arctan ( x ) 1 d x . \int\arctan(x)\cdot 1\,dx.
  57. u = arctan ( x ) d u = d x 1 + x 2 u=\arctan(x)\Rightarrow du=\frac{dx}{1+x^{2}}
  58. d v = d x v = x dv=dx\Rightarrow v=x
  59. arctan ( x ) d x \displaystyle\int\arctan(x)\,dx
  60. x cos x d x . \int x\cos x\,dx.\!
  61. x sin x - 1 sin x d x x\sin x-\int 1\sin x\,dx\!
  62. x sin x + cos x + C . x\sin x+\cos x+C.\!
  63. x 2 2 cos x + x 2 2 sin x d x , \frac{x^{2}}{2}\cos x+\int\frac{x^{2}}{2}\sin x\,dx,
  64. x 3 e x 2 d x , \int x^{3}e^{x^{2}}\,dx,
  65. u = x 2 , d v = x e x 2 d x , u=x^{2},\quad dv=xe^{x^{2}}\,dx,
  66. d u = 2 x d x , v = 1 2 e x 2 . du=2x\,dx,\quad v=\frac{1}{2}e^{x^{2}}.
  67. x 3 e x 2 d x = ( x 2 ) ( x e x 2 ) d x = u d v = u v - v d u = 1 2 x 2 e x 2 - x e x 2 d x . \int x^{3}e^{x^{2}}\,dx=\int\left(x^{2}\right)\left(xe^{x^{2}}\right)\,dx=\int u% \,dv=uv-\int v\,du=\frac{1}{2}x^{2}e^{x^{2}}-\int xe^{x^{2}}\,dx.
  68. x 3 e x 2 d x = 1 2 e x 2 ( x 2 - 1 ) + C . \int x^{3}e^{x^{2}}\,dx=\frac{1}{2}e^{x^{2}}(x^{2}-1)+C.
  69. Γ ( z ) \displaystyle\Gamma(z)
  70. Γ ( z ) = ( z - 1 ) Γ ( z - 1 ) . \Gamma(z)=(z-1)\Gamma(z-1)\,.
  71. Γ ( z + 1 ) = z ! \Gamma(z+1)=z!
  72. ( f ( k ) ) ( ξ ) = ( 2 π i ξ ) k f ( ξ ) , (\mathcal{F}f^{(k)})(\xi)=(2\pi i\xi)^{k}\mathcal{F}f(\xi),
  73. d d y e - 2 π i y ξ = - 2 π i ξ e - 2 π i y ξ , \frac{d}{dy}e^{-2\pi iy\xi}=-2\pi i\xi e^{-2\pi iy\xi},
  74. ( f ) ( ξ ) = - e - 2 π i y ξ f ( y ) d y = [ e - 2 π i y ξ f ( y ) ] - - - ( - 2 π i ξ e - 2 π i y ξ ) f ( y ) d y = 2 π i ξ - e - 2 π i y ξ f ( y ) d y = 2 π i ξ f ( ξ ) . \begin{aligned}\displaystyle(\mathcal{F}f^{\prime})(\xi)&\displaystyle=\int_{-% \infty}^{\infty}e^{-2\pi iy\xi}f^{\prime}(y)\,dy\\ &\displaystyle=\left[e^{-2\pi iy\xi}f(y)\right]_{-\infty}^{\infty}-\int_{-% \infty}^{\infty}(-2\pi i\xi e^{-2\pi iy\xi})f(y)\,dy\\ &\displaystyle=2\pi i\xi\int_{-\infty}^{\infty}e^{-2\pi iy\xi}f(y)\,dy\\ &\displaystyle=2\pi i\xi\mathcal{F}f(\xi).\end{aligned}
  75. | f ( ξ ) | I ( f ) 1 + | 2 π ξ | k |\mathcal{F}f(\xi)|\leq\frac{I(f)}{1+|2\pi\xi|^{k}}
  76. I ( f ) = - ( | f ( y ) | + | f ( k ) ( y ) | ) d y I(f)=\int_{-\infty}^{\infty}\Bigl(|f(y)|+|f^{(k)}(y)|\Bigr)dy
  77. | f ( ξ ) | - | f ( y ) | d y . |\mathcal{F}f(\xi)|\leq\int_{-\infty}^{\infty}|f(y)|\,dy.
  78. | ( 2 π i ξ ) k f ( ξ ) | - | f ( k ) ( y ) | d y . |(2\pi i\xi)^{k}\mathcal{F}f(\xi)|\leq\int_{-\infty}^{\infty}|f^{(k)}(y)|\,dy.
  79. - Δ f , f L 2 = - - f ′′ ( x ) f ( x ) ¯ d x = - [ f ( x ) f ( x ) ¯ ] - + - f ( x ) f ( x ) ¯ d x = - | f ( x ) | 2 d x 0. \begin{aligned}\displaystyle\langle-\Delta f,f\rangle_{L^{2}}&\displaystyle=-% \int_{-\infty}^{\infty}f^{\prime\prime}(x)\overline{f(x)}\,dx\\ &\displaystyle=-\left[f^{\prime}(x)\overline{f(x)}\right]_{-\infty}^{\infty}+% \int_{-\infty}^{\infty}f^{\prime}(x)\overline{f^{\prime}(x)}\,dx\\ &\displaystyle=\int_{-\infty}^{\infty}|f^{\prime}(x)|^{2}\,dx\geq 0.\end{aligned}
  80. u ( n ) v = u ( n - 1 ) v - u ( n - 2 ) v + u ( n - 3 ) v ′′ - + ( - 1 ) n - 1 u v ( n - 1 ) + ( - 1 ) n u v ( n ) . \int u^{(n)}v=u^{(n-1)}v-u^{(n-2)}v^{\prime}+u^{(n-3)}v^{\prime\prime}-\cdots+% (-1)^{n-1}uv^{(n-1)}+(-1)^{n}\int uv^{(n)}.
  81. x 3 cos x d x . \int x^{3}\cos x\,dx.\!
  82. x 3 x^{3}\,
  83. cos x \cos x\,
  84. 3 x 2 3x^{2}\,
  85. sin x \sin x\,
  86. 6 x 6x\,
  87. - cos x -\cos x\,
  88. 6 6\,
  89. - sin x -\sin x\,
  90. 0 0\,
  91. cos x \cos x\,
  92. ( + ) ( x 3 ) ( sin x ) - ( 3 x 2 ) ( - cos x ) + ( 6 x ) ( - sin x ) - ( 6 ) ( cos x ) + C . (+)(x^{3})(\sin x)-(3x^{2})(-\cos x)+(6x)(-\sin x)-(6)(\cos x)+C\,.
  93. x 3 sin x + 3 x 2 cos x - 6 x sin x - 6 cos x + C . x^{3}\sin x+3x^{2}\cos x-6x\sin x-6\cos x+C.\,
  94. e x cos x d x . \int e^{x}\cos x\,dx.
  95. e x e^{x}\,
  96. cos x \cos x\,
  97. e x e^{x}\,
  98. sin x \sin x\,
  99. e x e^{x}\,
  100. - cos x -\cos x\,
  101. e x cos x d x = e x sin x + e x cos x - e x cos x d x , \int e^{x}\cos x\,dx=e^{x}\sin x+e^{x}\cos x-\int e^{x}\cos x\,dx,
  102. 2 e x cos x d x = e x sin x + e x cos x , 2\,\int e^{x}\cos x\,dx=e^{x}\sin x+e^{x}\cos x,
  103. e x cos x d x = e x ( sin x + cos x ) 2 + C . \int e^{x}\cos x\,dx={e^{x}(\sin x+\cos x)\over 2}+C.\!
  104. Ω φ div v d V = Ω φ v d S - Ω v grad φ d V \int_{\Omega}\varphi\,\operatorname{div}\,\vec{v}\;\mathrm{d}V=\int_{\partial% \Omega}\varphi\,\vec{v}\cdot\mathrm{d}\vec{S}-\int_{\Omega}\vec{v}\cdot% \operatorname{grad}\,\varphi\;\mathrm{d}V
  105. Ω u x i v d Ω = Γ u v ν ^ i d Γ - Ω u v x i d Ω , \int_{\Omega}\frac{\partial u}{\partial x_{i}}v\,d\Omega=\int_{\Gamma}uv\,\hat% {\nu}_{i}\,d\Gamma-\int_{\Omega}u\frac{\partial v}{\partial x_{i}}\,d\Omega,
  106. ν ^ \hat{\mathbf{\nu}}
  107. ν ^ i \hat{\nu}_{i}
  108. Ω u 𝐯 d Ω = Γ u ( 𝐯 ν ^ ) d Γ - Ω u 𝐯 d Ω , \int_{\Omega}\nabla u\cdot\mathbf{v}\,d\Omega=\int_{\Gamma}u(\mathbf{v}\cdot% \hat{\nu})\,d\Gamma-\int_{\Omega}u\,\nabla\cdot\mathbf{v}\,d\Omega,
  109. Γ 𝐯 ν ^ d Γ = Ω 𝐯 d Ω . \int_{\Gamma}\mathbf{v}\cdot\hat{\nu}\,d\Gamma=\int_{\Omega}\nabla\cdot\mathbf% {v}\,d\Omega.
  110. 𝐯 = v \mathbf{v}=\nabla v
  111. v C 2 ( Ω ¯ ) v\in C^{2}(\bar{\Omega})
  112. Ω u v d Ω = Γ u v ν ^ d Γ - Ω u 2 v d Ω , \int_{\Omega}\nabla u\cdot\nabla v\,d\Omega=\int_{\Gamma}u\,\nabla v\cdot\hat{% \nu}\,d\Gamma-\int_{\Omega}u\,\nabla^{2}v\,d\Omega,

Integration_by_substitution.html

  1. I I⊆ℝ
  2. φ : a a , b I φ:aa,b→I
  3. [ u U n i c o d e , u 192 ] : I [u^{\prime}Unicode^{\prime},u^{\prime}\u{0}192^{\prime}]:I→ℝ
  4. φ ( a ) φ ( b ) f ( x ) d x = a b f ( φ ( t ) ) φ ( t ) d t . \int_{\varphi(a)}^{\varphi(b)}f(x)\,dx=\int_{a}^{b}f(\varphi(t))\varphi^{% \prime}(t)\,dt.
  5. x = φ ( t ) x=φ(t)
  6. d x d t = φ ( t ) \frac{dx}{dt}=φ′(t)
  7. d x = φ ( t ) d t dx=φ′(t)dt
  8. d x dx
  9. ƒ ƒ
  10. φ φ
  11. ƒ ƒ
  12. I I
  13. φ φ′
  14. a a , b aa,b
  15. ƒ ( φ ( t ) ) φ ( t ) ƒ(φ(t))φ′(t)
  16. a a , b aa,b
  17. φ ( a ) φ ( b ) f ( x ) d x \int_{\varphi(a)}^{\varphi(b)}f(x)\,dx
  18. a b f ( φ ( t ) ) φ ( t ) d t \int_{a}^{b}f(\varphi(t))\varphi^{\prime}(t)\,dt
  19. ƒ ƒ
  20. F F
  21. F φ F∘φ
  22. F F
  23. φ φ
  24. ( F φ ) ( t ) = F ( φ ( t ) ) φ ( t ) = f ( φ ( t ) ) φ ( t ) . (F\circ\varphi)^{\prime}(t)=F^{\prime}(\varphi(t))\varphi^{\prime}(t)=f(% \varphi(t))\varphi^{\prime}(t).
  25. a b f ( φ ( t ) ) φ ( t ) d t \displaystyle\int_{a}^{b}f(\varphi(t))\varphi^{\prime}(t)\,dt
  26. 0 2 x cos ( x 2 + 1 ) d x \int_{0}^{2}x\cos(x^{2}+1)\,dx
  27. x = 0 x = 2 x cos ( x 2 + 1 ) d x \displaystyle\int_{x=0}^{x=2}x\cos(x^{2}+1)\,dx
  28. 0 1 1 - x 2 d x \int_{0}^{1}\sqrt{1-x^{2}}\;dx
  29. ( 1 - sin 2 ( u ) ) = cos ( u ) \sqrt{(1-\sin^{2}(u))}=\cos(u)
  30. 0 1 1 - x 2 d x = 0 π 2 1 - sin 2 ( u ) cos ( u ) d u = 0 π 2 cos 2 ( u ) d u = π 4 \int_{0}^{1}\sqrt{1-x^{2}}\;dx=\int_{0}^{\frac{\pi}{2}}\sqrt{1-\sin^{2}(u)}% \cos(u)\;du=\int_{0}^{\frac{\pi}{2}}\cos^{2}(u)\;du=\frac{\pi}{4}
  31. x cos ( x 2 + 1 ) d x = 1 2 2 x cos ( x 2 + 1 ) d x \displaystyle{}\quad\int x\cos(x^{2}+1)\,dx=\frac{1}{2}\int 2x\cos(x^{2}+1)\,dx
  32. d v 1 d v n = | det ( D φ ) ( u 1 , , u n ) | d u 1 d u n dv_{1}\cdots dv_{n}=|\det(\operatorname{D}\varphi)(u_{1},\ldots,u_{n})|\,du_{1% }\cdots du_{n}
  33. φ ( U ) f ( 𝐯 ) d 𝐯 = U f ( φ ( 𝐮 ) ) | det ( D φ ) ( 𝐮 ) | d 𝐮 . \int_{\varphi(U)}f(\mathbf{v})\,d\mathbf{v}=\int_{U}f(\varphi(\mathbf{u}))% \left|\det(\operatorname{D}\varphi)(\mathbf{u})\right|\,d\mathbf{u}.
  34. φ ( U ) f ( v ) d v = U f ( φ ( u ) ) | det φ ( u ) | d u \int_{\varphi(U)}f(v)\,dv\;=\;\int_{U}f(\varphi(u))\;\left|\det\varphi^{\prime% }(u)\right|\,du
  35. \circ
  36. Y f ( y ) d ρ ( y ) = X f φ ( x ) w ( x ) d μ ( x ) . \int_{Y}f(y)\,d\rho(y)=\int_{X}f\circ\varphi(x)w(x)\,d\mu(x).
  37. w ( x ) = g φ ( x ) w(x)=g\circ\varphi(x)
  38. U ( f φ ) | det D φ | = φ ( U ) f \int_{U}(f\circ\varphi)|\det D\varphi|=\int_{\varphi(U)}f
  39. X X
  40. p x p_{x}
  41. Y Y
  42. X X
  43. y = ϕ ( x ) y=\phi(x)
  44. Y Y
  45. Y Y
  46. S S
  47. P ( Y S ) P(Y\in S)
  48. Y Y
  49. p y p_{y}
  50. P ( Y S ) = S p y ( y ) d y , P(Y\in S)=\int_{S}p_{y}(y)\,dy,
  51. X X
  52. Y Y
  53. ϕ - 1 ( S ) \phi^{-1}(S)
  54. P ( Y S ) = ϕ - 1 ( S ) p x ( x ) d x . P(Y\in S)=\int_{\phi^{-1}(S)}p_{x}(x)\,dx.
  55. P ( Y S ) = ϕ - 1 ( S ) p x ( x ) d x = S p x ( ϕ - 1 ( y ) ) | d ϕ - 1 d y | d y . P(Y\in S)=\int_{\phi^{-1}(S)}p_{x}(x)~{}dx=\int_{S}p_{x}(\phi^{-1}(y))~{}\left% |\frac{d\phi^{-1}}{dy}\right|~{}dy.
  56. S p y ( y ) d y = S p x ( ϕ - 1 ( y ) ) | d ϕ - 1 d y | d y \int_{S}p_{y}(y)~{}dy=\int_{S}p_{x}(\phi^{-1}(y))~{}\left|\frac{d\phi^{-1}}{dy% }\right|~{}dy
  57. p y ( y ) = p x ( ϕ - 1 ( y ) ) | d ϕ - 1 d y | . p_{y}(y)=p_{x}(\phi^{-1}(y))~{}\left|\frac{d\phi^{-1}}{dy}\right|.
  58. X X
  59. Y Y
  60. p x = p x ( x 1 x n ) p_{x}=p_{x}(x_{1}\ldots x_{n})
  61. y = ϕ ( x ) y=\phi(x)
  62. p y p_{y}
  63. p y ( y ) = p x ( ϕ - 1 ( y ) ) | det [ D ϕ - 1 ( y ) ] | . p_{y}(y)=p_{x}(\phi^{-1}(y))~{}\left|\det\left[D\phi^{-1}(y)\right]\right|.

Intensive_and_extensive_properties.html

  1. { a i } \{a_{i}\}
  2. { A j } \{A_{j}\}
  3. F ( { a i } , { A j } ) F(\{a_{i}\},\{A_{j}\})
  4. α \alpha
  5. F ( { α a i } , { A j } ) = F ( { a i } , { A j } ) . F(\{\alpha a_{i}\},\{A_{j}\})=F(\{a_{i}\},\{A_{j}\}).\,
  6. { a i } \{a_{i}\}
  7. { A j } \{A_{j}\}
  8. F ( { a i } , { A j } ) F(\{a_{i}\},\{A_{j}\})
  9. α \alpha
  10. F ( { a i } , { α A j } ) = α F ( { a i } , { A j } ) . F(\{a_{i}\},\{\alpha A_{j}\})=\alpha F(\{a_{i}\},\{A_{j}\}).\,
  11. { A j } \{A_{j}\}
  12. F ( { a i } , { A j } ) = j A j ( F A j ) , F(\{a_{i}\},\{A_{j}\})=\sum_{j}A_{j}\left(\frac{\partial F}{\partial A_{j}}% \right),
  13. A j A_{j}

Interest.html

  1. I simple = r B 0 m t I\text{simple}=r\cdot B_{0}\cdot m_{t}
  2. I simple = ( 0.1299 12 $ 2500 ) 3 = $ 81.19 I\text{simple}=\left(\frac{0.1299}{12}\cdot\$2500\right)\cdot 3=\$81.19
  3. I = ( 0.1299 12 $ 2500 ) 3 = ( $ 27.0625 / month ) 3 = $ 81.19 I=\left(\frac{0.1299}{12}\cdot\$2500\right)\cdot 3=(\$27.0625/\,\text{month})% \cdot 3=\$81.19
  4. i = r + π i=r+\pi
  5. π \pi
  6. i t = r ( t + 1 ) + π ( t + 1 ) + σ i_{t}=r_{(t+1)}+\pi_{(t+1)}+\sigma
  7. π \pi
  8. σ \sigma
  9. lim n ( 1 + 1 n ) n = e , \lim_{n\rightarrow\infty}\left(1+\dfrac{1}{n}\right)^{n}=e,
  10. B k + 1 = ( 1 + r ) B k - p , B_{k+1}=\big(1+r\big)B_{k}-p,
  11. B k = ( 1 + r ) k B 0 - ( 1 + r ) k - 1 r p B_{k}=(1+r)^{k}B_{0}-\frac{(1+r)^{k}-1}{r}p
  12. p = r [ ( 1 + r ) n B 0 - B n ( 1 + r ) n - 1 ] p=r\left[\frac{(1+r)^{n}B_{0}-B_{n}}{(1+r)^{n}-1}\right]
  13. p = PMT ( rate , num , PV , FV , ) = PMT ( r , n , - B 0 , B n , ) p=\mathrm{PMT}(\,\text{rate},\,\text{num},\,\text{PV},\,\text{FV},)=\mathrm{% PMT}(r,n,-B_{0},B_{n},)\;
  14. p I = r B . p_{I}=rB.\,
  15. I T = n p - B 0 . I_{T}=np-B_{0}.\,
  16. B k = B 0 = ( 1 + r k ) B 0 - p k . B_{k}=B^{\prime}_{0}=(1+r_{k})B_{0}-p_{k}.\,
  17. r k = ( 1 + r ) k - 1 r_{k}=(1+r)^{k}-1
  18. p k = p r r k . p_{k}=\frac{p}{r}r_{k}.\,
  19. B * = p r = p k r k B^{*}=\frac{p}{r}=\frac{p_{k}}{r_{k}}
  20. B k = ( 1 + r k ) B 0 - r k B * . B_{k}=(1+r_{k})B_{0}-r_{k}B^{*}.
  21. r k = B 0 - B k B * - B 0 r_{k}=\frac{B_{0}-B_{k}}{B^{*}-B_{0}}
  22. λ k = p k p = r k r = ( 1 + r ) k - 1 r = k [ 1 + ( k - 1 ) r 2 + ] \lambda_{k}=\frac{p_{k}}{p}=\frac{r_{k}}{r}=\frac{(1+r)^{k}-1}{r}=k\left[1+% \frac{(k-1)r}{2}+\cdots\right]
  23. r k = λ k r r_{k}=\lambda_{k}r\;
  24. p k = λ k p p_{k}=\lambda_{k}p\;
  25. Δ B k = B - B = ( λ k r B - λ k p ) = λ k Δ B \Delta B_{k}=B^{\prime}-B=(\lambda_{k}rB-\lambda_{k}p)=\lambda_{k}\,\Delta B\;
  26. B k = B 0 - r k ( B * - B 0 ) B_{k}=B_{0}-r_{k}(B^{*}-B_{0})\;
  27. B * = B 0 ( 1 r n + 1 ) . B^{*}=B_{0}\left(\frac{1}{r_{n}}+1\right).
  28. B k = B 0 ( 1 - r k r n ) = B 0 ( 1 - λ k λ n ) B_{k}=B_{0}\left(1-\frac{r_{k}}{r_{n}}\right)=B_{0}\left(1-\frac{\lambda_{k}}{% \lambda_{n}}\right)
  29. λ k + 1 = 1 + ( 1 + r ) λ k \lambda_{k+1}=1+(1+r)\lambda_{k}\;
  30. p = ( r + 1 λ n ) B 0 p=\left(r+\frac{1}{\lambda_{n}}\right)B_{0}
  31. r loan = I T n B 0 = r + 1 λ n - 1 n , r\text{loan}=\frac{I_{T}}{nB_{0}}=r+\frac{1}{\lambda_{n}}-\frac{1}{n},

Interest_rate.html

  1. r = 1 + i 1 + p - 1 r=\frac{1+i}{1+p}-1\,\!
  2. r i - p r\approx i-p\,\!
  3. i r = i n - p e i_{r}=i_{n}-p_{e}\,\!
  4. i r i_{r}\,\!
  5. i n i_{n}\,\!
  6. p e p_{e}\,\!
  7. i n = i r + p e i_{n}=i_{r}+p_{e}\,\!
  8. i n i_{n}\,\!
  9. i r i_{r}\,\!
  10. p e p_{e}\,\!
  11. i n = i r + p e + r p + l p i_{n}=i_{r}+p_{e}+rp+lp\,\!
  12. i n = i n * + r p + l p i_{n}=i^{*}_{n}+rp+lp\,\!
  13. i n = i r + p e i_{n}=i_{r}+p_{e}\,\!
  14. ( 1 + i n ) = ( 1 + i r ) ( 1 + p e ) (1+i_{n})=(1+i_{r})(1+p_{e})\,\!
  15. i r = 1 + i n 1 + p e - 1 i_{r}=\frac{1+i_{n}}{1+p_{e}}-1\,\!
  16. ( 1 + x ) ( 1 + y ) \displaystyle(1+x)(1+y)

Interest_rate_cap_and_floor.html

  1. N α max ( L - K , 0 ) N\cdot\alpha\max(L-K,0)
  2. α \alpha
  3. $ 1 M 0.5 max ( 0.03 - 0.025 , 0 ) = $ 2500 \$1M\cdot 0.5\cdot\max(0.03-0.025,0)=\$2500
  4. σ \sigma
  5. V = α P ( 0 , T ) ( F N ( d 1 ) - K N ( d 2 ) ) , V=\alpha P(0,T)\left(FN(d_{1})-KN(d_{2})\right),
  6. 1 α ( P ( 0 , t ) P ( 0 , T ) - 1 ) {1\over\alpha}\left(\frac{P(0,t)}{P(0,T)}-1\right)
  7. d 1 = ln ( F / K ) + 0.5 σ 2 t σ t d_{1}=\frac{\ln(F/K)+0.5\sigma^{2}t}{\sigma\sqrt{t}}
  8. d 2 = d 1 - σ t d_{2}=d_{1}-\sigma\sqrt{t}

Interest_rate_swap.html

  1. P V fixed = N × C × i = 1 n ( δ ~ i × P D ( t ~ i ) ) PV\text{fixed}=N\times C\times\sum_{i=1}^{n}\left(\tilde{\delta}_{i}\times P^{% D}(\tilde{t}_{i})\right)
  2. δ ~ i \tilde{\delta}_{i}
  3. P D ( t ~ i ) P^{D}(\tilde{t}_{i})
  4. t ~ i \tilde{t}_{i}
  5. [ t ~ j - 1 , t j ] [\tilde{t}_{j-1},t_{j}]
  6. δ i \delta_{i}
  7. F j = 1 δ j ( P I ( t j - 1 ) P I ( t j ) - 1 ) F_{j}=\frac{1}{\delta_{j}}\left(\frac{P^{I}(t_{j-1})}{P^{I}(t_{j})}-1\right)
  8. P I ( t j ) P^{I}(t_{j})
  9. P V float = N × j = 1 m ( F j × δ j × P D ( t j ) ) PV\text{float}=N\times\sum_{j=1}^{m}(F_{j}\times\delta_{j}\times P^{D}(t_{j}))
  10. δ i \delta_{i}
  11. P D = P I P^{D}=P^{I}
  12. P V float = N × ( 1 - P D ( t ~ m ) ) PV\text{float}=N\times(1-P^{D}(\tilde{t}_{m}))
  13. P V float PV\text{float}
  14. P V fixed - P V float = N × ( B eq - 1 ) PV\text{fixed}-PV\text{float}=N\times(B\text{eq}-1)
  15. B eq B\text{eq}
  16. P V fixed - P V float = N × ( B eq - P D ( t r ) ) PV\text{fixed}-PV\text{float}=N\times(B\text{eq}-P^{D}(t_{r}))
  17. t r t_{r}
  18. C = P V float i = 1 n ( N × δ ~ i × P D ( t ~ i ) ) C=\frac{PV\text{float}}{\sum_{i=1}^{n}(N\times\tilde{\delta}_{i}\times P^{D}(% \tilde{t}_{i}))}
  19. P V fixed = P V float PV\text{fixed}=PV\text{float}\,

International_Standard_Serial_Number.html

  1. 0 8 + 3 7 + 7 6 + 8 5 + 5 4 + 9 3 + 5 2 0\cdot 8+3\cdot 7+7\cdot 6+8\cdot 5+5\cdot 4+9\cdot 3+5\cdot 2
  2. = 0 + 21 + 42 + 40 + 20 + 27 + 10 =0+21+42+40+20+27+10
  3. = 160 =160
  4. 160 11 = 14 remainder 6 = 14 + 6 11 \frac{160}{11}=14\mbox{ remainder }~{}6=14+\frac{6}{11}
  5. 11 - 6 = 5 11-6=5

Internationalization.html

  1. F i j = G M i M j D i j F_{ij}=G\frac{M_{i}M_{j}}{D_{ij}}
  2. F F\,
  3. i , j i,j\,
  4. M M\,
  5. D D\,
  6. G G\,

Intertemporal_choice.html

  1. N N
  2. U ( x 1 , x 2 ) U(x_{1},x_{2})
  3. x t = ( x t 1 , , x t N ) x_{t}=(x_{t1},\dots,x_{tN})
  4. t t
  5. Y t Y_{t}
  6. S 1 S_{1}
  7. t t
  8. C t C_{t}
  9. r r
  10. C 1 + S 1 Y 1 , C_{1}+S_{1}\leq Y_{1},
  11. C 2 Y 2 + S 1 ( 1 + r ) . C_{2}\leq Y_{2}+S_{1}(1+r).
  12. C 1 + C 2 1 + r = Y 1 + Y 2 1 + r . C_{1}+\frac{C_{2}}{1+r}=Y_{1}+\frac{Y_{2}}{1+r}.
  13. ( 1 + r ) (1+r)
  14. C 1 C_{1}
  15. C 2 C_{2}
  16. U ( C 1 , C 2 ) U(C_{1},C_{2})
  17. C 1 + C 2 / ( 1 + r ) = Y 1 + Y 2 / ( 1 + r ) . C_{1}+C_{2}/(1+r)=Y_{1}+Y_{2}/(1+r).
  18. max U t = t U ( C t ) ( 1 + δ ) - t \max U_{t}=\sum_{t}U(C_{t})(1+\delta)^{-t}
  19. t C t ( 1 + r ) - t = t Y t ( 1 + r ) - t + W 0 , \sum_{t}C_{t}(1+r)^{-t}=\sum_{t}Y_{t}(1+r)^{-t}+W_{0},
  20. f H ( D ) = 1 1 + k D f_{H}(D)=\frac{1}{1+kD}\,

Introduction_to_gauge_theory.html

  1. V V
  2. V V + C V\rightarrow V+C
  3. V V + C V\rightarrow V+C
  4. t t + C t\rightarrow t+C
  5. t t + t 3 / t 0 2 t\rightarrow t+t^{3}/t_{0}^{2}
  6. t 0 t_{0}

Intuitionistic_logic.html

  1. { , } \{\top,\bot\}
  2. { , } \{\top,\bot\}
  3. ϕ \phi
  4. ϕ ψ \phi\to\psi
  5. ψ \psi
  6. ϕ ( χ ϕ ) \phi\to(\chi\to\phi)
  7. ( ϕ ( χ ψ ) ) ( ( ϕ χ ) ( ϕ ψ ) ) (\phi\to(\chi\to\psi))\to((\phi\to\chi)\to(\phi\to\psi))
  8. ϕ χ ϕ \phi\land\chi\to\phi
  9. ϕ χ χ \phi\land\chi\to\chi
  10. ϕ ( χ ( ϕ χ ) ) \phi\to(\chi\to(\phi\land\chi))
  11. ϕ ϕ χ \phi\to\phi\lor\chi
  12. χ ϕ χ \chi\to\phi\lor\chi
  13. ( ϕ ψ ) ( ( χ ψ ) ( ϕ χ ψ ) ) (\phi\to\psi)\to((\chi\to\psi)\to(\phi\lor\chi\to\psi))
  14. ϕ \bot\to\phi
  15. \forall
  16. ψ ϕ \psi\to\phi
  17. ψ ( x ϕ ) \psi\to(\forall x\ \phi)
  18. x x
  19. ψ \psi
  20. \exists
  21. ϕ ψ \phi\to\psi
  22. ( x ϕ ) ψ (\exists x\ \phi)\to\psi
  23. x x
  24. ψ \psi
  25. ( x ϕ ( x ) ) ϕ ( t ) (\forall x\ \phi(x))\to\phi(t)
  26. ϕ \phi
  27. ϕ ( t ) \phi(t)
  28. ϕ ( t ) ( x ϕ ( x ) ) \phi(t)\to(\exists x\ \phi(x))
  29. ¬ \lnot
  30. ϕ \phi\to\bot
  31. ( ϕ ) ¬ ϕ (\phi\to\bot)\to\lnot\phi
  32. ¬ ϕ ( ϕ ) \lnot\phi\to(\phi\to\bot)
  33. \bot
  34. ( ϕ χ ) ( ( ϕ ¬ χ ) ¬ ϕ ) (\phi\to\chi)\to((\phi\to\lnot\chi)\to\lnot\phi)
  35. ϕ ( ¬ ϕ χ ) \phi\to(\lnot\phi\to\chi)
  36. ( ϕ ¬ χ ) ( χ ¬ ϕ ) (\phi\to\lnot\chi)\to(\chi\to\lnot\phi)
  37. ( ϕ ¬ ϕ ) ¬ ϕ (\phi\to\lnot\phi)\to\lnot\phi
  38. \leftrightarrow
  39. ϕ χ \phi\leftrightarrow\chi
  40. ( ϕ χ ) ( χ ϕ ) (\phi\to\chi)\land(\chi\to\phi)
  41. ( ϕ χ ) ( ϕ χ ) (\phi\leftrightarrow\chi)\to(\phi\to\chi)
  42. ( ϕ χ ) ( χ ϕ ) (\phi\leftrightarrow\chi)\to(\chi\to\phi)
  43. ( ϕ χ ) ( ( χ ϕ ) ( ϕ χ ) ) (\phi\to\chi)\to((\chi\to\phi)\to(\phi\leftrightarrow\chi))
  44. ( ϕ χ ) ( ( ϕ χ ) ( χ ϕ ) ) (\phi\leftrightarrow\chi)\to((\phi\to\chi)\land(\chi\to\phi))
  45. ϕ ¬ ϕ \phi\lor\lnot\phi
  46. ( ϕ χ ) ( ( ¬ ϕ χ ) χ ) (\phi\to\chi)\to((\lnot\phi\to\chi)\to\chi)
  47. ¬ ¬ ϕ ϕ \lnot\lnot\phi\to\phi
  48. ( ( ϕ χ ) ϕ ) ϕ ((\phi\to\chi)\to\phi)\to\phi
  49. \circ{\longrightarrow}\circ
  50. ( ϕ ψ ) ¬ ( ¬ ϕ ¬ ψ ) (\phi\wedge\psi)\to\neg(\neg\phi\vee\neg\psi)
  51. ( ϕ ψ ) ¬ ( ¬ ϕ ¬ ψ ) (\phi\vee\psi)\to\neg(\neg\phi\wedge\neg\psi)
  52. ( ¬ ϕ ¬ ψ ) ¬ ( ϕ ψ ) (\neg\phi\vee\neg\psi)\to\neg(\phi\wedge\psi)
  53. ( ¬ ϕ ¬ ψ ) ¬ ( ϕ ψ ) (\neg\phi\wedge\neg\psi)\leftrightarrow\neg(\phi\vee\psi)
  54. ( ϕ ψ ) ¬ ( ϕ ¬ ψ ) (\phi\wedge\psi)\to\neg(\phi\to\neg\psi)
  55. ( ϕ ψ ) ¬ ( ϕ ¬ ψ ) (\phi\to\psi)\to\neg(\phi\wedge\neg\psi)
  56. ( ϕ ¬ ψ ) ¬ ( ϕ ψ ) (\phi\wedge\neg\psi)\to\neg(\phi\to\psi)
  57. ( ϕ ¬ ψ ) ¬ ( ϕ ψ ) (\phi\to\neg\psi)\leftrightarrow\neg(\phi\wedge\psi)
  58. ( ϕ ψ ) ( ¬ ϕ ψ ) (\phi\vee\psi)\to(\neg\phi\to\psi)
  59. ( ¬ ϕ ψ ) ( ϕ ψ ) (\neg\phi\vee\psi)\to(\phi\to\psi)
  60. ( x ϕ ( x ) ) ¬ ( x ¬ ϕ ( x ) ) (\forall x\ \phi(x))\to\neg(\exists x\ \neg\phi(x))
  61. ( x ϕ ( x ) ) ¬ ( x ¬ ϕ ( x ) ) (\exists x\ \phi(x))\to\neg(\forall x\ \neg\phi(x))
  62. ( x ¬ ϕ ( x ) ) ¬ ( x ϕ ( x ) ) (\exists x\ \neg\phi(x))\to\neg(\forall x\ \phi(x))
  63. ( x ¬ ϕ ( x ) ) ¬ ( x ϕ ( x ) ) (\forall x\ \neg\phi(x))\leftrightarrow\neg(\exists x\ \phi(x))
  64. ( ϕ ψ ) ( ( ϕ ψ ) ( ψ ϕ ) ) (\phi\leftrightarrow\psi)\leftrightarrow((\phi\to\psi)\land(\psi\to\phi))
  65. ( ϕ ψ ) ( ( ϕ ψ ) ψ ) (\phi\to\psi)\leftrightarrow((\phi\lor\psi)\leftrightarrow\psi)
  66. ( ϕ ψ ) ( ( ϕ ψ ) ϕ ) (\phi\to\psi)\leftrightarrow((\phi\land\psi)\leftrightarrow\phi)
  67. ( ϕ ψ ) ( ( ϕ ψ ) ϕ ) (\phi\land\psi)\leftrightarrow((\phi\to\psi)\leftrightarrow\phi)
  68. ( ϕ ψ ) ( ( ( ϕ ψ ) ψ ) ϕ ) (\phi\land\psi)\leftrightarrow(((\phi\lor\psi)\leftrightarrow\psi)% \leftrightarrow\phi)
  69. ( ( p q ) ¬ r ) ( ¬ p ( q r ) ) , ((p\lor q)\land\neg r)\lor(\neg p\land(q\leftrightarrow r)),
  70. p ( q ¬ r ( s t ) ) . p\to(q\land\neg r\land(s\lor t)).
  71. * \bot^{*}
  72. \ \bot
  73. A * A^{*}
  74. A if A is prime (a positive literal) \ \Box A\qquad\hbox{if }A\hbox{ is prime (a positive literal)}
  75. ( A B ) * (A\wedge B)^{*}
  76. A * B * \ A^{*}\wedge B^{*}
  77. ( A B ) * (A\vee B)^{*}
  78. A * B * \ A^{*}\vee B^{*}
  79. ( A B ) * (A\rightarrow B)^{*}
  80. ( A * B * ) \ \Box(A^{*}\rightarrow B^{*})
  81. ( ¬ A ) * (\neg A)^{*}
  82. ( ¬ ( A * ) ) since ¬ A := A \ \Box(\neg(A^{*}))\qquad\hbox{since }\neg A:=A\rightarrow\bot

Invariant_mass.html

  1. E E
  2. 𝐩 \mathbf{p}
  3. m 0 2 c 2 = ( E c ) 2 - 𝐩 2 m_{0}^{2}c^{2}=\left(\frac{E}{c}\right)^{2}-\left\|\mathbf{p}\right\|^{2}
  4. c = 1 c=1
  5. m 0 2 = E 2 - 𝐩 2 . m_{0}^{2}=E^{2}-\left\|\mathbf{p}\right\|^{2}.
  6. ( E , 𝐩 ) (E,\mathbf{p})
  7. ( W c 2 ) 2 = ( E ) 2 - 𝐩 c 2 , \left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum\mathbf{p}c\right\|% ^{2},
  8. W W
  9. E \sum E
  10. 𝐩 \sum\mathbf{p}
  11. W W
  12. W 2 = ( E in - E out ) 2 - 𝐩 in - 𝐩 out 2 . W^{2}=\left(\sum E\text{in}-\sum E\text{out}\right)^{2}-\left\|\sum\mathbf{p}% \text{in}-\sum\mathbf{p}\text{out}\right\|^{2}.
  13. M 2 M^{2}
  14. = ( E 1 + E 2 ) 2 - 𝐩 1 + 𝐩 2 2 =(E_{1}+E_{2})^{2}-\left\|\,\textbf{p}_{1}+\,\textbf{p}_{2}\right\|^{2}
  15. = m 1 2 + m 2 2 + 2 ( E 1 E 2 - 𝐩 1 𝐩 2 ) . =m_{1}^{2}+m_{2}^{2}+2\left(E_{1}E_{2}-\,\textbf{p}_{1}\cdot\,\textbf{p}_{2}% \right).
  16. θ \theta
  17. M 2 M^{2}
  18. = ( E 1 + E 2 ) 2 - 𝐩 1 + 𝐩 2 2 =(E_{1}+E_{2})^{2}-\left\|\,\textbf{p}_{1}+\,\textbf{p}_{2}\right\|^{2}
  19. = [ ( p 1 , 0 , 0 , p 1 ) + ( p 2 , 0 , p 2 sin θ , p 2 cos θ ) ] 2 = ( p 1 + p 2 ) 2 - p 2 2 sin 2 θ - ( p 1 + p 2 cos θ ) 2 =[(p_{1},0,0,p_{1})+(p_{2},0,p_{2}\sin\theta,p_{2}\cos\theta)]^{2}=(p_{1}+p_{2% })^{2}-p_{2}^{2}\sin^{2}\theta-(p_{1}+p_{2}\cos\theta)^{2}
  20. = 2 p 1 p 2 ( 1 - cos θ ) . =2p_{1}p_{2}(1-\cos\theta).
  21. ϕ \phi
  22. η \eta
  23. p T p_{T}
  24. E m E>>m
  25. M 2 M^{2}\,
  26. = 2 p T 1 p T 2 ( cosh ( η 1 - η 2 ) - cos ( ϕ 1 - ϕ 2 ) ) . =2p_{T1}p_{T2}(\cosh(\eta_{1}-\eta_{2})-\cos(\phi_{1}-\phi_{2})).\,
  27. E 0 E_{0}
  28. E 0 = m 0 c 2 \ E_{0}=m_{0}c^{2}
  29. c c

Inverse_(logic).html

  1. P Q P\rightarrow Q
  2. ¬ P ¬ Q \neg P\rightarrow\neg Q
  3. ¬ P ¬ Q \neg P\rightarrow\neg Q
  4. ¬ ¬ P ¬ ¬ Q \neg\neg P\rightarrow\neg\neg Q
  5. P Q P\rightarrow Q
  6. ¬ P ¬ Q \neg P\rightarrow\neg Q
  7. P Q P\rightarrow Q
  8. P ¬ Q P\rightarrow\neg Q
  9. ¬ P Q \neg P\rightarrow Q

Inverse_function_theorem.html

  1. f f
  2. a a
  3. f f
  4. a a
  5. ( f - 1 ) ( f ( a ) ) = 1 f ( a ) , \bigl(f^{-1}\bigr)^{\prime}(f(a))=\frac{1}{f^{\prime}(a)},
  6. F F
  7. n \mathbb{R}^{n}
  8. n \mathbb{R}^{n}
  9. p p
  10. F F
  11. p p
  12. F F
  13. p p
  14. F F
  15. F ( p ) F(p)
  16. F - 1 F^{-1}
  17. p p
  18. J F - 1 ( F ( p ) ) = [ J F ( p ) ] - 1 , J_{F^{-1}}(F(p))=[J_{F}(p)]^{-1},
  19. [ ] - 1 [\cdot]^{-1}
  20. J G ( q ) J_{G}(q)
  21. G G
  22. q q
  23. G G
  24. H H
  25. H ( p ) H(p)
  26. p p
  27. J G H ( p ) = J G ( H ( p ) ) J H ( p ) . J_{G\circ H}(p)=J_{G}(H(p))\cdot J_{H}(p).
  28. G G
  29. F - 1 F^{-1}
  30. H H
  31. F F
  32. G H G\circ H
  33. J F - 1 ( F ( p ) ) J_{F^{-1}}(F(p))
  34. H H
  35. F - 1 F^{-1}
  36. p p
  37. F F
  38. n n
  39. y i = F i ( x 1 , , x n ) y_{i}=F_{i}(x_{1},\dots,x_{n})
  40. x 1 , , x n x_{1},\dots,x_{n}
  41. y 1 , , y n y_{1},\dots,y_{n}
  42. x x
  43. y y
  44. p p
  45. F ( p ) F(p)
  46. F F
  47. 2 \mathbb{R}^{2}
  48. 2 \mathbb{R}^{2}
  49. 𝐅 ( x , y ) = [ e x cos y e x sin y ] . \mathbf{F}(x,y)=\begin{bmatrix}{e^{x}\cos y}\\ {e^{x}\sin y}\\ \end{bmatrix}.
  50. J F ( x , y ) = [ e x cos y - e x sin y e x sin y e x cos y ] J_{F}(x,y)=\begin{bmatrix}{e^{x}\cos y}&{-e^{x}\sin y}\\ {e^{x}\sin y}&{e^{x}\cos y}\\ \end{bmatrix}
  51. det J F ( x , y ) = e 2 x cos 2 y + e 2 x sin 2 y = e 2 x . \det J_{F}(x,y)=e^{2x}\cos^{2}y+e^{2x}\sin^{2}y=e^{2x}.\,\!
  52. e 2 x e^{2x}
  53. p p
  54. 2 \mathbb{R}^{2}
  55. p p
  56. F F
  57. F F
  58. F F
  59. f ( x , y ) = f ( x , y + 2 π ) f(x,y)=f(x,y+2\pi)
  60. F : M N F:M\to N
  61. F F
  62. d F p : T p M T F ( p ) N dF_{p}:T_{p}M\to T_{F(p)}N
  63. p p
  64. M M
  65. U U
  66. p p
  67. F | U : U F ( U ) F|_{U}:U\to F(U)
  68. M M
  69. N N
  70. p p
  71. F F
  72. p p
  73. M M
  74. F F
  75. X X
  76. Y Y
  77. U U
  78. X X
  79. F : U Y F:U\to Y
  80. d F 0 : X Y dF_{0}:X\to Y
  81. F F
  82. X X
  83. Y Y
  84. V V
  85. F ( 0 ) F(0)
  86. Y Y
  87. G : V X G:V\to X
  88. F ( G ( y ) ) = y F(G(y))=y
  89. y y
  90. V V
  91. G ( y ) G(y)
  92. x x
  93. F ( x ) = y F(x)=y
  94. F : M N F:M\to N
  95. p M p\in M
  96. U U
  97. p p
  98. V V
  99. F ( p ) F(p)
  100. u : T p M U u:T_{p}M\to U
  101. v : T F ( p ) N V v:T_{F(p)}N\to V
  102. F ( U ) V F(U)\subseteq V
  103. d F p : T p M T F ( p ) N dF_{p}:T_{p}M\to T_{F(p)}N
  104. v - 1 F u v^{-1}\circ F\circ u
  105. F F
  106. p p
  107. F F
  108. p p
  109. p p
  110. F F
  111. F F
  112. U U
  113. n \mathbb{C}^{n}
  114. n \mathbb{C}^{n}
  115. p p
  116. F F
  117. p p

Inverse_functions_and_differentiation.html

  1. y = f ( x ) y=f(x)
  2. f f
  3. f f
  4. f - 1 f^{-1}
  5. d x d y d y d x = 1. \frac{dx}{dy}\,\cdot\,\frac{dy}{dx}=1.
  6. d x d y d y d x = d x d x \frac{dx}{dy}\,\cdot\,\frac{dy}{dx}=\frac{dx}{dx}
  7. x x
  8. x x
  9. y y
  10. x x
  11. [ f - 1 ] ( a ) = 1 f ( f - 1 ( a ) ) \left[f^{-1}\right]^{\prime}(a)=\frac{1}{f^{\prime}\left(f^{-1}(a)\right)}
  12. f f
  13. x x
  14. x x
  15. y = x 2 \,y=x^{2}
  16. x x
  17. x = y x=\sqrt{y}
  18. d y d x = 2 x ; d x d y = 1 2 y = 1 2 x \frac{dy}{dx}=2x\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\mbox{ }~{};\mbox{ }~{}\mbox{% }~{}\mbox{ }~{}\mbox{ }~{}\frac{dx}{dy}=\frac{1}{2\sqrt{y}}=\frac{1}{2x}
  19. d y d x d x d y = 2 x 1 2 x = 1. \frac{dy}{dx}\,\cdot\,\frac{dx}{dy}=2x\cdot\frac{1}{2x}=1.
  20. y = e x \,y=e^{x}
  21. x x
  22. x = ln y \,x=\ln{y}
  23. y y
  24. d y d x = e x ; d x d y = 1 y \frac{dy}{dx}=e^{x}\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\mbox{ }~{};\mbox{ }~{}% \mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\frac{dx}{dy}=\frac{1}{y}
  25. d y d x d x d y = e x 1 y = e x e x = 1 \frac{dy}{dx}\,\cdot\,\frac{dx}{dy}=e^{x}\cdot\frac{1}{y}=\frac{e^{x}}{e^{x}}=1
  26. f - 1 ( x ) = 1 f ( f - 1 ( x ) ) d x + c . {f^{-1}}(x)=\int\frac{1}{f^{\prime}({f^{-1}}(x))}\,{dx}+c.
  27. f ( x ) f^{\prime}(x)
  28. d 2 y d x 2 d x d y + d 2 x d y 2 ( d y d x ) 2 = 0 \frac{d^{2}y}{dx^{2}}\,\cdot\,\frac{dx}{dy}+\frac{d^{2}x}{dy^{2}}\,\cdot\,% \left(\frac{dy}{dx}\right)^{2}=0
  29. d 2 y d x 2 = - d 2 x d y 2 ( d y d x ) 3 . \frac{d^{2}y}{dx^{2}}=-\frac{d^{2}x}{dy^{2}}\,\cdot\,\left(\frac{dy}{dx}\right% )^{3}.
  30. d 3 y d x 3 = - d 3 x d y 3 ( d y d x ) 4 - 3 d 2 x d y 2 d 2 y d x 2 ( d y d x ) 2 \frac{d^{3}y}{dx^{3}}=-\frac{d^{3}x}{dy^{3}}\,\cdot\,\left(\frac{dy}{dx}\right% )^{4}-3\frac{d^{2}x}{dy^{2}}\,\cdot\,\frac{d^{2}y}{dx^{2}}\,\cdot\,\left(\frac% {dy}{dx}\right)^{2}
  31. d 3 y d x 3 = - d 3 x d y 3 ( d y d x ) 4 + 3 ( d 2 x d y 2 ) 2 ( d y d x ) 5 \frac{d^{3}y}{dx^{3}}=-\frac{d^{3}x}{dy^{3}}\,\cdot\,\left(\frac{dy}{dx}\right% )^{4}+3\left(\frac{d^{2}x}{dy^{2}}\right)^{2}\,\cdot\,\left(\frac{dy}{dx}% \right)^{5}
  32. g ′′ ( x ) = - f ′′ ( g ( x ) ) [ f ( g ( x ) ) ] 3 g^{\prime\prime}(x)=\frac{-f^{\prime\prime}(g(x))}{[f^{\prime}(g(x))]^{3}}
  33. y = e x \,y=e^{x}
  34. x = ln y \,x=\ln y
  35. d y d x = d 2 y d x 2 = e x = y ; ( d y d x ) 3 = y 3 ; \frac{dy}{dx}=\frac{d^{2}y}{dx^{2}}=e^{x}=y\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}% \mbox{ }~{};\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\left(\frac{dy}{dx}% \right)^{3}=y^{3};
  36. d 2 x d y 2 y 3 + y = 0 ; d 2 x d y 2 = - 1 y 2 \frac{d^{2}x}{dy^{2}}\,\cdot\,y^{3}+y=0\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\mbox{% }~{};\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\mbox{ }~{}\frac{d^{2}x}{dy^{2}}=-\frac% {1}{y^{2}}

Inverse_gambler's_fallacy.html

  1. P ( M | U ) = P ( M ) P ( U | M ) P ( U ) P(M|U)=P(M)\frac{P(U|M)}{P(U)}

Inverse_Laplace_transform.html

  1. { f } ( s ) = { f ( t ) } ( s ) = F ( s ) , \mathcal{L}\{f\}(s)=\mathcal{L}\{f(t)\}(s)=F(s),
  2. \mathcal{L}
  3. f ( t ) = - 1 { F } ( t ) = - 1 { F ( s ) } ( t ) = 1 2 π i lim T γ - i T γ + i T e s t F ( s ) d s , f(t)=\mathcal{L}^{-1}\{F\}(t)=\mathcal{L}^{-1}\{F(s)\}(t)=\frac{1}{2\pi i}\lim% _{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,

Inverse_problem.html

  1. m m
  2. d = G ( m ) \ d=G(m)
  3. G G
  4. d d
  5. G G
  6. d d
  7. m m
  8. d = G m \ d=Gm
  9. G G
  10. d = a = K M r 2 d=a=\frac{KM}{r^{2}}
  11. a a
  12. K K
  13. M M
  14. r r
  15. d = G m , d=Gm,\,
  16. d = [ d 1 d 2 d 3 d 4 d 5 ] , d=\begin{bmatrix}d_{1}\\ d_{2}\\ d_{3}\\ d_{4}\\ d_{5}\end{bmatrix},
  17. m = [ M 1 M 2 M 3 M 4 M 5 ] , m=\begin{bmatrix}M_{1}\\ M_{2}\\ M_{3}\\ M_{4}\\ M_{5}\end{bmatrix},
  18. G = [ K r 11 2 K r 12 2 K r 13 2 K r 14 2 K r 15 2 K r 21 2 K r 22 2 K r 23 2 K r 24 2 K r 25 2 K r 31 2 K r 32 2 K r 33 2 K r 34 2 K r 35 2 K r 41 2 K r 42 2 K r 43 2 K r 44 2 K r 45 2 K r 51 2 K r 52 2 K r 53 2 K r 54 2 K r 55 2 ] G=\begin{bmatrix}\frac{K}{r_{11}^{2}}&\frac{K}{r_{12}^{2}}&\frac{K}{r_{13}^{2}% }&\frac{K}{r_{14}^{2}}&\frac{K}{r_{15}^{2}}\\ \frac{K}{r_{21}^{2}}&\frac{K}{r_{22}^{2}}&\frac{K}{r_{23}^{2}}&\frac{K}{r_{24}% ^{2}}&\frac{K}{r_{25}^{2}}\\ \frac{K}{r_{31}^{2}}&\frac{K}{r_{32}^{2}}&\frac{K}{r_{33}^{2}}&\frac{K}{r_{34}% ^{2}}&\frac{K}{r_{35}^{2}}\\ \frac{K}{r_{41}^{2}}&\frac{K}{r_{42}^{2}}&\frac{K}{r_{43}^{2}}&\frac{K}{r_{44}% ^{2}}&\frac{K}{r_{45}^{2}}\\ \frac{K}{r_{51}^{2}}&\frac{K}{r_{52}^{2}}&\frac{K}{r_{53}^{2}}&\frac{K}{r_{54}% ^{2}}&\frac{K}{r_{55}^{2}}\end{bmatrix}
  19. G G
  20. m m
  21. G G
  22. m = G - 1 d m=G^{-1}d\,
  23. G G
  24. G G
  25. G G
  26. ϕ \phi
  27. ϕ = || d - G m || 2 2 \phi=||d-Gm||_{2}^{2}\,
  28. m ϕ = G T G m - G T d = 0 \nabla_{m}\phi=G^{\mathrm{T}}Gm-G^{\mathrm{T}}d=0\,
  29. G T G m = G T d G^{\mathrm{T}}Gm=G^{\mathrm{T}}d\,
  30. m = ( G T G ) - 1 G T d m=(G^{\mathrm{T}}G)^{-1}G^{T}d\,
  31. β ^ = ( X T X ) - 1 X T y \hat{\beta}=(X^{\mathrm{T}}X)^{-1}X^{\mathrm{T}}y
  32. d ( x ) = a b g ( x , y ) m ( y ) d y d(x)=\int_{a}^{b}g(x,y)\,m(y)\,dy
  33. g g
  34. d d
  35. m m
  36. m m
  37. d d
  38. d = G ( m ) . \ d=G(m).
  39. G G
  40. m m

Inversive_geometry.html

  1. O P × O P = r 2 . OP\times OP^{\prime}=r^{2}.
  2. O A B = O B A and O B A = O A B . \angle OAB=\angle OB^{\prime}A^{\prime}\ \,\text{ and }\ \angle OBA=\angle OA^% {\prime}B^{\prime}.
  3. O P × O P = R 2 \scriptstyle OP\times OP^{\prime}=R^{2}
  4. x R 2 x | x | 2 = y T 2 y | y | 2 = ( T R ) 2 x . x\mapsto R^{2}\frac{x}{|x|^{2}}=y\mapsto T^{2}\frac{y}{|y|^{2}}=\left(\frac{T}% {R}\right)^{2}\ x.
  5. z = x + i y z=x+iy\,
  6. z ¯ = x - i y \bar{z}=x-iy
  7. 1 z = z ¯ | z | 2 \scriptstyle\frac{1}{z}=\frac{\bar{z}}{|z|^{2}}
  8. z w z\mapsto w
  9. w = 1 z ¯ = ( 1 z ) ¯ w=\frac{1}{\bar{z}}=\overline{\left(\frac{1}{z}\right)}
  10. x i r 2 x i j x j 2 x_{i}\mapsto\frac{r^{2}x_{i}}{\sum_{j}x_{j}^{2}}
  11. J J T = k I \scriptstyle J\cdot J^{T}=kI
  12. det ( J ) = - k . \scriptstyle\det(J)=-\sqrt{k}.
  13. x 1 2 + + x n 2 + 2 a 1 x 1 + + 2 a n x n + c = 0 x_{1}^{2}+\cdots+x_{n}^{2}+2a_{1}x_{1}+\cdots+2a_{n}x_{n}+c=0
  14. x 1 2 + + x n 2 + 2 a 1 c x 1 + + 2 a n c x n + 1 c = 0. x_{1}^{2}+\cdots+x_{n}^{2}+2\frac{a_{1}}{c}x_{1}+\cdots+2\frac{a_{n}}{c}x_{n}+% \frac{1}{c}=0.
  15. x 1 2 + + x n 2 + 2 a 1 x 1 + + 2 a n x n + 1 = 0 , x_{1}^{2}+\cdots+x_{n}^{2}+2a_{1}x_{1}+\cdots+2a_{n}x_{n}+1=0,

Inverted_pendulum.html

  1. θ ¨ - g sin θ = 0 \ddot{\theta}-{g\over\ell}\sin\theta=0
  2. θ ¨ \ddot{\theta}
  3. g g
  4. \ell
  5. θ \theta
  6. θ ¨ = g sin θ \ddot{\theta}={g\over\ell}\sin\theta
  7. m m
  8. \ell
  9. s y m b o l τ net = I θ ¨ symbol{\tau}_{\mathrm{net}}=I\ddot{\theta}
  10. s y m b o l τ net = m g sin θ symbol{\tau}_{\mathrm{net}}=mg\ell\sin\theta\,\!
  11. θ \theta
  12. I θ ¨ = m g sin θ I\ddot{\theta}=mg\ell\sin\theta\,\!
  13. I = m R 2 I=mR^{2}
  14. \ell
  15. I = m 2 I=m\ell^{2}
  16. m 2 θ ¨ = m g sin θ m\ell^{2}\ddot{\theta}=mg\ell\sin\theta\,\!
  17. 2 \ell^{2}
  18. θ ¨ = g sin θ \ddot{\theta}={g\over\ell}\sin\theta
  19. m m
  20. \ell
  21. θ \theta
  22. x x
  23. = θ + k x =\theta+kx
  24. k k
  25. ω p = g / \omega_{p}=\sqrt{g/\ell}
  26. ω p \omega_{p}
  27. θ ( t ) \theta(t)
  28. l l
  29. x ( t ) x(t)
  30. L = T - V L=T-V
  31. L = 1 2 M v 1 2 + 1 2 m v 2 2 - m g cos θ L=\frac{1}{2}Mv_{1}^{2}+\frac{1}{2}mv_{2}^{2}-mg\ell\cos\theta
  32. v 1 v_{1}
  33. v 2 v_{2}
  34. m m
  35. v 1 v_{1}
  36. v 2 v_{2}
  37. θ \theta
  38. v 1 2 = x ˙ 2 v_{1}^{2}=\dot{x}^{2}
  39. v 2 2 = ( d d t ( x - sin θ ) ) 2 + ( d d t ( cos θ ) ) 2 v_{2}^{2}=\left({\frac{d}{dt}}{\left(x-\ell\sin\theta\right)}\right)^{2}+\left% ({\frac{d}{dt}}{\left(\ell\cos\theta\right)}\right)^{2}
  40. v 2 v_{2}
  41. v 2 2 = x ˙ 2 - 2 x ˙ θ ˙ cos θ + 2 θ ˙ 2 v_{2}^{2}=\dot{x}^{2}-2\ell\dot{x}\dot{\theta}\cos\theta+\ell^{2}\dot{\theta}^% {2}
  42. L = 1 2 ( M + m ) x ˙ 2 - m x ˙ θ ˙ cos θ + 1 2 m 2 θ ˙ 2 - m g cos θ L=\frac{1}{2}\left(M+m\right)\dot{x}^{2}-m\ell\dot{x}\dot{\theta}\cos\theta+% \frac{1}{2}m\ell^{2}\dot{\theta}^{2}-mg\ell\cos\theta
  43. d d t L x ˙ - L x = F \frac{\mathrm{d}}{\mathrm{d}t}{\partial{L}\over\partial{\dot{x}}}-{\partial{L}% \over\partial x}=F
  44. d d t L θ ˙ - L θ = 0 \frac{\mathrm{d}}{\mathrm{d}t}{\partial{L}\over\partial{\dot{\theta}}}-{% \partial{L}\over\partial\theta}=0
  45. L L
  46. ( M + m ) x ¨ - m θ ¨ cos θ + m θ ˙ 2 sin θ = F \left(M+m\right)\ddot{x}-m\ell\ddot{\theta}\cos\theta+m\ell\dot{\theta}^{2}% \sin\theta=F
  47. θ ¨ - g sin θ = x ¨ cos θ \ell\ddot{\theta}-g\sin\theta=\ddot{x}\cos\theta
  48. θ 0 \theta\approx 0
  49. ( - sin θ , y + cos θ ) \left(-\ell\sin\theta,y+\ell\cos\theta\right)
  50. v 2 = y ˙ 2 - 2 y ˙ θ ˙ sin θ + 2 θ ˙ 2 . v^{2}=\dot{y}^{2}-2\ell\dot{y}\dot{\theta}\sin\theta+\ell^{2}\dot{\theta}^{2}.
  51. L = 1 2 m ( y ˙ 2 - 2 y ˙ θ ˙ sin θ + 2 θ ˙ 2 ) - m g ( y + cos θ ) L=\frac{1}{2}m\left(\dot{y}^{2}-2\ell\dot{y}\dot{\theta}\sin\theta+\ell^{2}% \dot{\theta}^{2}\right)-mg\left(y+\ell\cos\theta\right)
  52. d d t L θ ˙ - L θ = 0 {\mathrm{d}\over\mathrm{d}t}{\partial{L}\over\partial{\dot{\theta}}}-{\partial% {L}\over\partial\theta}=0
  53. θ ¨ - y ¨ sin θ = g sin θ . \ell\ddot{\theta}-\ddot{y}\sin\theta=g\sin\theta.
  54. y = A sin ω t y=A\sin\omega t
  55. θ ¨ - g sin θ = - A ω 2 sin ω t sin θ . \ddot{\theta}-{g\over\ell}\sin\theta=-{A\over\ell}\omega^{2}\sin\omega t\sin\theta.
  56. y y
  57. θ \theta
  58. y y
  59. θ = 0 \theta=0

Invertible_matrix.html

  1. 𝐀𝐁 = 𝐁𝐀 = 𝐈 n \mathbf{AB}=\mathbf{BA}=\mathbf{I}_{n}
  2. A A
  3. A A
  4. A A
  5. n × n n\times n
  6. A - 1 = 1 det ( A ) adj ( A ) A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)
  7. 𝐀𝐁 = 𝐈 \mathbf{AB}=\mathbf{I}
  8. 𝐁𝐀 = 𝐈 \mathbf{BA}=\mathbf{I}
  9. X k + 1 = 2 X k - X k A X k . X_{k+1}=2X_{k}-X_{k}AX_{k}.
  10. x + i y x+iy
  11. x - i y x-iy
  12. x + i y x+iy
  13. x 2 + y 2 \sqrt{x^{2}+y^{2}}
  14. max I J | A ( I , J ) | \max_{I}\sum_{J}\left|A(I,J)\right|
  15. X k + 1 = 2 X k - X k A k + 1 X k , X_{k+1}=2X_{k}-X_{k}A_{k+1}X_{k},
  16. A 0 = S , A_{0}=S,
  17. X 0 = S - 1 , X_{0}=S^{-1},
  18. A N = A A_{N}=A
  19. ( a b - b a ) \begin{pmatrix}a&b\\ -b&a\\ \end{pmatrix}
  20. 𝐀 - 1 = 1 det ( 𝐀 ) s = 0 n - 1 𝐀 s k 1 , k 2 , , k n - 1 l = 1 n - 1 ( - 1 ) k l + 1 l k l k l ! tr ( 𝐀 l ) k l , \mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\sum_{s=0}^{n-1}\mathbf{A}^{s}\sum_{% k_{1},k_{2},\ldots,k_{n-1}}\prod_{l=1}^{n-1}\frac{(-1)^{k_{l}+1}}{l^{k_{l}}k_{% l}!}\mathrm{tr}(\mathbf{A}^{l})^{k_{l}},
  21. s + l = 1 n - 1 l k l = n - 1. s+\sum_{l=1}^{n-1}lk_{l}=n-1.
  22. 𝐀 - 1 = 𝐐 𝚲 - 1 𝐐 - 1 \mathbf{A}^{-1}=\mathbf{Q}\mathbf{\Lambda}^{-1}\mathbf{Q}^{-1}
  23. q i q_{i}
  24. Λ i i = λ i \Lambda_{ii}=\lambda_{i}
  25. [ Λ - 1 ] i i = 1 λ i \left[\Lambda^{-1}\right]_{ii}=\frac{1}{\lambda_{i}}
  26. 𝐀 - 1 = ( 𝐋 * ) - 1 𝐋 - 1 , \mathbf{A}^{-1}=(\mathbf{L}^{*})^{-1}\mathbf{L}^{-1},
  27. 𝐀 - 1 = 1 | 𝐀 | 𝐂 T = 1 | 𝐀 | ( 𝐂 11 𝐂 21 𝐂 n 1 𝐂 12 𝐂 22 𝐂 n 2 𝐂 1 n 𝐂 2 n 𝐂 n n ) \mathbf{A}^{-1}={1\over\begin{vmatrix}\mathbf{A}\end{vmatrix}}\mathbf{C}^{% \mathrm{T}}={1\over\begin{vmatrix}\mathbf{A}\end{vmatrix}}\begin{pmatrix}% \mathbf{C}_{11}&\mathbf{C}_{21}&\cdots&\mathbf{C}_{n1}\\ \mathbf{C}_{12}&\mathbf{C}_{22}&\cdots&\mathbf{C}_{n2}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{C}_{1n}&\mathbf{C}_{2n}&\cdots&\mathbf{C}_{nn}\\ \end{pmatrix}
  28. ( 𝐀 - 1 ) i j = 1 | 𝐀 | ( 𝐂 T ) i j = 1 | 𝐀 | ( 𝐂 j i ) \left(\mathbf{A}^{-1}\right)_{ij}={1\over\begin{vmatrix}\mathbf{A}\end{vmatrix% }}\left(\mathbf{C}^{\mathrm{T}}\right)_{ij}={1\over\begin{vmatrix}\mathbf{A}% \end{vmatrix}}\left(\mathbf{C}_{ji}\right)
  29. 𝐀 - 1 = [ a b c d ] - 1 = 1 det ( 𝐀 ) [ d - b - c a ] = 1 a d - b c [ d - b - c a ] . \mathbf{A}^{-1}=\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}^{-1}=\frac{1}{\det(\mathbf{A})}\begin{bmatrix}\,\,\,d&\!\!-b\\ -c&\,a\\ \end{bmatrix}=\frac{1}{ad-bc}\begin{bmatrix}\,\,\,d&\!\!-b\\ -c&\,a\\ \end{bmatrix}.
  30. 𝐀 - 1 = 1 det ( 𝐀 ) [ ( tr 𝐀 ) 𝐈 - 𝐀 ] . \mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\left[\left(\mathrm{tr}\mathbf{A}% \right)\mathbf{I}-\mathbf{A}\right].
  31. 𝐀 - 1 = [ a b c d e f g h i ] - 1 = 1 det ( 𝐀 ) [ A B C D E F G H I ] T = 1 det ( 𝐀 ) [ A D G B E H C F I ] \mathbf{A}^{-1}=\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\\ \end{bmatrix}^{-1}=\frac{1}{\det(\mathbf{A})}\begin{bmatrix}\,A&\,B&\,C\\ \,D&\,E&\,F\\ \,G&\,H&\,I\\ \end{bmatrix}^{T}=\frac{1}{\det(\mathbf{A})}\begin{bmatrix}\,A&\,D&\,G\\ \,B&\,E&\,H\\ \,C&\,F&\,I\\ \end{bmatrix}
  32. A = ( e i - f h ) D = - ( b i - c h ) G = ( b f - c e ) B = - ( d i - f g ) E = ( a i - c g ) H = - ( a f - c d ) C = ( d h - e g ) F = - ( a h - b g ) I = ( a e - b d ) \begin{matrix}A=(ei-fh)&D=-(bi-ch)&G=(bf-ce)\\ B=-(di-fg)&E=(ai-cg)&H=-(af-cd)\\ C=(dh-eg)&F=-(ah-bg)&I=(ae-bd)\\ \end{matrix}
  33. det ( 𝐀 ) = a A + b B + c C . \det(\mathbf{A})=aA+bB+cC.
  34. 𝐀 - 1 = 1 det ( 𝐀 ) [ 1 2 ( ( tr 𝐀 ) 2 - tr 𝐀 2 ) 𝐈 - 𝐀 tr 𝐀 + 𝐀 2 ] . \mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\left[\frac{1}{2}\left((\mathrm{tr}% \mathbf{A})^{2}-\mathrm{tr}\mathbf{A}^{2}\right)\mathbf{I}-\mathbf{A}\mathrm{% tr}\mathbf{A}+\mathbf{A}^{2}\right].
  35. 𝐀 = [ 𝐱 𝟎 , 𝐱 𝟏 , 𝐱 𝟐 ] \mathbf{A}=\left[\mathbf{x_{0}},\;\mathbf{x_{1}},\;\mathbf{x_{2}}\right]
  36. 𝐱 𝟎 \mathbf{x_{0}}
  37. 𝐱 𝟏 \mathbf{x_{1}}
  38. 𝐱 𝟐 \mathbf{x_{2}}
  39. 𝐀 - 1 = 1 det ( 𝐀 ) [ ( 𝐱 𝟏 × 𝐱 𝟐 ) T ( 𝐱 𝟐 × 𝐱 𝟎 ) T ( 𝐱 𝟎 × 𝐱 𝟏 ) T ] . \mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\begin{bmatrix}{(\mathbf{x_{1}}% \times\mathbf{x_{2}})}^{T}\\ {(\mathbf{x_{2}}\times\mathbf{x_{0}})}^{T}\\ {(\mathbf{x_{0}}\times\mathbf{x_{1}})}^{T}\\ \end{bmatrix}.
  40. det ( A ) \det(A)
  41. 𝐱 𝟎 \mathbf{x_{0}}
  42. 𝐱 𝟏 \mathbf{x_{1}}
  43. 𝐱 𝟐 \mathbf{x_{2}}
  44. det ( 𝐀 ) = 𝐱 𝟎 ( 𝐱 𝟏 × 𝐱 𝟐 ) . \det(\mathbf{A})=\mathbf{x_{0}}\cdot(\mathbf{x_{1}}\times\mathbf{x_{2}}).
  45. 𝐀 - 1 \mathbf{A}^{-1}
  46. 𝐀 \mathbf{A}
  47. 𝐈 = 𝐀 - 1 𝐀 \mathbf{I}=\mathbf{A}^{-1}\mathbf{A}
  48. det ( 𝐀 ) = 𝐱 𝟎 ( 𝐱 𝟏 × 𝐱 𝟐 ) \det(\mathbf{A})=\mathbf{x_{0}}\cdot(\mathbf{x_{1}}\times\mathbf{x_{2}})
  49. 𝐈 = 𝐀 - 1 𝐀 \mathbf{I}=\mathbf{A}^{-1}\mathbf{A}
  50. 1 = 1 𝐱 𝟎 ( 𝐱 𝟏 × 𝐱 𝟐 ) 𝐱 𝟎 ( 𝐱 𝟏 × 𝐱 𝟐 ) . 1=\frac{1}{\mathbf{x_{0}}\cdot(\mathbf{x_{1}}\times\mathbf{x_{2}})}\mathbf{x_{% 0}}\cdot(\mathbf{x_{1}}\times\mathbf{x_{2}}).
  51. 𝐀 - 1 = 1 det ( 𝐀 ) [ 1 6 ( ( tr 𝐀 ) 3 - 3 t r 𝐀 tr 𝐀 2 + 2 t r 𝐀 3 ) 𝐈 - 1 2 𝐀 ( ( tr 𝐀 ) 2 - tr 𝐀 2 ) + 𝐀 2 tr 𝐀 - 𝐀 3 ] . \mathbf{A}^{-1}=\frac{1}{\det(\mathbf{A})}\left[\frac{1}{6}\left((\mathrm{tr}% \mathbf{A})^{3}-3\mathrm{tr}\mathbf{A}\mathrm{tr}\mathbf{A}^{2}+2\mathrm{tr}% \mathbf{A}^{3}\right)\mathbf{I}-\frac{1}{2}\mathbf{A}\left((\mathrm{tr}\mathbf% {A})^{2}-\mathrm{tr}\mathbf{A}^{2}\right)+\mathbf{A}^{2}\mathrm{tr}\mathbf{A}-% \mathbf{A}^{3}\right].
  52. [ 𝐀 𝐁 𝐂 𝐃 ] - 1 = [ 𝐀 - 1 + 𝐀 - 1 𝐁 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 𝐂𝐀 - 1 - 𝐀 - 1 𝐁 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 - ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 𝐂𝐀 - 1 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 ] , \begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}^{-1}=\begin{bmatrix}\mathbf{A}^{-1}+\mathbf% {A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}% &-\mathbf{A}^{-1}\mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\\ -(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}&(\mathbf{D}-% \mathbf{CA}^{-1}\mathbf{B})^{-1}\end{bmatrix},
  53. ( 1 ) (1)\,
  54. [ 𝐀 𝐁 𝐂 𝐃 ] - 1 = [ ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 - ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐁𝐃 - 1 - 𝐃 - 1 𝐂 ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐃 - 1 + 𝐃 - 1 𝐂 ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐁𝐃 - 1 ] . \begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}^{-1}=\begin{bmatrix}(\mathbf{A}-\mathbf{BD}% ^{-1}\mathbf{C})^{-1}&-(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}% ^{-1}\\ -\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}&\quad% \mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C% })^{-1}\mathbf{BD}^{-1}\end{bmatrix}.
  55. ( 2 ) (2)\,
  56. ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 = 𝐀 - 1 + 𝐀 - 1 𝐁 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 𝐂𝐀 - 1 (\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}=\mathbf{A}^{-1}+\mathbf{A}^{-1}% \mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}\,
  57. ( 3 ) (3)\,
  58. ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐁𝐃 - 1 = 𝐀 - 1 𝐁 ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 (\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}\mathbf{BD}^{-1}=\mathbf{A}^{-1}% \mathbf{B}(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\,
  59. 𝐃 - 1 𝐂 ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 = ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 𝐂𝐀 - 1 \mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C})^{-1}=(\mathbf% {D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\mathbf{CA}^{-1}\,
  60. 𝐃 - 1 + 𝐃 - 1 𝐂 ( 𝐀 - 𝐁𝐃 - 1 𝐂 ) - 1 𝐁𝐃 - 1 = ( 𝐃 - 𝐂𝐀 - 1 𝐁 ) - 1 \mathbf{D}^{-1}+\mathbf{D}^{-1}\mathbf{C}(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C% })^{-1}\mathbf{BD}^{-1}=(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B})^{-1}\,
  61. Ω ( n 2 l o g n ) Ω(n^{2}logn)
  62. lim n ( 𝐈 - 𝐀 ) n = 0 \lim_{n\to\infty}(\mathbf{I}-\mathbf{A})^{n}=0
  63. 𝐀 - 1 = n = 0 ( 𝐈 - 𝐀 ) n . \mathbf{A}^{-1}=\sum_{n=0}^{\infty}(\mathbf{I}-\mathbf{A})^{n}.
  64. 2 L 2^{L}
  65. A , A 2 , A 4 , , A 2 L A,A^{2},A^{4},...,A^{2^{L}}
  66. 2 L 2^{L}
  67. lim n ( 𝐈 - 𝐗 - 1 𝐀 ) n = 0 or lim n ( 𝐈 - 𝐀𝐗 - 1 ) n = 0 \lim_{n\to\infty}(\mathbf{I}-\mathbf{X}^{-1}\mathbf{A})^{n}=0\mathrm{~{}~{}or~% {}~{}}\lim_{n\to\infty}(\mathbf{I}-\mathbf{A}\mathbf{X}^{-1})^{n}=0
  68. 𝐀 - 1 = n = 0 ( 𝐗 - 1 ( 𝐗 - 𝐀 ) ) n 𝐗 - 1 . \mathbf{A}^{-1}=\sum_{n=0}^{\infty}\left(\mathbf{X}^{-1}(\mathbf{X}-\mathbf{A}% )\right)^{n}\mathbf{X}^{-1}~{}.
  69. 𝐀 - 1 = 𝐗 - 1 - 𝐗 - 1 ( 𝐀 - 𝐗 ) 𝐗 - 1 1 + tr ( 𝐗 - 1 ( 𝐀 - 𝐗 ) ) . \mathbf{A}^{-1}=\mathbf{X}^{-1}-\frac{\mathbf{X}^{-1}(\mathbf{A}-\mathbf{X})% \mathbf{X}^{-1}}{1+\operatorname{tr}(\mathbf{X}^{-1}(\mathbf{A}-\mathbf{X}))}~% {}.
  70. O ( n 4 log 2 n ) O(n^{4}\log^{2}n)
  71. O ( n 3 ) O(n^{3})
  72. O ( n 3 log 2 n ) O(n^{3}\log^{2}n)
  73. d 𝐀 - 1 d t = - 𝐀 - 1 d 𝐀 d t 𝐀 - 1 . \frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t}=-\mathbf{A}^{-1}\frac{\mathrm{d}% \mathbf{A}}{\mathrm{d}t}\mathbf{A}^{-1}.
  74. 𝐀 - 1 𝐀 = 𝐈 \mathbf{A}^{-1}\mathbf{A}=\mathbf{I}
  75. d 𝐀 - 1 𝐀 d t = d 𝐀 - 1 d t 𝐀 + 𝐀 - 1 d 𝐀 d t = d 𝐈 d t = 0. \frac{\mathrm{d}\mathbf{A}^{-1}\mathbf{A}}{\mathrm{d}t}=\frac{\mathrm{d}% \mathbf{A}^{-1}}{\mathrm{d}t}\mathbf{A}+\mathbf{A}^{-1}\frac{\mathrm{d}\mathbf% {A}}{\mathrm{d}t}=\frac{\mathrm{d}\mathbf{I}}{\mathrm{d}t}=\mathbf{0}.
  76. 𝐀 - 1 d 𝐀 d t \mathbf{A}^{-1}\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}
  77. 𝐀 - 1 \mathbf{A}^{-1}
  78. d 𝐀 - 1 d t = - 𝐀 - 1 d 𝐀 d t 𝐀 - 1 . \frac{\mathrm{d}\mathbf{A}^{-1}}{\mathrm{d}t}=-\mathbf{A}^{-1}\frac{\mathrm{d}% \mathbf{A}}{\mathrm{d}t}\mathbf{A}^{-1}.
  79. ϵ \epsilon
  80. ( 𝐀 + ϵ 𝐗 ) - 1 = 𝐀 - 1 - ϵ 𝐀 - 1 𝐗𝐀 - 1 + 𝒪 ( ϵ 2 ) . \left(\mathbf{A}+\epsilon\mathbf{X}\right)^{-1}=\mathbf{A}^{-1}-\epsilon% \mathbf{A}^{-1}\mathbf{X}\mathbf{A}^{-1}+\mathcal{O}(\epsilon^{2})\,.

Involution_(mathematics).html

  1. f f
  2. f ( f ( x ) ) = x f(f(x))=x
  3. x x
  4. f f
  5. g f g\circ f
  6. g f = f g g\circ f=f\circ g
  7. f ( x ) = - x f(x)=-x
  8. f ( x ) = 1 x f(x)=\frac{1}{x}
  9. f ( x ) = - 1 x f(x)=\frac{-1}{x}
  10. R + R^{+}
  11. f ( x ) = ln ( e x + 1 e x - 1 ) : x > 0 f(x)=\ln\left(\frac{e^{x}+1}{e^{x}-1}\right):x>0
  12. T 2 = I T^{2}=I
  13. x f ( x ) \begin{aligned}\displaystyle x&\displaystyle\mapsto f(x)\end{aligned}
  14. f ( f ( x ) ) = x f(f(x))=x
  15. f ( x 1 + x 2 ) = f ( x 1 ) + f ( x 2 ) f(x_{1}+x_{2})=f(x_{1})+f(x_{2})
  16. f ( λ x ) = λ f ( x ) f(\lambda x)=\lambda f(x)
  17. f ( x 1 x 2 ) = f ( x 1 ) f ( x 2 ) f(x_{1}x_{2})=f(x_{1})f(x_{2})
  18. f ( x 1 x 2 ) = f ( x 2 ) f ( x 1 ) f(x_{1}x_{2})=f(x_{2})f(x_{1})

Irreducible_polynomial.html

  1. ( x - 2 ) ( x + 2 ) (x-\sqrt{2})(x+\sqrt{2})
  2. p 1 ( x ) = x 2 + 4 x + 4 = ( x + 2 ) ( x + 2 ) p_{1}(x)=x^{2}+4x+4\,={(x+2)(x+2)}
  3. p 2 ( x ) = x 2 - 4 = ( x - 2 ) ( x + 2 ) p_{2}(x)=x^{2}-4\,={(x-2)(x+2)}
  4. p 3 ( x ) = 9 x 2 - 3 = 3 ( 3 x 2 - 1 ) = 3 ( x 3 - 1 ) ( x 3 + 1 ) p_{3}(x)=9x^{2}-3\,=3(3x^{2}-1)\,=3(x\sqrt{3}-1)(x\sqrt{3}+1)
  5. p 4 ( x ) = x 2 - 4 / 9 = ( x - 2 / 3 ) ( x + 2 / 3 ) p_{4}(x)=x^{2}-4/9\,=(x-2/3)(x+2/3)
  6. p 5 ( x ) = x 2 - 2 = ( x - 2 ) ( x + 2 ) p_{5}(x)=x^{2}-2\,=(x-\sqrt{2})(x+\sqrt{2})
  7. p 6 ( x ) = x 2 + 1 = ( x - i ) ( x + i ) p_{6}(x)=x^{2}+1\,={(x-i)(x+i)}
  8. \mathbb{Z}
  9. \mathbb{Q}
  10. \mathbb{R}
  11. p 6 ( x ) p_{6}(x)
  12. \mathbb{C}
  13. a ( x - z 1 ) ( x - z n ) a(x-z_{1})\cdots(x-z_{n})
  14. n n
  15. a a
  16. z 1 , , z n z_{1},\dots,z_{n}
  17. x n + y n - 1 , x^{n}+y^{n}-1,
  18. a x 2 + b x + c ax^{2}+bx+c
  19. b 2 - 4 a c . b^{2}-4ac.
  20. x 4 + 1 x^{4}+1
  21. ( x 2 + 2 x + 1 ) ( x 2 - 2 x + 1 ) , (x^{2}+\sqrt{2}x+1)(x^{2}-\sqrt{2}x+1),
  22. ( ± 2 ) 2 - 4 = - 2 < 0. (\pm\sqrt{2})^{2}-4=-2<0.
  23. F F
  24. F F
  25. F F
  26. F F
  27. F F
  28. F F
  29. F F
  30. F F
  31. F F
  32. F F
  33. F F
  34. \mathbb{Z}
  35. 𝔽 p \mathbb{F}_{p}
  36. p p
  37. p p
  38. \mathbb{Z}
  39. 𝔽 p \mathbb{F}_{p}
  40. p p
  41. \mathbb{Z}
  42. p 2 p^{2}
  43. x 4 + 1. x^{4}+1.
  44. P ( X ) K [ X ] P(X)\in K[X]
  45. L = K [ X ] / P ( X ) L=K[X]/P(X)
  46. K [ X ] K[X]
  47. P P
  48. L L
  49. P P
  50. K K
  51. x x
  52. X X
  53. L L
  54. x x
  55. P P
  56. = [ X ] / ( X 2 + 1 ) . \mathbb{C}=\mathbb{R}[X]/(X^{2}+1).
  57. P P
  58. Q Q
  59. K K
  60. Q Q
  61. P P
  62. K K
  63. P P
  64. P P