wpmath0000006_4

Fibonacci_prime.html

  1. F a F_{a}
  2. F b F_{b}
  3. F n F_{n}
  4. F n F_{n}
  5. F n F_{n}
  6. F p F_{p}
  7. L p L_{p}
  8. L n L_{n}
  9. L 2 n - 1 L_{2^{n-1}}

Fibred_category.html

  1. f * ( g * ( z ) ) ( g f ) * ( z ) . f^{*}(g^{*}(z))\neq(g\circ f)^{*}(z).
  2. Lim ( F / E ) = Cart E ( E , F ) . \underset{\longleftarrow}{\mathrm{Lim}}(F/E)=\mathrm{Cart}_{E}(E,F).
  3. ϵ : Lim ( F / E ) F e , s s ( e ) \epsilon\colon\underset{\longleftarrow}{\mathrm{Lim}}(F/E)\to F_{e},\qquad s% \mapsto s(e)
  4. c f , g : g * f * ( f g ) * . c_{f,g}\colon\quad g^{*}f^{*}\to(f\circ g)^{*}.
  5. c f , id T = c id S , f = id f * c_{f,\mathrm{id}_{T}}=c_{\mathrm{id}_{S},f}=\mathrm{id}_{f^{*}}
  6. h , g , f : V U T S h,g,f\colon\quad V\to U\to T\to S
  7. x F S x\in F_{S}
  8. c f , g h c g , h ( f * ( x ) ) = c f g , h ( x ) h * ( c f , g ( x ) ) . c_{f,g\circ h}\cdot c_{g,h}(f^{*}(x))=c_{f\circ g,h}(x)\cdot h^{*}(c_{f,g}(x)).

File:Cube_root_of_positive_X.gif.html

  1. y = x 3 y=\sqrt[3]{x}

File:Cube_root_of_positive_X.png.html

  1. y = x 3 y=\sqrt[3]{x}

File:Logistic_pdf_(single).png.html

  1. m = 0 m=0
  2. s = 1 s=1
  3. σ = π / 3 \sigma=\pi/\sqrt{3}

File:Null-balance_voltmeter.png.html

  1. V t = V k R e R w Vt=\frac{Vk}{Re}Rw

File:Pith_helmet.gif.html

  1. z ( r ) = 10 - 125 - r 2 , 0 < r < 5 z(r)=10-\sqrt{125-r^{2}},\;0<r<5
  2. z ( r ) = - 4 + 2 r - 1 , 5 < r < z(r)=-4+2\sqrt{r-1},\;5<r<\infty

File:Psi_(large).png.html

  1. ψ \psi
  2. Ψ \Psi

Filtered_algebra.html

  1. k k
  2. ( A , ) (A,\cdot)
  3. k k
  4. { 0 } F 0 F 1 F i A \{0\}\subset F_{0}\subset F_{1}\subset\cdots\subset F_{i}\subset\cdots\subset A
  5. A A
  6. A = i F i A=\cup_{i\in\mathbb{N}}F_{i}
  7. m , n , F m F n F n + m . \forall m,n\in\mathbb{N},\qquad F_{m}\cdot F_{n}\subset F_{n+m}.
  8. A A
  9. 𝒢 ( A ) \mathcal{G}(A)
  10. G 0 = F 0 , G_{0}=F_{0},
  11. n > 0 , G n = F n / F n - 1 , \forall n>0,\quad G_{n}=F_{n}/F_{n-1}\,,
  12. ( x + F n - 1 ) ( y + F m - 1 ) = x y + F n + m - 1 (x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}
  13. x F n x\in F_{n}
  14. y F m y\in F_{m}
  15. 𝒢 ( A ) × 𝒢 ( A ) 𝒢 ( A ) \mathcal{G}(A)\times\mathcal{G}(A)\to\mathcal{G}(A)
  16. ( F n / F n - 1 ) × ( F m / F m - 1 ) F n + m / F n + m - 1 , ( x + F n - 1 , y + F m - 1 ) x y + F n + m - 1 (F_{n}/F_{n-1})\times(F_{m}/F_{m-1})\to F_{n+m}/F_{n+m-1},\ \ \ \ \ \left(x+F_% {n-1},y+F_{m-1}\right)\mapsto x\cdot y+F_{n+m-1}
  17. n 0 n\geq 0
  18. m 0 m\geq 0
  19. 𝒢 ( A ) \mathcal{G}(A)
  20. { G n } n . \{G_{n}\}_{n\in\mathbb{N}}.
  21. A A
  22. 𝒢 ( A ) \mathcal{G}(A)
  23. A A
  24. F 0 F_{0}
  25. 𝒢 ( A ) \mathcal{G}(A)
  26. A A
  27. 𝒢 ( A ) \mathcal{G}(A)
  28. A A
  29. A = n A n A=\oplus_{n\in\mathbb{N}}A_{n}
  30. F n = i = 0 n A i F_{n}=\oplus_{i=0}^{n}A_{i}
  31. Cliff ( V , q ) \mathrm{Cliff}(V,q)
  32. V V
  33. q . q.
  34. V \bigwedge V
  35. V . V.
  36. 𝔤 \mathfrak{g}
  37. Sym ( 𝔤 ) \mathrm{Sym}(\mathfrak{g})
  38. M M
  39. T * M T^{*}M
  40. π : T * M M \pi\colon T^{*}M\rightarrow M

Final_stellation_of_the_icosahedron.html

  1. 3 2 ( 3 + 5 ) : 1 2 ( 25 + 11 5 ) : 1 2 ( 97 + 43 5 ) . \sqrt{\frac{3}{2}\left(3+\sqrt{5}\right)}\,:\,\sqrt{\frac{1}{2}\left(25+11% \sqrt{5}\right)}\,:\,\sqrt{\frac{1}{2}\left(97+43\sqrt{5}\right)}\,.
  2. S = 1 20 ( 13211 + 174306161 ) a 2 , S=\frac{1}{20}(13211+\sqrt{174306161})a^{2}\,,
  3. V = ( 210 + 90 5 ) a 3 . V=(210+90\sqrt{5})a^{3}\,.

Finger_binary.html

  1. 1 2 x \tfrac{1}{2^{x}}

Finite_character.html

  1. \mathcal{F}
  2. A A\in\mathcal{F}
  3. A A
  4. \mathcal{F}
  5. A A
  6. \mathcal{F}
  7. A A
  8. \mathcal{F}
  9. \mathcal{F}
  10. A A\in\mathcal{F}
  11. A A
  12. \mathcal{F}
  13. \mathcal{F}
  14. \mathcal{F}
  15. \mathcal{F}
  16. \mathcal{F}

Finite_morphism.html

  1. f : X Y f:X\rightarrow Y
  2. Y Y
  3. V i = Spec B i V_{i}=\mbox{Spec}~{}\;B_{i}
  4. i i
  5. f - 1 ( V i ) = U i f^{-1}(V_{i})=U_{i}
  6. Spec A i \mbox{Spec}~{}\;A_{i}
  7. U i U_{i}
  8. B i A i , B_{i}\rightarrow A_{i},
  9. A i A_{i}
  10. B i B_{i}
  11. g : Z Y g:Z\rightarrow Y
  12. X × Y Z Z X\times_{Y}Z\rightarrow Z
  13. A B C A\otimes_{B}C
  14. C B C\rightarrow B
  15. a i 1 a_{i}\otimes 1
  16. a i a_{i}
  17. A A / I A\rightarrow A/I
  18. y 3 = x 4 - z y^{3}=x^{4}-z
  19. Spec [ x , y , z ] / y 3 - x 4 + z Spec \mbox{Spec}~{}\;\mathbb{Z}[x,y,z]/\langle y^{3}-x^{4}+z\rangle\rightarrow\mbox% {Spec}~{}\;\mathbb{Z}
  20. [ x , y , z ] / y 3 - x 4 + z \mathbb{Z}\rightarrow\mathbb{Z}[x,y,z]/\langle y^{3}-x^{4}+z\rangle
  21. { V i = Spec B i } \{V_{i}=\mbox{Spec}~{}\;B_{i}\}
  22. Y Y
  23. i i
  24. { U i j = Spec A i j } \{U_{ij}=\,\text{Spec}\;A_{ij}\}
  25. f - 1 ( V i ) f^{-1}(V_{i})
  26. U i j U_{ij}
  27. B i A i j B_{i}\rightarrow A_{ij}
  28. A i j A_{ij}
  29. B i B_{i}
  30. f - 1 ( V i ) = j U i j f^{-1}(V_{i})=\bigcup_{j}U_{ij}
  31. k k
  32. 𝔸 n ( k ) \mathbb{A}^{n}(k)
  33. Spec k \,\text{Spec}\;k
  34. k k [ X 1 , , X n ] . k\to k[X_{1},\ldots,X_{n}].
  35. n 1 n\geq 1
  36. Spec k [ X , Y ] / Y 2 - X 3 - X {\mbox{Spec}~{}}\;k[X,Y]/\langle Y^{2}-X^{3}-X\rangle
  37. 𝔸 1 \mathbb{A}^{1}
  38. k [ X ] k [ X , Y ] / Y 2 - X 3 - X . k[X]\to k[X,Y]/\langle Y^{2}-X^{3}-X\rangle.

Finite_potential_well.html

  1. - 2 2 m d 2 ψ d x 2 + V ( x ) ψ = E ψ ( 1 ) -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+V(x)\psi=E\psi\quad(1)
  2. = h 2 π \hbar=\frac{h}{2\pi}
  3. h h\,
  4. m m\,
  5. ψ \psi\,
  6. V ( x ) V\left(x\right)\,
  7. E E\,
  8. - 2 2 m d 2 ψ d x 2 + V ( x ) ψ = E ψ ( 1 ) -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+V(x)\psi=E\psi\quad(1)
  9. = h 2 π \hbar=\frac{h}{2\pi}
  10. h h\,
  11. m m\,
  12. ψ \psi\,
  13. V ( x ) V\left(x\right)\,
  14. E E\,
  15. V o V_{o}
  16. ψ = { ψ 1 , if x < - L / 2 (the region outside the box) ψ 2 , if - L / 2 < x < L / 2 (the region inside the box) ψ 3 if x > L / 2 (the region outside the box) \psi=\begin{cases}\psi_{1},&\mbox{if }~{}x<-L/2\mbox{ (the region outside the % box)}\\ \psi_{2},&\mbox{if }~{}-L/2<x<L/2\mbox{ (the region inside the box)}\\ \psi_{3}&\mbox{if }~{}x>L/2\mbox{ (the region outside the box)}\end{cases}
  17. - 2 2 m d 2 ψ 2 d x 2 = E ψ 2 . -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi_{2}}{dx^{2}}=E\psi_{2}.
  18. k = 2 m E , k=\frac{\sqrt{2mE}}{\hbar},
  19. d 2 ψ 2 d x 2 = - k 2 ψ 2 . \frac{d^{2}\psi_{2}}{dx^{2}}=-k^{2}\psi_{2}.
  20. ψ 2 = A sin ( k x ) + B cos ( k x ) . \psi_{2}=A\sin(kx)+B\cos(kx)\quad.
  21. E = k 2 2 2 m . E=\frac{k^{2}\hbar^{2}}{2m}.
  22. V o V_{o}
  23. - 2 2 m d 2 ψ 1 d x 2 = ( E - V o ) ψ 1 -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi_{1}}{dx^{2}}=(E-V_{o})\psi_{1}
  24. V o V_{o}
  25. V o V_{o}
  26. V o V_{o}
  27. k = 2 m ( E - V o ) k^{\prime}=\frac{\sqrt{2m(E-V_{o})}}{\hbar}
  28. d 2 ψ 1 d x 2 = - k 2 ψ 1 \frac{d^{2}\psi_{1}}{dx^{2}}=-k^{\prime 2}\psi_{1}
  29. ψ 1 = C sin ( k x ) + D cos ( k x ) \psi_{1}=C\sin(k^{\prime}x)+D\cos(k^{\prime}x)\quad
  30. V o V_{o}
  31. α = 2 m ( V o - E ) \alpha=\frac{\sqrt{2m(V_{o}-E)}}{\hbar}
  32. d 2 ψ 1 d x 2 = α 2 ψ 1 \frac{d^{2}\psi_{1}}{dx^{2}}=\alpha^{2}\psi_{1}
  33. ψ 1 = F e - α x + G e α x \psi_{1}=Fe^{-\alpha x}+Ge^{\alpha x}\,\!
  34. ψ 3 = H e - α x + I e α x \psi_{3}=He^{-\alpha x}+Ie^{\alpha x}\,\!
  35. ψ = { ψ 1 , if x < - L / 2 (the region outside the box) ψ 2 , if - L / 2 < x < L / 2 (the region inside the box) ψ 3 if x > L / 2 (the region outside the box) \psi=\begin{cases}\psi_{1},&\mbox{if }~{}x<-L/2\mbox{ (the region outside the % box)}\\ \psi_{2},&\mbox{if }~{}-L/2<x<L/2\mbox{ (the region inside the box)}\\ \psi_{3}&\mbox{if }~{}x>L/2\mbox{ (the region outside the box)}\end{cases}
  36. ψ 1 , ψ 2 \psi_{1},\psi_{2}\,\!
  37. ψ 3 \psi_{3}\,\!
  38. ψ 1 = F e - α x + G e α x \psi_{1}=Fe^{-\alpha x}+Ge^{\alpha x}\,\!
  39. ψ 2 = A sin ( k x ) + B cos ( k x ) \psi_{2}=A\sin(kx)+B\cos(kx)\quad
  40. ψ 3 = H e - α x + I e α x \psi_{3}=He^{-\alpha x}+Ie^{\alpha x}\,\!
  41. x x
  42. - -\infty
  43. F F
  44. x x
  45. + +\infty
  46. I I
  47. F = I = 0 F=I=0
  48. ψ 1 = G e α x \psi_{1}=Ge^{\alpha x}\,\!
  49. ψ 3 = H e - α x \psi_{3}=He^{-\alpha x}\,\!
  50. ψ \psi\,\!
  51. ψ 1 ( - L / 2 ) = ψ 2 ( - L / 2 ) \psi_{1}(-L/2)=\psi_{2}(-L/2)\,\!
  52. ψ 2 ( L / 2 ) = ψ 3 ( L / 2 ) \psi_{2}(L/2)=\psi_{3}(L/2)\,\!
  53. d ψ 1 d x ( - L / 2 ) = d ψ 2 d x ( - L / 2 ) \frac{d\psi_{1}}{dx}(-L/2)=\frac{d\psi_{2}}{dx}(-L/2)\,\!
  54. d ψ 2 d x ( L / 2 ) = d ψ 3 d x ( L / 2 ) \frac{d\psi_{2}}{dx}(L/2)=\frac{d\psi_{3}}{dx}(L/2)\,\!
  55. A = 0 A=0
  56. G = H G=H
  57. B = 0 B=0
  58. G = - H G=-H
  59. H e - α L / 2 = B cos ( k L / 2 ) He^{-\alpha L/2}=B\cos(kL/2)
  60. - α H e - α L / 2 = - k B sin ( k L / 2 ) -\alpha He^{-\alpha L/2}=-kB\sin(kL/2)
  61. α = k tan ( k L / 2 ) \alpha=k\tan(kL/2)
  62. α = - k cot ( k L / 2 ) \alpha=-k\cot(kL/2)
  63. α \alpha
  64. k k
  65. u = α L / 2 u=\alpha L/2
  66. v = k L / 2 v=kL/2
  67. α \alpha
  68. k k
  69. u 2 = u 0 2 - v 2 u^{2}=u_{0}^{2}-v^{2}
  70. u 0 2 = m L 2 V 0 / 2 2 u_{0}^{2}=mL^{2}V_{0}/2\hbar^{2}
  71. u 0 2 - v 2 = { v tan v , (symmetric case) - v cot v , (antisymmetric case) \sqrt{u_{0}^{2}-v^{2}}=\begin{cases}v\tan v,&\mbox{(symmetric case) }\\ -v\cot v,&\mbox{(antisymmetric case) }\end{cases}
  72. u 0 2 = 20 u_{0}^{2}=20
  73. v tan v v\tan v
  74. - v cot v -v\cot v
  75. v i v_{i}
  76. π 2 ( i - 1 ) v i < π 2 i \frac{\pi}{2}(i-1)\leq v_{i}<\frac{\pi}{2}i
  77. N N
  78. u 0 u_{0}
  79. π / 2 \pi/2
  80. N = 2 u 0 π N=\left\lceil\frac{2u_{0}}{\pi}\right\rceil
  81. N = 2 20 / π = 2.85 = 3 N=\lceil 2\sqrt{20}/\pi\rceil=\lceil 2.85\rceil=3
  82. v 1 = 1.28 , v 2 = 2.54 v_{1}=1.28,v_{2}=2.54
  83. v 3 = 3.73 v_{3}=3.73
  84. E n = 2 2 v n 2 m L 2 E_{n}={2\hbar^{2}v_{n}^{2}\over mL^{2}}
  85. A , B , G , H A,B,G,H
  86. x 0 / 2 m V 0 x_{0}\equiv\hbar/\sqrt{2mV_{0}}
  87. u 0 u_{0}
  88. V 0 V_{0}\to\infty
  89. v n = n π / 2 v_{n}=n\pi/2
  90. V 0 V_{0}\to\infty
  91. L 0 L\to 0
  92. V 0 L V_{0}L
  93. u 0 V 0 L 2 u_{0}\propto V_{0}L^{2}
  94. v 2 = u 0 2 - u 0 4 v^{2}=u_{0}^{2}-u_{0}^{4}
  95. E = - m L 2 V 0 2 / 2 2 E=-mL^{2}V_{0}^{2}/2\hbar^{2}
  96. V 0 L V_{0}L
  97. U ( r ) r ψ ( r ) U(r)\equiv r\psi(r)
  98. - 2 2 m d 2 U d r 2 + V ( r ) U ( r ) = E U ( r ) -\frac{\hbar^{2}}{2m}{d^{2}U\over dr^{2}}+V(r)U(r)=EU(r)
  99. U ( r ) U(r)
  100. r = R r=R
  101. ψ ( 0 ) \psi(0)
  102. U ( 0 ) = 0 U(0)=0
  103. α = - k cot ( k R ) \alpha=-k\cot(kR)

Finite_strain_theory.html

  1. κ 0 ( ) \kappa_{0}(\mathcal{B})\,\!
  2. κ t ( ) \kappa_{t}(\mathcal{B})\,\!
  3. i i
  4. P i P_{i}\,\!
  5. p i p_{i}\,\!
  6. 𝐗 \mathbf{X}\,\!
  7. P i P_{i}\,\!
  8. 𝐱 \mathbf{x}\,\!
  9. p i p_{i}\,\!
  10. 𝐮 ( 𝐗 , t ) = u i 𝐞 i \mathbf{u}(\mathbf{X},t)=u_{i}\mathbf{e}_{i}\,\!
  11. 𝐞 i \mathbf{e}_{i}\,\!
  12. 𝐮 ( 𝐗 , t ) = 𝐛 ( t ) + 𝐱 ( 𝐗 , t ) - 𝐗 or u i = α i J b J + x i - α i J X J \mathbf{u}(\mathbf{X},t)=\mathbf{b}(t)+\mathbf{x}(\mathbf{X},t)-\mathbf{X}% \qquad\,\text{or}\qquad u_{i}=\alpha_{iJ}b_{J}+x_{i}-\alpha_{iJ}X_{J}\,\!
  13. 𝐗 𝐮 \nabla_{\mathbf{X}}\mathbf{u}\,\!
  14. 𝐗 𝐮 = 𝐗 𝐱 - 𝐈 = 𝐅 - 𝐈 or u i X K = x i X K - δ i K = F i K - δ i K \begin{aligned}\displaystyle\nabla_{\mathbf{X}}\mathbf{u}&\displaystyle=\nabla% _{\mathbf{X}}\mathbf{x}-\mathbf{I}=\mathbf{F}-\mathbf{I}&\displaystyle\,\text{% or}&\displaystyle\qquad\frac{\partial u_{i}}{\partial X_{K}}=\frac{\partial x_% {i}}{\partial X_{K}}-\delta_{iK}=F_{iK}-\delta_{iK}\end{aligned}
  15. 𝐅 \mathbf{F}\,\!
  16. P P\,\!
  17. 𝐔 ( 𝐱 , t ) = U i 𝐄 i \mathbf{U}(\mathbf{x},t)=U_{i}\mathbf{E}_{i}\,\!
  18. 𝐄 i \mathbf{E}_{i}\,\!
  19. 𝐔 ( 𝐱 , t ) = 𝐛 ( t ) + 𝐱 - 𝐗 ( 𝐱 , t ) or U J = b J + α J i x i - X J \mathbf{U}(\mathbf{x},t)=\mathbf{b}(t)+\mathbf{x}-\mathbf{X}(\mathbf{x},t)% \qquad\,\text{or}\qquad U_{J}=b_{J}+\alpha_{Ji}x_{i}-X_{J}\,\!
  20. 𝐱 𝐔 \nabla_{\mathbf{x}}\mathbf{U}\,\!
  21. 𝐱 𝐔 = 𝐈 - 𝐱 𝐗 = 𝐈 - 𝐅 - 1 or U J x k = δ J k - X J x k = δ J k - F J k - 1 . \begin{aligned}\displaystyle\nabla_{\mathbf{x}}\mathbf{U}&\displaystyle=% \mathbf{I}-\nabla_{\mathbf{x}}\mathbf{X}=\mathbf{I}-\mathbf{F}^{-1}&% \displaystyle\,\text{or}&\displaystyle\qquad\frac{\partial U_{J}}{\partial x_{% k}}=\delta_{Jk}-\frac{\partial X_{J}}{\partial x_{k}}=\delta_{Jk}-F^{-1}_{Jk}% \,.\end{aligned}
  22. α J i \alpha_{Ji}\,\!
  23. 𝐄 J \mathbf{E}_{J}\,\!
  24. 𝐞 i \mathbf{e}_{i}\,\!
  25. 𝐄 J 𝐞 i = α J i = α i J \mathbf{E}_{J}\cdot\mathbf{e}_{i}=\alpha_{Ji}=\alpha_{iJ}\,\!
  26. u i u_{i}\,\!
  27. U J U_{J}\,\!
  28. u i = α i J U J or U J = α J i u i u_{i}=\alpha_{iJ}U_{J}\qquad\,\text{or}\qquad U_{J}=\alpha_{Ji}u_{i}\,\!
  29. 𝐞 i = α i J 𝐄 J \mathbf{e}_{i}=\alpha_{iJ}\mathbf{E}_{J}\,\!
  30. 𝐮 ( 𝐗 , t ) = u i 𝐞 i = u i ( α i J 𝐄 J ) = U J 𝐄 J = 𝐔 ( 𝐱 , t ) \mathbf{u}(\mathbf{X},t)=u_{i}\mathbf{e}_{i}=u_{i}(\alpha_{iJ}\mathbf{E}_{J})=% U_{J}\mathbf{E}_{J}=\mathbf{U}(\mathbf{x},t)\,\!
  31. 𝐛 = 0 \mathbf{b}=0\,\!
  32. 𝐄 J 𝐞 i = δ J i = δ i J \mathbf{E}_{J}\cdot\mathbf{e}_{i}=\delta_{Ji}=\delta_{iJ}\,\!
  33. 𝐮 ( 𝐗 , t ) = 𝐱 ( 𝐗 , t ) - 𝐗 or u i = x i - δ i J X J \mathbf{u}(\mathbf{X},t)=\mathbf{x}(\mathbf{X},t)-\mathbf{X}\qquad\,\text{or}% \qquad u_{i}=x_{i}-\delta_{iJ}X_{J}\,\!
  34. 𝐔 ( 𝐱 , t ) = 𝐱 - 𝐗 ( 𝐱 , t ) or U J = δ J i x i - X J \mathbf{U}(\mathbf{x},t)=\mathbf{x}-\mathbf{X}(\mathbf{x},t)\qquad\,\text{or}% \qquad U_{J}=\delta_{Ji}x_{i}-X_{J}\,\!
  35. 𝐅 ( 𝐗 , t ) = F j K 𝐞 j 𝐈 K \mathbf{F}(\mathbf{X},t)=F_{jK}\mathbf{e}_{j}\otimes\mathbf{I}_{K}\,\!
  36. 𝐞 j \mathbf{e}_{j}\,\!
  37. 𝐈 K \mathbf{I}_{K}\,\!
  38. χ ( 𝐗 , t ) \chi(\mathbf{X},t)\,\!
  39. 𝐅 \mathbf{F}\,\!
  40. 𝐇 = 𝐅 - 1 \mathbf{H}=\mathbf{F}^{-1}\,\!
  41. 𝐇 \mathbf{H}\,\!
  42. J ( 𝐗 , t ) J(\mathbf{X},t)\,\!
  43. J ( 𝐗 , t ) = det 𝐅 ( 𝐗 , t ) 0 J(\mathbf{X},t)=\det\mathbf{F}(\mathbf{X},t)\neq 0\,\!
  44. 𝐅 ( 𝐗 , t ) = F j K 𝐞 j 𝐈 K \mathbf{F}(\mathbf{X},t)=F_{jK}\mathbf{e}_{j}\otimes\mathbf{I}_{K}\,\!
  45. χ ( 𝐗 , t ) \chi(\mathbf{X},t)\,\!
  46. 𝐗 \mathbf{X}\,\!
  47. χ ( 𝐗 , t ) \chi(\mathbf{X},t)\,\!
  48. 𝐗 \mathbf{X}\,\!
  49. t t\,\!
  50. d 𝐱 = 𝐱 𝐗 d 𝐗 or d x j = x j X K d X K = χ ( 𝐗 , t ) d 𝐗 = 𝐅 ( 𝐗 , t ) d 𝐗 or d x j = F j K d X K . \begin{aligned}\displaystyle d\mathbf{x}&\displaystyle=\frac{\partial\mathbf{x% }}{\partial\mathbf{X}}\,d\mathbf{X}&\displaystyle\,\text{or}&\displaystyle% \qquad dx_{j}=\frac{\partial x_{j}}{\partial X_{K}}\,dX_{K}\\ &\displaystyle=\nabla\chi(\mathbf{X},t)\,d\mathbf{X}=\mathbf{F}(\mathbf{X},t)% \,d\mathbf{X}&\displaystyle\,\text{or}&\displaystyle\qquad dx_{j}=F_{jK}\,dX_{% K}\,.\end{aligned}\,\!
  51. P P\,\!
  52. 𝐗 = X I 𝐈 I \mathbf{X}=X_{I}\mathbf{I}_{I}\,\!
  53. p p\,\!
  54. 𝐱 = x i 𝐞 i \mathbf{x}=x_{i}\mathbf{e}_{i}\,\!
  55. Q Q\,\!
  56. P P\,\!
  57. 𝐗 + Δ 𝐗 = ( X I + Δ X I ) 𝐈 I \mathbf{X}+\Delta\mathbf{X}=(X_{I}+\Delta X_{I})\mathbf{I}_{I}\,\!
  58. q q\,\!
  59. 𝐱 + Δ 𝐱 \mathbf{x}+\Delta\mathbf{x}\,\!
  60. Δ X \Delta X\,\!
  61. Δ 𝐱 \Delta\mathbf{x}\,\!
  62. P P\,\!
  63. Q Q\,\!
  64. d 𝐗 d\mathbf{X}\,\!
  65. d 𝐱 d\mathbf{x}\,\!
  66. 𝐱 + d 𝐱 = 𝐗 + d 𝐗 + 𝐮 ( 𝐗 + d 𝐗 ) d 𝐱 = 𝐗 - 𝐱 + d 𝐗 + 𝐮 ( 𝐗 + d 𝐗 ) = d 𝐗 + 𝐮 ( 𝐗 + d 𝐗 ) - 𝐮 ( 𝐗 ) = d 𝐗 + d 𝐮 \begin{aligned}\displaystyle\mathbf{x}+d\mathbf{x}&\displaystyle=\mathbf{X}+d% \mathbf{X}+\mathbf{u}(\mathbf{X}+d\mathbf{X})\\ \displaystyle d\mathbf{x}&\displaystyle=\mathbf{X}-\mathbf{x}+d\mathbf{X}+% \mathbf{u}(\mathbf{X}+d\mathbf{X})\\ &\displaystyle=d\mathbf{X}+\mathbf{u}(\mathbf{X}+d\mathbf{X})-\mathbf{u}(% \mathbf{X})\\ &\displaystyle=d\mathbf{X}+d\mathbf{u}\\ \end{aligned}\,\!
  67. 𝐝𝐮 \mathbf{du}\,\!
  68. Q Q\,\!
  69. P P\,\!
  70. d 𝐗 d\mathbf{X}\,\!
  71. P P\,\!
  72. Q Q\,\!
  73. 𝐮 ( 𝐗 + d 𝐗 ) = 𝐮 ( 𝐗 ) + d 𝐮 or u i * = u i + d u i 𝐮 ( 𝐗 ) + 𝐗 𝐮 d 𝐗 or u i * u i + u i X J d X J . \begin{aligned}\displaystyle\mathbf{u}(\mathbf{X}+d\mathbf{X})&\displaystyle=% \mathbf{u}(\mathbf{X})+d\mathbf{u}&\displaystyle\,\text{or}&\displaystyle\quad u% _{i}^{*}=u_{i}+du_{i}\\ &\displaystyle\approx\mathbf{u}(\mathbf{X})+\nabla_{\mathbf{X}}\mathbf{u}\cdot d% \mathbf{X}&\displaystyle\,\text{or}&\displaystyle\quad u_{i}^{*}\approx u_{i}+% \frac{\partial u_{i}}{\partial X_{J}}dX_{J}\,.\end{aligned}\,\!
  74. d 𝐱 = d 𝐗 + d 𝐮 d\mathbf{x}=d\mathbf{X}+d\mathbf{u}\,\!
  75. d 𝐱 = d 𝐗 + d 𝐮 = d 𝐗 + 𝐗 𝐮 d 𝐗 = ( 𝐈 + 𝐗 𝐮 ) d 𝐗 = 𝐅 d 𝐗 \begin{aligned}\displaystyle d\mathbf{x}&\displaystyle=d\mathbf{X}+d\mathbf{u}% \\ &\displaystyle=d\mathbf{X}+\nabla_{\mathbf{X}}\mathbf{u}\cdot d\mathbf{X}\\ &\displaystyle=\left(\mathbf{I}+\nabla_{\mathbf{X}}\mathbf{u}\right)d\mathbf{X% }\\ &\displaystyle=\mathbf{F}d\mathbf{X}\end{aligned}\,\!
  76. 𝐅 \mathbf{F}
  77. 𝐅 ˙ = 𝐅 t = t [ 𝐱 ( 𝐗 , t ) 𝐗 ] = 𝐗 [ 𝐱 ( 𝐗 , t ) t ] = 𝐗 [ 𝐕 ( 𝐗 , t ) ] \dot{\mathbf{F}}=\frac{\partial\mathbf{F}}{\partial t}=\frac{\partial}{% \partial t}\left[\frac{\partial\mathbf{x}(\mathbf{X},t)}{\partial\mathbf{X}}% \right]=\frac{\partial}{\partial\mathbf{X}}\left[\frac{\partial\mathbf{x}(% \mathbf{X},t)}{\partial t}\right]=\frac{\partial}{\partial\mathbf{X}}\left[% \mathbf{V}(\mathbf{X},t)\right]
  78. 𝐕 \mathbf{V}
  79. 𝐅 ˙ = 𝐗 [ 𝐕 ( 𝐗 , t ) ] = 𝐱 [ 𝐯 ( 𝐱 , t ) ] 𝐱 ( 𝐗 , t ) 𝐗 = s y m b o l l 𝐅 \dot{\mathbf{F}}=\frac{\partial}{\partial\mathbf{X}}\left[\mathbf{V}(\mathbf{X% },t)\right]=\frac{\partial}{\partial\mathbf{x}}\left[\mathbf{v}(\mathbf{x},t)% \right]\cdot\frac{\partial\mathbf{x}(\mathbf{X},t)}{\partial\mathbf{X}}=symbol% {l}\cdot\mathbf{F}
  80. s y m b o l l symbol{l}
  81. 𝐅 = e s y m b o l l t \mathbf{F}=e^{symbol{l}\,t}
  82. 𝐅 = 𝟏 \mathbf{F}=\mathbf{1}
  83. t = 0 t=0
  84. s y m b o l d = 1 2 ( s y m b o l l + s y m b o l l T ) , s y m b o l w = 1 2 ( s y m b o l l - s y m b o l l T ) . symbol{d}=\tfrac{1}{2}\left(symbol{l}+symbol{l}^{T}\right)\,,~{}~{}symbol{w}=% \tfrac{1}{2}\left(symbol{l}-symbol{l}^{T}\right)\,.
  85. d a 𝐧 = J d A 𝐅 - T 𝐍 da~{}\mathbf{n}=J~{}dA~{}\mathbf{F}^{-T}\cdot\mathbf{N}\,\!
  86. d a da\,\!
  87. d A dA\,\!
  88. 𝐧 \mathbf{n}\,\!
  89. 𝐍 \mathbf{N}\,\!
  90. 𝐅 \mathbf{F}\,\!
  91. J = det 𝐅 J=\det\mathbf{F}\,\!
  92. d v = J d V dv=J~{}dV\,\!
  93. d 𝐀 = d A 𝐍 ; d 𝐚 = d a 𝐧 d\mathbf{A}=dA~{}\mathbf{N}~{};~{}~{}d\mathbf{a}=da~{}\mathbf{n}\,\!
  94. d V = d 𝐀 T d 𝐋 ; d v = d 𝐚 T d 𝐥 dV=d\mathbf{A}^{T}\cdot d\mathbf{L}~{};~{}~{}dv=d\mathbf{a}^{T}\cdot d\mathbf{% l}\,\!
  95. d 𝐥 = 𝐅 d 𝐋 d\mathbf{l}=\mathbf{F}\cdot d\mathbf{L}\,\!
  96. d 𝐚 T d 𝐥 = d v = J d V = J d 𝐀 T d 𝐋 d\mathbf{a}^{T}\cdot d\mathbf{l}=dv=J~{}dV=J~{}d\mathbf{A}^{T}\cdot d\mathbf{L% }\,\!
  97. d 𝐚 T 𝐅 d 𝐋 = d v = J d V = J d 𝐀 T d 𝐋 d\mathbf{a}^{T}\cdot\mathbf{F}\cdot d\mathbf{L}=dv=J~{}dV=J~{}d\mathbf{A}^{T}% \cdot d\mathbf{L}\,\!
  98. d 𝐚 T 𝐅 = J d 𝐀 T d\mathbf{a}^{T}\cdot\mathbf{F}=J~{}d\mathbf{A}^{T}\,\!
  99. d 𝐚 = J 𝐅 - T d 𝐀 d\mathbf{a}=J~{}\mathbf{F}^{-T}\cdot d\mathbf{A}\,\!
  100. d a 𝐧 = J d A 𝐅 - T 𝐍 da~{}\mathbf{n}=J~{}dA~{}\mathbf{F}^{-T}\cdot\mathbf{N}\qquad\qquad\square\,\!
  101. 𝐅 \mathbf{F}\,\!
  102. 𝐅 = 𝐑𝐔 = 𝐕𝐑 \mathbf{F}=\mathbf{R}\mathbf{U}=\mathbf{V}\mathbf{R}\,\!
  103. 𝐑 \mathbf{R}\,\!
  104. 𝐑 - 1 = 𝐑 T \mathbf{R}^{-1}=\mathbf{R}^{T}\,\!
  105. det 𝐑 = + 1 \det\mathbf{R}=+1\,\!
  106. 𝐔 \mathbf{U}\,\!
  107. 𝐕 \mathbf{V}\,\!
  108. 𝐑 \mathbf{R}\,\!
  109. 𝐔 \mathbf{U}\,\!
  110. 𝐕 \mathbf{V}\,\!
  111. 𝐱 𝐔 𝐱 0 \mathbf{x}\cdot\mathbf{U}\cdot\mathbf{x}\geq 0\,\!
  112. 𝐱 𝐕 𝐱 0 \mathbf{x}\cdot\mathbf{V}\cdot\mathbf{x}\geq 0\,\!
  113. 𝐔 = 𝐔 T \mathbf{U}=\mathbf{U}^{T}\,\!
  114. 𝐕 = 𝐕 T \mathbf{V}=\mathbf{V}^{T}\,\!
  115. d 𝐗 d\mathbf{X}\,\!
  116. d 𝐱 d\mathbf{x}\,\!
  117. d 𝐱 = 𝐅 d 𝐗 d\mathbf{x}=\mathbf{F}\,d\mathbf{X}\,\!
  118. 𝐔 \mathbf{U}\,\!
  119. d 𝐱 = 𝐔 d 𝐗 d\mathbf{x}^{\prime}=\mathbf{U}\,d\mathbf{X}\,\!
  120. 𝐑 \mathbf{R}\,\!
  121. d 𝐱 = 𝐑 d 𝐱 d\mathbf{x}=\mathbf{R}\,d\mathbf{x}^{\prime}\,\!
  122. 𝐑 \mathbf{R}\,\!
  123. d 𝐱 = 𝐑 d 𝐗 d\mathbf{x}^{\prime}=\mathbf{R}\,d\mathbf{X}\,\!
  124. 𝐕 \mathbf{V}\,\!
  125. d 𝐱 = 𝐕 d 𝐱 d\mathbf{x}=\mathbf{V}\,d\mathbf{x}^{\prime}\,\!
  126. 𝐕 = 𝐑 𝐔 𝐑 T \mathbf{V}=\mathbf{R}\cdot\mathbf{U}\cdot\mathbf{R}^{T}\,\!
  127. 𝐔 \mathbf{U}\,\!
  128. 𝐕 \mathbf{V}\,\!
  129. 𝐍 i \mathbf{N}_{i}\,\!
  130. 𝐧 i \mathbf{n}_{i}\,\!
  131. 𝐧 i = 𝐑𝐍 i . \mathbf{n}_{i}=\mathbf{R}\mathbf{N}_{i}.\,\!
  132. 𝐅 \mathbf{F}\,\!
  133. 𝐑𝐑 T = 𝐑 T 𝐑 = 𝟏 \mathbf{R}\mathbf{R}^{T}=\mathbf{R}^{T}\mathbf{R}=\mathbf{1}\,\!
  134. 𝐅 \mathbf{F}\,\!
  135. 𝐂 = 𝐅 T 𝐅 = 𝐔 2 or C I J = F k I F k J = x k X I x k X J . \mathbf{C}=\mathbf{F}^{T}\mathbf{F}=\mathbf{U}^{2}\qquad\,\text{or}\qquad C_{% IJ}=F_{kI}~{}F_{kJ}=\frac{\partial x_{k}}{\partial X_{I}}\frac{\partial x_{k}}% {\partial X_{J}}.\,\!
  136. d 𝐱 2 = d 𝐗 𝐂 d 𝐗 d\mathbf{x}^{2}=d\mathbf{X}\cdot\mathbf{C}d\mathbf{X}\,\!
  137. 𝐂 \mathbf{C}\,\!
  138. I 1 C : = tr ( 𝐂 ) = C I I = λ 1 2 + λ 2 2 + λ 3 2 I 2 C : = 1 2 [ ( tr 𝐂 ) 2 - tr ( 𝐂 2 ) ] = 1 2 [ ( C J J ) 2 - C I K C K I ] = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 I 3 C : = det ( 𝐂 ) = λ 1 2 λ 2 2 λ 3 2 . \begin{aligned}\displaystyle I_{1}^{C}&\displaystyle:=\,\text{tr}(\mathbf{C})=% C_{II}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\\ \displaystyle I_{2}^{C}&\displaystyle:=\tfrac{1}{2}\left[(\,\text{tr}~{}% \mathbf{C})^{2}-\,\text{tr}(\mathbf{C}^{2})\right]=\tfrac{1}{2}\left[(C_{JJ})^% {2}-C_{IK}C_{KI}\right]=\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_% {3}^{2}+\lambda_{3}^{2}\lambda_{1}^{2}\\ \displaystyle I_{3}^{C}&\displaystyle:=\det(\mathbf{C})=\lambda_{1}^{2}\lambda% _{2}^{2}\lambda_{3}^{2}.\end{aligned}\,\!
  139. λ i \lambda_{i}\,\!
  140. 𝐂 - 1 \mathbf{C}^{-1}
  141. 𝐟 = 𝐂 - 1 = 𝐅 - 1 𝐅 - T or f I J = X I x k X J x k \mathbf{f}=\mathbf{C}^{-1}=\mathbf{F}^{-1}\mathbf{F}^{-T}\qquad\,\text{or}% \qquad f_{IJ}=\frac{\partial X_{I}}{\partial x_{k}}\frac{\partial X_{J}}{% \partial x_{k}}\,\!
  142. 𝐁 = 𝐅𝐅 T = 𝐕 2 or B i j = x i X K x j X K \mathbf{B}=\mathbf{F}\mathbf{F}^{T}=\mathbf{V}^{2}\qquad\,\text{or}\qquad B_{% ij}=\frac{\partial x_{i}}{\partial X_{K}}\frac{\partial x_{j}}{\partial X_{K}}\,\!
  143. 𝐁 \mathbf{B}\,\!
  144. I 1 : = tr ( 𝐁 ) = B i i = λ 1 2 + λ 2 2 + λ 3 2 I 2 : = 1 2 [ ( tr 𝐁 ) 2 - tr ( 𝐁 2 ) ] = 1 2 ( B i i 2 - B j k B k j ) = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 I 3 : = det 𝐁 = J 2 = λ 1 2 λ 2 2 λ 3 2 \begin{aligned}\displaystyle I_{1}&\displaystyle:=\,\text{tr}(\mathbf{B})=B_{% ii}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\\ \displaystyle I_{2}&\displaystyle:=\tfrac{1}{2}\left[(\,\text{tr}~{}\mathbf{B}% )^{2}-\,\text{tr}(\mathbf{B}^{2})\right]=\tfrac{1}{2}\left(B_{ii}^{2}-B_{jk}B_% {kj}\right)=\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+% \lambda_{3}^{2}\lambda_{1}^{2}\\ \displaystyle I_{3}&\displaystyle:=\det\mathbf{B}=J^{2}=\lambda_{1}^{2}\lambda% _{2}^{2}\lambda_{3}^{2}\end{aligned}\,\!
  145. J := det 𝐅 J:=\det\mathbf{F}\,\!
  146. ( I ¯ 1 := J - 2 / 3 I 1 ; I ¯ 2 := J - 4 / 3 I 2 ; J = 1 ) . (\bar{I}_{1}:=J^{-2/3}I_{1}~{};~{}~{}\bar{I}_{2}:=J^{-4/3}I_{2}~{};~{}~{}J=1)~% {}.\,\!
  147. 𝐁 - 1 \mathbf{B}^{-1}\,\!
  148. 𝐜 = 𝐁 - 1 = 𝐅 - T 𝐅 - 1 or c i j = X K x i X K x j \mathbf{c}=\mathbf{B}^{-1}=\mathbf{F}^{-T}\mathbf{F}^{-1}\qquad\,\text{or}% \qquad c_{ij}=\frac{\partial X_{K}}{\partial x_{i}}\frac{\partial X_{K}}{% \partial x_{j}}\,\!
  149. λ i \lambda_{i}\,\!
  150. 𝐂 \mathbf{C}\,\!
  151. 𝐁 \mathbf{B}\,\!
  152. 𝐂 = i = 1 3 λ i 2 𝐍 i 𝐍 i and 𝐁 = i = 1 3 λ i 2 𝐧 i 𝐧 i \mathbf{C}=\sum_{i=1}^{3}\lambda_{i}^{2}\mathbf{N}_{i}\otimes\mathbf{N}_{i}% \qquad\,\text{and}\qquad\mathbf{B}=\sum_{i=1}^{3}\lambda_{i}^{2}\mathbf{n}_{i}% \otimes\mathbf{n}_{i}\,\!
  153. 𝐔 = i = 1 3 λ i 𝐍 i 𝐍 i ; 𝐕 = i = 1 3 λ i 𝐧 i 𝐧 i \mathbf{U}=\sum_{i=1}^{3}\lambda_{i}\mathbf{N}_{i}\otimes\mathbf{N}_{i}~{};~{}% ~{}\mathbf{V}=\sum_{i=1}^{3}\lambda_{i}\mathbf{n}_{i}\otimes\mathbf{n}_{i}\,\!
  154. 𝐑 = i = 1 3 𝐧 i 𝐍 i ; 𝐅 = i = 1 3 λ i 𝐧 i 𝐍 i \mathbf{R}=\sum_{i=1}^{3}\mathbf{n}_{i}\otimes\mathbf{N}_{i}~{};~{}~{}\mathbf{% F}=\sum_{i=1}^{3}\lambda_{i}\mathbf{n}_{i}\otimes\mathbf{N}_{i}\,\!
  155. 𝐕 = 𝐑 𝐔 𝐑 T = i = 1 3 λ i 𝐑 ( 𝐍 i 𝐍 i ) 𝐑 T = i = 1 3 λ i ( 𝐑 𝐍 i ) ( 𝐑 𝐍 i ) \mathbf{V}=\mathbf{R}~{}\mathbf{U}~{}\mathbf{R}^{T}=\sum_{i=1}^{3}\lambda_{i}~% {}\mathbf{R}~{}(\mathbf{N}_{i}\otimes\mathbf{N}_{i})~{}\mathbf{R}^{T}=\sum_{i=% 1}^{3}\lambda_{i}~{}(\mathbf{R}~{}\mathbf{N}_{i})\otimes(\mathbf{R}~{}\mathbf{% N}_{i})\,\!
  156. 𝐧 i = 𝐑 𝐍 i \mathbf{n}_{i}=\mathbf{R}~{}\mathbf{N}_{i}\,\!
  157. 𝐕 \mathbf{V}\,\!
  158. 𝐔 \mathbf{U}\,\!
  159. 𝐅 \mathbf{F}\,\!
  160. 𝐍 i \mathbf{N}_{i}\,\!
  161. λ i \lambda_{i}\,\!
  162. 𝐧 i \mathbf{n}_{i}\,\!
  163. 𝐅 𝐍 i = λ i ( 𝐑 𝐍 i ) = λ i 𝐧 i \mathbf{F}~{}\mathbf{N}_{i}=\lambda_{i}~{}(\mathbf{R}~{}\mathbf{N}_{i})=% \lambda_{i}~{}\mathbf{n}_{i}\,\!
  164. 𝐅 - T 𝐍 i = 1 λ i 𝐧 i ; 𝐅 T 𝐧 i = λ i 𝐍 i ; 𝐅 - 1 𝐧 i = 1 λ i 𝐍 i . \mathbf{F}^{-T}~{}\mathbf{N}_{i}=\cfrac{1}{\lambda_{i}}~{}\mathbf{n}_{i}~{};~{% }~{}\mathbf{F}^{T}~{}\mathbf{n}_{i}=\lambda_{i}~{}\mathbf{N}_{i}~{};~{}~{}% \mathbf{F}^{-1}~{}\mathbf{n}_{i}=\cfrac{1}{\lambda_{i}}~{}\mathbf{N}_{i}~{}.\,\!
  165. α = α 𝟏 \mathbf{\alpha=\alpha_{1}}\,\!
  166. α 𝟏 α 𝟐 α 𝟑 = 𝟏 \mathbf{\alpha_{1}\alpha_{2}\alpha_{3}=1}\,\!
  167. α 𝟐 = α 𝟑 = α - 0.5 \mathbf{\alpha_{2}=\alpha_{3}=\alpha^{-0.5}}\,\!
  168. 𝐅 = [ α 0 0 0 α - 0.5 0 0 0 α - 0.5 ] \mathbf{F}=\begin{bmatrix}\alpha&0&0\\ 0&\alpha^{-0.5}&0\\ 0&0&\alpha^{-0.5}\end{bmatrix}\,\!
  169. 𝐁 = 𝐂 = [ α 2 0 0 0 α - 1 0 0 0 α - 1 ] \mathbf{B}=\mathbf{C}=\begin{bmatrix}\alpha^{2}&0&0\\ 0&\alpha^{-1}&0\\ 0&0&\alpha^{-1}\end{bmatrix}\,\!
  170. 𝐅 = [ 1 γ 0 0 1 0 0 0 1 ] \mathbf{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\,\!
  171. 𝐁 = [ 1 + γ 2 γ 0 γ 1 0 0 0 1 ] \mathbf{B}=\begin{bmatrix}1+\gamma^{2}&\gamma&0\\ \gamma&1&0\\ 0&0&1\end{bmatrix}\,\!
  172. 𝐂 = [ 1 γ 0 γ 1 + γ 2 0 0 0 1 ] \mathbf{C}=\begin{bmatrix}1&\gamma&0\\ \gamma&1+\gamma^{2}&0\\ 0&0&1\end{bmatrix}\,\!
  173. 𝐅 = [ cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ] \mathbf{F}=\begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}\,\!
  174. 𝐁 = 𝐂 = [ 1 0 0 0 1 0 0 0 1 ] = 𝟏 \mathbf{B}=\mathbf{C}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}=\mathbf{1}\,\!
  175. λ i 𝐂 = 1 2 λ i 𝐍 i 𝐍 i = 1 2 λ i 𝐑 T ( 𝐧 i 𝐧 i ) 𝐑 ; i = 1 , 2 , 3 \cfrac{\partial\lambda_{i}}{\partial\mathbf{C}}=\cfrac{1}{2\lambda_{i}}~{}% \mathbf{N}_{i}\otimes\mathbf{N}_{i}=\cfrac{1}{2\lambda_{i}}~{}\mathbf{R}^{T}~{% }(\mathbf{n}_{i}\otimes\mathbf{n}_{i})~{}\mathbf{R}~{};~{}~{}i=1,2,3\,\!
  176. 𝐂 : ( 𝐍 i 𝐍 i ) = λ i 2 ; 𝐂 𝐂 = 𝖨 ( s ) ; 𝖨 ( s ) : ( 𝐍 i 𝐍 i ) = 𝐍 i 𝐍 i . \mathbf{C}:(\mathbf{N}_{i}\otimes\mathbf{N}_{i})=\lambda_{i}^{2}~{};~{}~{}~{}~% {}\cfrac{\partial\mathbf{C}}{\partial\mathbf{C}}=\mathsf{I}^{(s)}~{};~{}~{}~{}% ~{}\mathsf{I}^{(s)}:(\mathbf{N}_{i}\otimes\mathbf{N}_{i})=\mathbf{N}_{i}% \otimes\mathbf{N}_{i}.\,\!
  177. 𝐗 = X i s y m b o l E i \mathbf{X}=X^{i}~{}symbol{E}_{i}
  178. 𝐱 = x i s y m b o l E i \mathbf{x}=x^{i}~{}symbol{E}_{i}
  179. 𝐗 ( s ) \mathbf{X}(s)
  180. s [ 0 , 1 ] s\in[0,1]
  181. 𝐱 ( 𝐗 ( s ) ) \mathbf{x}(\mathbf{X}(s))
  182. l X = 0 1 | d 𝐗 d s | d s = 0 1 d 𝐗 d s d 𝐗 d s d s = 0 1 d 𝐗 d s \cdotsymbol I d 𝐗 d s d s l_{X}=\int_{0}^{1}\left|\cfrac{d\mathbf{X}}{ds}\right|~{}ds=\int_{0}^{1}\sqrt{% \cfrac{d\mathbf{X}}{ds}\cdot\cfrac{d\mathbf{X}}{ds}}~{}ds=\int_{0}^{1}\sqrt{% \cfrac{d\mathbf{X}}{ds}\cdotsymbol{I}\cdot\cfrac{d\mathbf{X}}{ds}}~{}ds
  183. l x = 0 1 | d 𝐱 d s | d s = 0 1 d 𝐱 d s d 𝐱 d s d s = 0 1 ( d 𝐱 d 𝐗 d 𝐗 d s ) ( d 𝐱 d 𝐗 d 𝐗 d s ) d s = 0 1 d 𝐗 d s [ ( d 𝐱 d 𝐗 ) T d 𝐱 d 𝐗 ] d 𝐗 d s d s \begin{aligned}\displaystyle l_{x}&\displaystyle=\int_{0}^{1}\left|\cfrac{d% \mathbf{x}}{ds}\right|~{}ds=\int_{0}^{1}\sqrt{\cfrac{d\mathbf{x}}{ds}\cdot% \cfrac{d\mathbf{x}}{ds}}~{}ds=\int_{0}^{1}\sqrt{\left(\cfrac{d\mathbf{x}}{d% \mathbf{X}}\cdot\cfrac{d\mathbf{X}}{ds}\right)\cdot\left(\cfrac{d\mathbf{x}}{d% \mathbf{X}}\cdot\cfrac{d\mathbf{X}}{ds}\right)}~{}ds\\ &\displaystyle=\int_{0}^{1}\sqrt{\cfrac{d\mathbf{X}}{ds}\cdot\left[\left(% \cfrac{d\mathbf{x}}{d\mathbf{X}}\right)^{T}\cdot\cfrac{d\mathbf{x}}{d\mathbf{X% }}\right]\cdot\cfrac{d\mathbf{X}}{ds}}~{}ds\end{aligned}
  184. s y m b o l C := s y m b o l F T \cdotsymbol F = ( d 𝐱 d 𝐗 ) T d 𝐱 d 𝐗 symbol{C}:=symbol{F}^{T}\cdotsymbol{F}=\left(\cfrac{d\mathbf{x}}{d\mathbf{X}}% \right)^{T}\cdot\cfrac{d\mathbf{x}}{d\mathbf{X}}
  185. l x = 0 1 d 𝐗 d s \cdotsymbol C d 𝐗 d s d s l_{x}=\int_{0}^{1}\sqrt{\cfrac{d\mathbf{X}}{ds}\cdotsymbol{C}\cdot\cfrac{d% \mathbf{X}}{ds}}~{}ds
  186. s y m b o l C symbol{C}
  187. 𝐄 = 1 2 ( 𝐂 - 𝐈 ) or E K L = 1 2 ( x j X K x j X L - δ K L ) \mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})\qquad\,\text{or}\qquad E_{KL}=% \frac{1}{2}\left(\frac{\partial x_{j}}{\partial X_{K}}\frac{\partial x_{j}}{% \partial X_{L}}-\delta_{KL}\right)\,\!
  188. 𝐄 = 1 2 [ ( 𝐗 𝐮 ) T + 𝐗 𝐮 + ( 𝐗 𝐮 ) T 𝐗 𝐮 ] \mathbf{E}=\frac{1}{2}\left[(\nabla_{\mathbf{X}}\mathbf{u})^{T}+\nabla_{% \mathbf{X}}\mathbf{u}+(\nabla_{\mathbf{X}}\mathbf{u})^{T}\cdot\nabla_{\mathbf{% X}}\mathbf{u}\right]\,\!
  189. E K L = 1 2 ( u K X L + u L X K + u M X K u M X L ) 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)\,\!
  190. 𝐂 \mathbf{C}\,\!
  191. 𝐈 \mathbf{I}\,\!
  192. 𝐞 = 1 2 ( 𝐈 - 𝐜 ) or e r s = 1 2 ( δ r s - X M x r X M x s ) \mathbf{e}=\frac{1}{2}(\mathbf{I}-\mathbf{c})\qquad\,\text{or}\qquad e_{rs}=% \frac{1}{2}\left(\delta_{rs}-\frac{\partial X_{M}}{\partial x_{r}}\frac{% \partial X_{M}}{\partial x_{s}}\right)\,\!
  193. e i j = 1 2 ( u i x j + u j x i - u k x i u k x j ) e_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u% _{j}}{\partial x_{i}}-\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{k% }}{\partial x_{j}}\right)\,\!
  194. d 𝐗 d\mathbf{X}\,\!
  195. d 𝐱 d\mathbf{x}\,\!
  196. d 𝐱 2 - d 𝐗 2 = d 𝐱 d 𝐱 - d 𝐗 d 𝐗 or ( d x ) 2 - ( d X ) 2 = d x j d x j - d X M d X M d\mathbf{x}^{2}-d\mathbf{X}^{2}=d\mathbf{x}\cdot d\mathbf{x}-d\mathbf{X}\cdot d% \mathbf{X}\qquad\,\text{or}\qquad(dx)^{2}-(dX)^{2}=dx_{j}dx_{j}-dX_{M}\,dX_{M}\,\!
  197. d 𝐱 = 𝐱 𝐗 d 𝐗 = 𝐅 d 𝐗 or d x j = x j X M d X M d\mathbf{x}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}}\,d\mathbf{X}=\mathbf% {F}\,d\mathbf{X}\qquad\,\text{or}\qquad dx_{j}=\frac{\partial x_{j}}{\partial X% _{M}}\,dX_{M}\,\!
  198. d 𝐱 2 = d 𝐱 d 𝐱 = 𝐅 d 𝐗 𝐅 d 𝐗 = d 𝐗 𝐅 T 𝐅 d 𝐗 = d 𝐗 𝐂 d 𝐗 or ( d x ) 2 = d x j d x j = x j X K x j X L d X K d X L = C K L d X K d X L \begin{aligned}\displaystyle d\mathbf{x}^{2}&\displaystyle=d\mathbf{x}\cdot d% \mathbf{x}\\ &\displaystyle=\mathbf{F}\cdot d\mathbf{X}\cdot\mathbf{F}\cdot d\mathbf{X}\\ &\displaystyle=d\mathbf{X}\cdot\mathbf{F}^{T}\mathbf{F}\cdot d\mathbf{X}\\ &\displaystyle=d\mathbf{X}\cdot\mathbf{C}\cdot d\mathbf{X}\end{aligned}\qquad% \,\text{or}\qquad\begin{aligned}\displaystyle(dx)^{2}&\displaystyle=dx_{j}\,dx% _{j}\\ &\displaystyle=\frac{\partial x_{j}}{\partial X_{K}}\frac{\partial x_{j}}{% \partial X_{L}}\,dX_{K}\,dX_{L}\\ &\displaystyle=C_{KL}\,dX_{K}\,dX_{L}\\ \end{aligned}\,\!
  199. C K L C_{KL}\,\!
  200. 𝐂 = 𝐅 T 𝐅 \mathbf{C}=\mathbf{F}^{T}\mathbf{F}\,\!
  201. d 𝐱 2 - d 𝐗 2 = d 𝐗 𝐂 d 𝐗 - d 𝐗 d 𝐗 = d 𝐗 ( 𝐂 - 𝐈 ) d 𝐗 = d 𝐗 2 𝐄 d 𝐗 \begin{aligned}\displaystyle d\mathbf{x}^{2}-d\mathbf{X}^{2}&\displaystyle=d% \mathbf{X}\cdot\mathbf{C}\cdot d\mathbf{X}-d\mathbf{X}\cdot d\mathbf{X}\\ &\displaystyle=d\mathbf{X}\cdot(\mathbf{C}-\mathbf{I})\cdot d\mathbf{X}\\ &\displaystyle=d\mathbf{X}\cdot 2\mathbf{E}\cdot d\mathbf{X}\\ \end{aligned}\,\!
  202. ( d x ) 2 - ( d X ) 2 = x j X K x j X L d X K d X L - d X M d X M = ( x j X K x j X L - δ K L ) d X K d X L = 2 E K L d X K d X L \begin{aligned}\displaystyle(dx)^{2}-(dX)^{2}&\displaystyle=\frac{\partial x_{% j}}{\partial X_{K}}\frac{\partial x_{j}}{\partial X_{L}}\,dX_{K}\,dX_{L}-dX_{M% }\,dX_{M}\\ &\displaystyle=\left(\frac{\partial x_{j}}{\partial X_{K}}\frac{\partial x_{j}% }{\partial X_{L}}-\delta_{KL}\right)\,dX_{K}\,dX_{L}\\ &\displaystyle=2E_{KL}\,dX_{K}\,dX_{L}\end{aligned}\,\!
  203. E K L E_{KL}\,\!
  204. 𝐄 = 1 2 ( 𝐂 - 𝐈 ) or E K L = 1 2 ( x j X K x j X L - δ K L ) \mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})\qquad\,\text{or}\qquad E_{KL}=% \frac{1}{2}\left(\frac{\partial x_{j}}{\partial X_{K}}\frac{\partial x_{j}}{% \partial X_{L}}-\delta_{KL}\right)\,\!
  205. d 𝐗 = 𝐗 𝐱 d 𝐱 = 𝐅 - 1 d 𝐱 = 𝐇 d 𝐱 or d X M = X M x n d x n d\mathbf{X}=\frac{\partial\mathbf{X}}{\partial\mathbf{x}}d\mathbf{x}=\mathbf{F% }^{-1}\,d\mathbf{x}=\mathbf{H}\,d\mathbf{x}\qquad\,\text{or}\qquad dX_{M}=% \frac{\partial X_{M}}{\partial x_{n}}\,dx_{n}\,\!
  206. X M x n \frac{\partial X_{M}}{\partial x_{n}}\,\!
  207. 𝐇 \mathbf{H}\,\!
  208. d 𝐗 2 = d 𝐗 d 𝐗 = 𝐅 - 1 d 𝐱 𝐅 - 1 d 𝐱 = d 𝐱 𝐅 - T 𝐅 - 1 d 𝐱 = d 𝐱 𝐜 d 𝐱 or ( d X ) 2 = d X M d X M = X M x r X M x s d x r d x s = c r s d x r d x s \begin{aligned}\displaystyle d\mathbf{X}^{2}&\displaystyle=d\mathbf{X}\cdot d% \mathbf{X}\\ &\displaystyle=\mathbf{F}^{-1}\cdot d\mathbf{x}\cdot\mathbf{F}^{-1}\cdot d% \mathbf{x}\\ &\displaystyle=d\mathbf{x}\cdot\mathbf{F}^{-T}\mathbf{F}^{-1}\cdot d\mathbf{x}% \\ &\displaystyle=d\mathbf{x}\cdot\mathbf{c}\cdot d\mathbf{x}\end{aligned}\qquad% \,\text{or}\qquad\begin{aligned}\displaystyle(dX)^{2}&\displaystyle=dX_{M}\,dX% _{M}\\ &\displaystyle=\frac{\partial X_{M}}{\partial x_{r}}\frac{\partial X_{M}}{% \partial x_{s}}\,dx_{r}\,dx_{s}\\ &\displaystyle=c_{rs}\,dx_{r}\,dx_{s}\\ \end{aligned}\,\!
  209. c r s c_{rs}\,\!
  210. 𝐜 = 𝐅 - T 𝐅 - 1 \mathbf{c}=\mathbf{F}^{-T}\mathbf{F}^{-1}\,\!
  211. d 𝐱 2 - d 𝐗 2 = d 𝐱 d 𝐱 - d 𝐱 𝐜 d 𝐱 = d 𝐱 ( 𝐈 - 𝐜 ) d 𝐱 = d 𝐱 2 𝐞 d 𝐱 \begin{aligned}\displaystyle d\mathbf{x}^{2}-d\mathbf{X}^{2}&\displaystyle=d% \mathbf{x}\cdot d\mathbf{x}-d\mathbf{x}\cdot\mathbf{c}\cdot d\mathbf{x}\\ &\displaystyle=d\mathbf{x}\cdot(\mathbf{I}-\mathbf{c})\cdot d\mathbf{x}\\ &\displaystyle=d\mathbf{x}\cdot 2\mathbf{e}\cdot d\mathbf{x}\\ \end{aligned}\,\!
  212. ( d x ) 2 - ( d X ) 2 = d x j d x j - X M x r X M x s d x r d x s = ( δ r s - X M x r X M x s ) d x r d x s = 2 e r s d x r d x s \begin{aligned}\displaystyle(dx)^{2}-(dX)^{2}&\displaystyle=dx_{j}\,dx_{j}-% \frac{\partial X_{M}}{\partial x_{r}}\frac{\partial X_{M}}{\partial x_{s}}\,dx% _{r}\,dx_{s}\\ &\displaystyle=\left(\delta_{rs}-\frac{\partial X_{M}}{\partial x_{r}}\frac{% \partial X_{M}}{\partial x_{s}}\right)\,dx_{r}\,dx_{s}\\ &\displaystyle=2e_{rs}\,dx_{r}\,dx_{s}\end{aligned}\,\!
  213. e r s e_{rs}\,\!
  214. 𝐞 = 1 2 ( 𝐈 - 𝐜 ) or e r s = 1 2 ( δ r s - X M x r X M x s ) \mathbf{e}=\frac{1}{2}(\mathbf{I}-\mathbf{c})\qquad\,\text{or}\qquad e_{rs}=% \frac{1}{2}\left(\delta_{rs}-\frac{\partial X_{M}}{\partial x_{r}}\frac{% \partial X_{M}}{\partial x_{s}}\right)\,\!
  215. 𝐮 ( 𝐗 , t ) \mathbf{u}(\mathbf{X},t)\,\!
  216. X M X_{M}\,\!
  217. 𝐗 𝐮 \nabla_{\mathbf{X}}\mathbf{u}\,\!
  218. 𝐮 ( 𝐗 , t ) = 𝐱 ( 𝐗 , t ) - 𝐗 𝐗 𝐮 = 𝐅 - 𝐈 𝐅 = 𝐗 𝐮 + 𝐈 or u i = x i - δ i J X J δ i J U J = x i - δ i J X J x i = δ i J ( U J + X J ) x i X K = δ i J ( U J X K + δ J K ) \begin{aligned}\displaystyle\mathbf{u}(\mathbf{X},t)&\displaystyle=\mathbf{x}(% \mathbf{X},t)-\mathbf{X}\\ \displaystyle\nabla_{\mathbf{X}}\mathbf{u}&\displaystyle=\mathbf{F}-\mathbf{I}% \\ \displaystyle\mathbf{F}&\displaystyle=\nabla_{\mathbf{X}}\mathbf{u}+\mathbf{I}% \\ \end{aligned}\qquad\,\text{or}\qquad\begin{aligned}\displaystyle u_{i}&% \displaystyle=x_{i}-\delta_{iJ}X_{J}\\ \displaystyle\delta_{iJ}U_{J}&\displaystyle=x_{i}-\delta_{iJ}X_{J}\\ \displaystyle x_{i}&\displaystyle=\delta_{iJ}\left(U_{J}+X_{J}\right)\\ \displaystyle\frac{\partial x_{i}}{\partial X_{K}}&\displaystyle=\delta_{iJ}% \left(\frac{\partial U_{J}}{\partial X_{K}}+\delta_{JK}\right)\\ \end{aligned}\,\!
  219. 𝐄 = 1 2 ( 𝐅 T 𝐅 - 𝐈 ) = 1 2 [ { ( 𝐗 𝐮 ) T + 𝐈 } ( 𝐗 𝐮 + 𝐈 ) - 𝐈 ] = 1 2 [ ( 𝐗 𝐮 ) T + 𝐗 𝐮 + ( 𝐗 𝐮 ) T 𝐗 𝐮 ] \begin{aligned}\displaystyle\mathbf{E}&\displaystyle=\frac{1}{2}\left(\mathbf{% F}^{T}\mathbf{F}-\mathbf{I}\right)\\ &\displaystyle=\frac{1}{2}\left[\left\{(\nabla_{\mathbf{X}}\mathbf{u})^{T}+% \mathbf{I}\right\}\left(\nabla_{\mathbf{X}}\mathbf{u}+\mathbf{I}\right)-% \mathbf{I}\right]\\ &\displaystyle=\frac{1}{2}\left[(\nabla_{\mathbf{X}}\mathbf{u})^{T}+\nabla_{% \mathbf{X}}\mathbf{u}+(\nabla_{\mathbf{X}}\mathbf{u})^{T}\cdot\nabla_{\mathbf{% X}}\mathbf{u}\right]\\ \end{aligned}\,\!
  220. E K L = 1 2 ( x j X K x j X L - δ K L ) = 1 2 [ δ j M ( U M X K + δ M K ) δ j N ( U N X L + δ N L ) - δ K L ] = 1 2 [ δ M N ( U M X K + δ M K ) ( U N X L + δ N L ) - δ K L ] = 1 2 [ ( U M X K + δ M K ) ( U M X L + δ M L ) - δ K L ] = 1 2 ( U K X L + U L X K + U M X K U M X L ) \begin{aligned}\displaystyle E_{KL}&\displaystyle=\frac{1}{2}\left(\frac{% \partial x_{j}}{\partial X_{K}}\frac{\partial x_{j}}{\partial X_{L}}-\delta_{% KL}\right)\\ &\displaystyle=\frac{1}{2}\left[\delta_{jM}\left(\frac{\partial U_{M}}{% \partial X_{K}}+\delta_{MK}\right)\delta_{jN}\left(\frac{\partial U_{N}}{% \partial X_{L}}+\delta_{NL}\right)-\delta_{KL}\right]\\ &\displaystyle=\frac{1}{2}\left[\delta_{MN}\left(\frac{\partial U_{M}}{% \partial X_{K}}+\delta_{MK}\right)\left(\frac{\partial U_{N}}{\partial X_{L}}+% \delta_{NL}\right)-\delta_{KL}\right]\\ &\displaystyle=\frac{1}{2}\left[\left(\frac{\partial U_{M}}{\partial X_{K}}+% \delta_{MK}\right)\left(\frac{\partial U_{M}}{\partial X_{L}}+\delta_{ML}% \right)-\delta_{KL}\right]\\ &\displaystyle=\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)\end{aligned}\,\!
  221. e i j = 1 2 ( u i x j + u j x i - u k x i u k x j ) e_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u% _{j}}{\partial x_{i}}-\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{k% }}{\partial x_{j}}\right)\,\!
  222. 𝐄 ( m ) = 1 2 m ( 𝐔 2 m - 𝐈 ) = 1 2 m [ 𝐂 m - 𝐈 ] \mathbf{E}_{(m)}=\frac{1}{2m}(\mathbf{U}^{2m}-\mathbf{I})=\frac{1}{2m}\left[% \mathbf{C}^{m}-\mathbf{I}\right]\,\!
  223. m m\,\!
  224. 𝐄 ( 1 ) = 1 2 ( 𝐔 2 - 𝐈 ) = 1 2 ( 𝐂 - 𝐈 ) Green-Lagrangian strain tensor 𝐄 ( 1 / 2 ) = ( 𝐔 - 𝐈 ) = 𝐂 1 / 2 - 𝐈 Biot strain tensor 𝐄 ( 0 ) = ln 𝐔 = 1 2 ln 𝐂 Logarithmic strain, Natural strain, True strain, or Hencky strain 𝐄 ( - 1 ) = 1 2 [ 𝐈 - 𝐔 - 2 ] Almansi strain \begin{aligned}\displaystyle\mathbf{E}_{(1)}&\displaystyle=\frac{1}{2}(\mathbf% {U}^{2}-\mathbf{I})=\frac{1}{2}(\mathbf{C}-\mathbf{I})&\displaystyle\qquad\,% \text{Green-Lagrangian strain tensor}\\ \displaystyle\mathbf{E}_{(1/2)}&\displaystyle=(\mathbf{U}-\mathbf{I})=\mathbf{% C}^{1/2}-\mathbf{I}&\displaystyle\qquad\,\text{Biot strain tensor}\\ \displaystyle\mathbf{E}_{(0)}&\displaystyle=\ln\mathbf{U}=\frac{1}{2}\,\ln% \mathbf{C}&\displaystyle\qquad\,\text{Logarithmic strain, Natural strain, True% strain, or Hencky strain}\\ \displaystyle\mathbf{E}_{(-1)}&\displaystyle=\frac{1}{2}\left[\mathbf{I}-% \mathbf{U}^{-2}\right]&\displaystyle\qquad\,\text{Almansi strain}\end{aligned}\,\!
  225. 𝐄 ( m ) = s y m b o l ε + 1 2 ( 𝐮 ) T 𝐮 - ( 1 - m ) s y m b o l ε T \cdotsymbol ε \mathbf{E}_{(m)}=symbol{\varepsilon}+{\tfrac{1}{2}}(\nabla\mathbf{u})^{T}\cdot% \nabla\mathbf{u}-(1-m)symbol{\varepsilon}^{T}\cdotsymbol{\varepsilon}
  226. s y m b o l ε symbol{\varepsilon}
  227. 𝐄 \mathbf{E}
  228. 𝐄 \mathbf{E}
  229. 𝐄 \mathbf{E}
  230. 𝐮 \nabla\mathbf{u}
  231. 𝐄 \mathbf{E}
  232. s y m b o l ε symbol{\varepsilon}
  233. | 𝐮 | 0 |\nabla\mathbf{u}|\to 0
  234. 𝐄 ( n ) = ( 𝐔 n - 𝐔 - n ) / 2 n \mathbf{E}^{(n)}=\left({\mathbf{U}}^{n}-{\mathbf{U}}^{-n}\right)/2n
  235. m = 0 m=0
  236. n n
  237. d 𝐗 = d X 𝐍 d\mathbf{X}=dX\mathbf{N}\,\!
  238. 𝐍 \mathbf{N}\,\!
  239. P P\,\!
  240. Λ ( 𝐍 ) = d x d X \Lambda_{(\mathbf{N})}=\frac{dx}{dX}\,\!
  241. d x dx\,\!
  242. d 𝐗 d\mathbf{X}\,\!
  243. d 𝐱 = d x 𝐧 d\mathbf{x}=dx\mathbf{n}\,\!
  244. 𝐧 \mathbf{n}\,\!
  245. p p\,\!
  246. 1 Λ ( 𝐧 ) = d X d x . \frac{1}{\Lambda_{(\mathbf{n})}}=\frac{dX}{dx}.\,\!
  247. e 𝐍 e_{\mathbf{N}}\,\!
  248. 𝐍 \mathbf{N}\,\!
  249. e ( 𝐍 ) = d x - d X d X = Λ ( 𝐍 ) - 1. e_{(\mathbf{N})}=\frac{dx-dX}{dX}=\Lambda_{(\mathbf{N})}-1.\,\!
  250. E K L E_{KL}\,\!
  251. E 11 = e ( 𝐈 1 ) + 1 2 e ( 𝐈 1 ) 2 E_{11}=e_{(\mathbf{I}_{1})}+\frac{1}{2}e_{(\mathbf{I}_{1})}^{2}\,\!
  252. e ( 𝐈 1 ) e_{(\mathbf{I}_{1})}\,\!
  253. 𝐈 1 \mathbf{I}_{1}\,\!
  254. E K L E_{KL}\,\!
  255. E 12 = 1 2 2 E 11 + 1 2 E 22 + 1 sin ϕ 12 E_{12}=\frac{1}{2}\sqrt{2E_{11}+1}\sqrt{2E_{22}+1}\sin\phi_{12}\,\!
  256. ϕ 12 \phi_{12}\,\!
  257. 𝐈 1 \mathbf{I}_{1}\,\!
  258. 𝐈 2 \mathbf{I}_{2}\,\!
  259. d 𝐗 = d X 𝐍 d\mathbf{X}=dX\mathbf{N}\,\!
  260. 𝐍 \mathbf{N}\,\!
  261. P P\,\!
  262. Λ ( 𝐍 ) = d x d X \Lambda_{(\mathbf{N})}=\frac{dx}{dX}\,\!
  263. d x dx\,\!
  264. d 𝐗 d\mathbf{X}\,\!
  265. d 𝐱 = d x 𝐧 d\mathbf{x}=dx\mathbf{n}\,\!
  266. 𝐧 \mathbf{n}\,\!
  267. p p\,\!
  268. 1 Λ ( 𝐧 ) = d X d x \frac{1}{\Lambda_{(\mathbf{n})}}=\frac{dX}{dx}\,\!
  269. Λ ( 𝐍 ) 2 = ( d x d X ) 2 \Lambda_{(\mathbf{N})}^{2}=\left(\frac{dx}{dX}\right)^{2}\,\!
  270. ( d x ) 2 = C K L d X K d X L (dx)^{2}=C_{KL}dX_{K}dX_{L}\,\!
  271. Λ ( 𝐍 ) 2 = C K L N K N L \Lambda_{(\mathbf{N})}^{2}=C_{KL}N_{K}N_{L}\,\!
  272. N K N_{K}\,\!
  273. N L N_{L}\,\!
  274. e 𝐍 e_{\mathbf{N}}\,\!
  275. 𝐍 \mathbf{N}\,\!
  276. e ( 𝐍 ) = d x - d X d X = Λ ( 𝐍 ) - 1 e_{(\mathbf{N})}=\frac{dx-dX}{dX}=\Lambda_{(\mathbf{N})}-1\,\!
  277. 𝐈 1 \mathbf{I}_{1}\,\!
  278. P P\,\!
  279. e ( 𝐈 1 ) = d x 1 - d X 1 d X 1 = Λ ( 𝐈 1 ) - 1 = C 11 - 1 = δ 11 + 2 E 11 - 1 = 1 + 2 E 11 - 1 \begin{aligned}\displaystyle e_{(\mathbf{I}_{1})}=\frac{dx_{1}-dX_{1}}{dX_{1}}% &\displaystyle=\Lambda_{(\mathbf{I}_{1})}-1\\ &\displaystyle=\sqrt{C_{11}}-1=\sqrt{\delta_{11}+2E_{11}}-1\\ &\displaystyle=\sqrt{1+2E_{11}}-1\end{aligned}\,\!
  280. E 11 E_{11}\,\!
  281. 2 E 11 = ( d x 1 ) 2 - ( d X 1 ) 2 ( d X 1 ) 2 E 11 = ( d x 1 - d X 1 d X 1 ) + 1 2 ( d x 1 - d X 1 d X 1 ) 2 = e ( 𝐈 1 ) + 1 2 e ( 𝐈 1 ) 2 \begin{aligned}\displaystyle 2E_{11}&\displaystyle=\frac{(dx_{1})^{2}-(dX_{1})% ^{2}}{(dX_{1})^{2}}\\ \displaystyle E_{11}&\displaystyle=\left(\frac{dx_{1}-dX_{1}}{dX_{1}}\right)+% \frac{1}{2}\left(\frac{dx_{1}-dX_{1}}{dX_{1}}\right)^{2}\\ &\displaystyle=e_{(\mathbf{I}_{1})}+\frac{1}{2}e_{(\mathbf{I}_{1})}^{2}\end{% aligned}\,\!
  282. d 𝐗 1 d\mathbf{X}_{1}\,\!
  283. d 𝐗 2 d\mathbf{X}_{2}\,\!
  284. 𝐈 1 \mathbf{I}_{1}\,\!
  285. 𝐈 2 \mathbf{I}_{2}\,\!
  286. d 𝐱 1 d\mathbf{x}_{1}\,\!
  287. d 𝐱 2 d\mathbf{x}_{2}\,\!
  288. d 𝐱 1 d 𝐱 2 = d x 1 d x 2 cos θ 12 𝐅 d 𝐗 1 𝐅 d 𝐗 2 = d 𝐗 1 𝐅 T 𝐅 d 𝐗 1 d 𝐗 2 𝐅 T 𝐅 d 𝐗 2 cos θ 12 d 𝐗 1 𝐅 T 𝐅 d 𝐗 2 d X 1 d X 2 = d 𝐗 1 𝐅 T 𝐅 d 𝐗 1 d 𝐗 2 𝐅 T 𝐅 d 𝐗 2 d X 1 d X 2 cos θ 12 𝐈 1 𝐂 𝐈 2 = Λ 𝐈 1 Λ 𝐈 2 cos θ 12 \begin{aligned}\displaystyle d\mathbf{x}_{1}\cdot d\mathbf{x}_{2}&% \displaystyle=dx_{1}dx_{2}\cos\theta_{12}\\ \displaystyle\mathbf{F}\cdot d\mathbf{X}_{1}\cdot\mathbf{F}\cdot d\mathbf{X}_{% 2}&\displaystyle=\sqrt{d\mathbf{X}_{1}\cdot\mathbf{F}^{T}\cdot\mathbf{F}\cdot d% \mathbf{X}_{1}}\cdot\sqrt{d\mathbf{X}_{2}\cdot\mathbf{F}^{T}\cdot\mathbf{F}% \cdot d\mathbf{X}_{2}}\cos\theta_{12}\\ \displaystyle\frac{d\mathbf{X}_{1}\cdot\mathbf{F}^{T}\cdot\mathbf{F}\cdot d% \mathbf{X}_{2}}{dX_{1}dX_{2}}&\displaystyle=\frac{\sqrt{d\mathbf{X}_{1}\cdot% \mathbf{F}^{T}\cdot\mathbf{F}\cdot d\mathbf{X}_{1}}\cdot\sqrt{d\mathbf{X}_{2}% \cdot\mathbf{F}^{T}\cdot\mathbf{F}\cdot d\mathbf{X}_{2}}}{dX_{1}dX_{2}}\cos% \theta_{12}\\ \displaystyle\mathbf{I}_{1}\cdot\mathbf{C}\cdot\mathbf{I}_{2}&\displaystyle=% \Lambda_{\mathbf{I}_{1}}\Lambda_{\mathbf{I}_{2}}\cos\theta_{12}\\ \end{aligned}\,\!
  289. θ 12 \theta_{12}\,\!
  290. d 𝐱 1 d\mathbf{x}_{1}\,\!
  291. d 𝐱 2 d\mathbf{x}_{2}\,\!
  292. ϕ 12 \phi_{12}\,\!
  293. ϕ 12 = π 2 - θ 12 \phi_{12}=\frac{\pi}{2}-\theta_{12}\,\!
  294. cos θ 12 = sin ϕ 12 \cos\theta_{12}=\sin\phi_{12}\,\!
  295. 𝐈 1 𝐂 𝐈 2 = Λ 𝐈 1 Λ 𝐈 2 sin ϕ 12 \mathbf{I}_{1}\cdot\mathbf{C}\cdot\mathbf{I}_{2}=\Lambda_{\mathbf{I}_{1}}% \Lambda_{\mathbf{I}_{2}}\sin\phi_{12}\,\!
  296. C 12 = C 11 C 22 sin ϕ 12 2 E 12 + δ 12 = 2 E 11 + 1 2 E 22 + 1 sin ϕ 12 E 12 = 1 2 2 E 11 + 1 2 E 22 + 1 sin ϕ 12 \begin{aligned}\displaystyle C_{12}&\displaystyle=\sqrt{C_{11}}\sqrt{C_{22}}% \sin\phi_{12}\\ \displaystyle 2E_{12}+\delta_{12}&\displaystyle=\sqrt{2E_{11}+1}\sqrt{2E_{22}+% 1}\sin\phi_{12}\\ \displaystyle E_{12}&\displaystyle=\frac{1}{2}\sqrt{2E_{11}+1}\sqrt{2E_{22}+1}% \sin\phi_{12}\end{aligned}\,\!
  297. 𝐱 = 𝐱 ( ξ 1 , ξ 2 , ξ 3 ) \mathbf{x}=\mathbf{x}(\xi^{1},\xi^{2},\xi^{3})
  298. ( ξ 1 , ξ 2 , ξ 3 ) (\xi^{1},\xi^{2},\xi^{3})
  299. ξ i \xi^{i}
  300. 𝐱 \mathbf{x}
  301. 𝐠 i = 𝐱 ξ i \mathbf{g}_{i}=\frac{\partial\mathbf{x}}{\partial\xi^{i}}
  302. 𝐱 \mathbf{x}
  303. 𝐠 i 𝐠 j = δ i j \mathbf{g}_{i}\cdot\mathbf{g}^{j}=\delta_{i}^{j}
  304. s y m b o l g symbol{g}
  305. g i j := 𝐱 ξ i 𝐱 ξ j = 𝐠 i 𝐠 j g_{ij}:=\frac{\partial\mathbf{x}}{\partial\xi^{i}}\cdot\frac{\partial\mathbf{x% }}{\partial\xi^{j}}=\mathbf{g}_{i}\cdot\mathbf{g}_{j}
  306. Γ i j k = 1 2 [ ( 𝐠 i 𝐠 k ) , j + ( 𝐠 j 𝐠 k ) , i - ( 𝐠 i 𝐠 j ) , k ] \Gamma_{ijk}=\tfrac{1}{2}[(\mathbf{g}_{i}\cdot\mathbf{g}_{k})_{,j}+(\mathbf{g}% _{j}\cdot\mathbf{g}_{k})_{,i}-(\mathbf{g}_{i}\cdot\mathbf{g}_{j})_{,k}]
  307. 𝐆 i := 𝐗 ξ i ; 𝐆 i 𝐆 j = δ i j ; 𝐠 i := 𝐱 ξ i ; 𝐠 i 𝐠 j = δ i j \mathbf{G}_{i}:=\frac{\partial\mathbf{X}}{\partial\xi^{i}}~{};~{}~{}\mathbf{G}% _{i}\cdot\mathbf{G}^{j}=\delta_{i}^{j}~{};~{}~{}\mathbf{g}_{i}:=\frac{\partial% \mathbf{x}}{\partial\xi^{i}}~{};~{}~{}\mathbf{g}_{i}\cdot\mathbf{g}^{j}=\delta% _{i}^{j}
  308. s y m b o l F = s y m b o l 𝐗 𝐱 = 𝐱 ξ i 𝐆 i = 𝐠 i 𝐆 i symbol{F}=symbol{\nabla}_{\mathbf{X}}\mathbf{x}=\frac{\partial\mathbf{x}}{% \partial\xi^{i}}\otimes\mathbf{G}^{i}=\mathbf{g}_{i}\otimes\mathbf{G}^{i}
  309. s y m b o l C = s y m b o l F T \cdotsymbol F = ( 𝐆 i 𝐠 i ) ( 𝐠 j 𝐆 j ) = ( 𝐠 i 𝐠 j ) ( 𝐆 i 𝐆 j ) symbol{C}=symbol{F}^{T}\cdotsymbol{F}=(\mathbf{G}^{i}\otimes\mathbf{g}_{i})% \cdot(\mathbf{g}_{j}\otimes\mathbf{G}^{j})=(\mathbf{g}_{i}\cdot\mathbf{g}_{j})% (\mathbf{G}^{i}\otimes\mathbf{G}^{j})
  310. s y m b o l C symbol{C}
  311. 𝐆 i \mathbf{G}^{i}
  312. s y m b o l C = C i j 𝐆 i 𝐆 j symbol{C}=C_{ij}~{}\mathbf{G}^{i}\otimes\mathbf{G}^{j}
  313. C i j = 𝐠 i 𝐠 j = g i j C_{ij}=\mathbf{g}_{i}\cdot\mathbf{g}_{j}=g_{ij}
  314. Γ i j k = 1 2 [ C i k , j + C j k , i - C i j , k ] = 1 2 [ ( 𝐆 i \cdotsymbol C 𝐆 k ) , j + ( 𝐆 j \cdotsymbol C 𝐆 k ) , i - ( 𝐆 i \cdotsymbol C 𝐆 j ) , k ] \Gamma_{ijk}=\tfrac{1}{2}[C_{ik,j}+C_{jk,i}-C_{ij,k}]=\tfrac{1}{2}[(\mathbf{G}% _{i}\cdotsymbol{C}\cdot\mathbf{G}_{k})_{,j}+(\mathbf{G}_{j}\cdotsymbol{C}\cdot% \mathbf{G}_{k})_{,i}-(\mathbf{G}_{i}\cdotsymbol{C}\cdot\mathbf{G}_{j})_{,k}]
  315. 𝐗 = { X 1 , X 2 , X 3 } \mathbf{X}=\{X^{1},X^{2},X^{3}\}
  316. 𝐱 = { x 1 , x 2 , x 3 } \mathbf{x}=\{x^{1},x^{2},x^{3}\}
  317. s y m b o l G symbol{G}
  318. s y m b o l g symbol{g}
  319. G i j = X α x i X β x j g α β G_{ij}=\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{% \partial x^{j}}~{}g_{\alpha\beta}
  320. G i j x k = ( 2 X α x i x k X β x j + X α x i 2 X β x j x k ) g α β + X α x i X β x j g α β x k \frac{\partial G_{ij}}{\partial x^{k}}=\left(\frac{\partial^{2}X^{\alpha}}{% \partial x^{i}\partial x^{k}}~{}\frac{\partial X^{\beta}}{\partial x^{j}}+% \frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial^{2}X^{\beta}}{% \partial x^{j}\partial x^{k}}\right)~{}g_{\alpha\beta}+\frac{\partial X^{% \alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{\partial x^{j}}~{}\frac{% \partial g_{\alpha\beta}}{\partial x^{k}}
  321. g α β x k = X γ x k g α β X γ \frac{\partial g_{\alpha\beta}}{\partial x^{k}}=\frac{\partial X^{\gamma}}{% \partial x^{k}}~{}\frac{\partial g_{\alpha\beta}}{\partial X^{\gamma}}
  322. g α β = g β α g_{\alpha\beta}=g_{\beta\alpha}
  323. G i j x k = ( 2 X α x i x k X β x j + 2 X α x j x k X β x i ) g α β + X α x i X β x j X γ x k g α β X γ G i k x j = ( 2 X α x i x j X β x k + 2 X α x j x k X β x i ) g α β + X α x i X β x k X γ x j g α β X γ G j k x i = ( 2 X α x i x j X β x k + 2 X α x i x k X β x j ) g α β + X α x j X β x k X γ x i g α β X γ \begin{aligned}\displaystyle\frac{\partial G_{ij}}{\partial x^{k}}&% \displaystyle=\left(\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{k}% }~{}\frac{\partial X^{\beta}}{\partial x^{j}}+\frac{\partial^{2}X^{\alpha}}{% \partial x^{j}\partial x^{k}}~{}\frac{\partial X^{\beta}}{\partial x^{i}}% \right)~{}g_{\alpha\beta}+\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{% \partial X^{\beta}}{\partial x^{j}}~{}\frac{\partial X^{\gamma}}{\partial x^{k% }}~{}\frac{\partial g_{\alpha\beta}}{\partial X^{\gamma}}\\ \displaystyle\frac{\partial G_{ik}}{\partial x^{j}}&\displaystyle=\left(\frac{% \partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}~{}\frac{\partial X^{% \beta}}{\partial x^{k}}+\frac{\partial^{2}X^{\alpha}}{\partial x^{j}\partial x% ^{k}}~{}\frac{\partial X^{\beta}}{\partial x^{i}}\right)~{}g_{\alpha\beta}+% \frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{% \partial x^{k}}~{}\frac{\partial X^{\gamma}}{\partial x^{j}}~{}\frac{\partial g% _{\alpha\beta}}{\partial X^{\gamma}}\\ \displaystyle\frac{\partial G_{jk}}{\partial x^{i}}&\displaystyle=\left(\frac{% \partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}~{}\frac{\partial X^{% \beta}}{\partial x^{k}}+\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x% ^{k}}~{}\frac{\partial X^{\beta}}{\partial x^{j}}\right)~{}g_{\alpha\beta}+% \frac{\partial X^{\alpha}}{\partial x^{j}}~{}\frac{\partial X^{\beta}}{% \partial x^{k}}~{}\frac{\partial X^{\gamma}}{\partial x^{i}}~{}\frac{\partial g% _{\alpha\beta}}{\partial X^{\gamma}}\end{aligned}
  324. Γ i j k ( x ) : = 1 2 ( G i k x j + G j k x i - G i j x k ) Γ α β γ ( X ) : = 1 2 ( g α γ X β + g β γ X α - g α β X γ ) \begin{aligned}{}_{(x)}\Gamma_{ijk}&\displaystyle:=\frac{1}{2}\left(\frac{% \partial G_{ik}}{\partial x^{j}}+\frac{\partial G_{jk}}{\partial x^{i}}-\frac{% \partial G_{ij}}{\partial x^{k}}\right)\\ {}_{(X)}\Gamma_{\alpha\beta\gamma}&\displaystyle:=\frac{1}{2}\left(\frac{% \partial g_{\alpha\gamma}}{\partial X^{\beta}}+\frac{\partial g_{\beta\gamma}}% {\partial X^{\alpha}}-\frac{\partial g_{\alpha\beta}}{\partial X^{\gamma}}% \right)\\ \end{aligned}
  325. Γ i j k ( x ) = X α x i X β x j X γ x k ( X ) Γ α β γ + 2 X α x i x j X β x k g α β {}_{(x)}\Gamma_{ijk}=\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{% \partial X^{\beta}}{\partial x^{j}}~{}\frac{\partial X^{\gamma}}{\partial x^{k% }}\,_{(X)}\Gamma_{\alpha\beta\gamma}+\frac{\partial^{2}X^{\alpha}}{\partial x^% {i}\partial x^{j}}~{}\frac{\partial X^{\beta}}{\partial x^{k}}~{}g_{\alpha\beta}
  326. [ G i j ] = [ G i j ] - 1 ; [ g α β ] = [ g α β ] - 1 [G^{ij}]=[G_{ij}]^{-1}~{};~{}~{}[g^{\alpha\beta}]=[g_{\alpha\beta}]^{-1}
  327. G i j = x i X α x j X β g α β G^{ij}=\frac{\partial x^{i}}{\partial X^{\alpha}}~{}\frac{\partial x^{j}}{% \partial X^{\beta}}~{}g^{\alpha\beta}
  328. Γ i j m ( x ) := G ( x ) m k Γ i j k ; Γ α β ν ( X ) := g ( X ) ν γ Γ α β γ {}_{(x)}\Gamma^{m}_{ij}:=G^{mk}\,_{(x)}\Gamma_{ijk}~{};~{}~{}_{(X)}\Gamma^{\nu% }_{\alpha\beta}:=g^{\nu\gamma}\,_{(X)}\Gamma_{\alpha\beta\gamma}
  329. Γ i j m ( x ) = G m k X α x i X β x j X γ x k ( X ) Γ α β γ + G m k 2 X α x i x j X β x k g α β = x m X ν x k X ρ g ν ρ X α x i X β x j X γ x k ( X ) Γ α β γ + x m X ν x k X ρ g ν ρ 2 X α x i x j X β x k g α β = x m X ν δ ρ γ g ν ρ X α x i X β x j ( X ) Γ α β γ + x m X ν δ ρ β g ν ρ 2 X α x i x j g α β = x m X ν g ν γ X α x i X β x j ( X ) Γ α β γ + x m X ν g ν β 2 X α x i x j g α β = x m X ν X α x i X β x j ( X ) Γ α β ν + x m X ν δ α ν 2 X α x i x j \begin{aligned}{}_{(x)}\Gamma^{m}_{ij}&\displaystyle=G^{mk}~{}\frac{\partial X% ^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{\partial x^{j}}~{}\frac% {\partial X^{\gamma}}{\partial x^{k}}\,_{(X)}\Gamma_{\alpha\beta\gamma}+G^{mk}% ~{}\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}~{}\frac{% \partial X^{\beta}}{\partial x^{k}}~{}g_{\alpha\beta}\\ &\displaystyle=\frac{\partial x^{m}}{\partial X^{\nu}}~{}\frac{\partial x^{k}}% {\partial X^{\rho}}~{}g^{\nu\rho}~{}\frac{\partial X^{\alpha}}{\partial x^{i}}% ~{}\frac{\partial X^{\beta}}{\partial x^{j}}~{}\frac{\partial X^{\gamma}}{% \partial x^{k}}\,_{(X)}\Gamma_{\alpha\beta\gamma}+\frac{\partial x^{m}}{% \partial X^{\nu}}~{}\frac{\partial x^{k}}{\partial X^{\rho}}~{}g^{\nu\rho}~{}% \frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}~{}\frac{\partial X% ^{\beta}}{\partial x^{k}}~{}g_{\alpha\beta}\\ &\displaystyle=\frac{\partial x^{m}}{\partial X^{\nu}}~{}\delta^{\gamma}_{\rho% }~{}g^{\nu\rho}~{}\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X% ^{\beta}}{\partial x^{j}}\,_{(X)}\Gamma_{\alpha\beta\gamma}+\frac{\partial x^{% m}}{\partial X^{\nu}}~{}\delta^{\beta}_{\rho}~{}g^{\nu\rho}~{}\frac{\partial^{% 2}X^{\alpha}}{\partial x^{i}\partial x^{j}}~{}g_{\alpha\beta}\\ &\displaystyle=\frac{\partial x^{m}}{\partial X^{\nu}}~{}g^{\nu\gamma}~{}\frac% {\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{\partial x^{% j}}\,_{(X)}\Gamma_{\alpha\beta\gamma}+\frac{\partial x^{m}}{\partial X^{\nu}}~% {}g^{\nu\beta}~{}\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}~{% }g_{\alpha\beta}\\ &\displaystyle=\frac{\partial x^{m}}{\partial X^{\nu}}~{}\frac{\partial X^{% \alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{\partial x^{j}}\,_{(X)}% \Gamma^{\nu}_{\alpha\beta}+\frac{\partial x^{m}}{\partial X^{\nu}}~{}\delta^{% \nu}_{\alpha}~{}\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}% \end{aligned}
  330. Γ i j m ( x ) = x m X ν X α x i X β x j ( X ) Γ α β ν + x m X α 2 X α x i x j {}_{(x)}\Gamma^{m}_{ij}=\frac{\partial x^{m}}{\partial X^{\nu}}~{}\frac{% \partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{\partial x^{j% }}\,_{(X)}\Gamma^{\nu}_{\alpha\beta}+\frac{\partial x^{m}}{\partial X^{\alpha}% }~{}\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}
  331. X μ x m ( x ) Γ i j m = X μ x m x m X ν X α x i X β x j ( X ) Γ α β ν + X μ x m x m X α 2 X α x i x j = δ ν μ X α x i X β x j ( X ) Γ α β ν + δ α μ 2 X α x i x j = X α x i X β x j ( X ) Γ α β μ + 2 X μ x i x j \begin{aligned}\displaystyle\frac{\partial X^{\mu}}{\partial x^{m}}\,_{(x)}% \Gamma^{m}_{ij}&\displaystyle=\frac{\partial X^{\mu}}{\partial x^{m}}~{}\frac{% \partial x^{m}}{\partial X^{\nu}}~{}\frac{\partial X^{\alpha}}{\partial x^{i}}% ~{}\frac{\partial X^{\beta}}{\partial x^{j}}\,_{(X)}\Gamma^{\nu}_{\alpha\beta}% +\frac{\partial X^{\mu}}{\partial x^{m}}~{}\frac{\partial x^{m}}{\partial X^{% \alpha}}~{}\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x^{j}}\\ &\displaystyle=\delta^{\mu}_{\nu}~{}\frac{\partial X^{\alpha}}{\partial x^{i}}% ~{}\frac{\partial X^{\beta}}{\partial x^{j}}\,_{(X)}\Gamma^{\nu}_{\alpha\beta}% +\delta^{\mu}_{\alpha}~{}\frac{\partial^{2}X^{\alpha}}{\partial x^{i}\partial x% ^{j}}\\ &\displaystyle=\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{% \beta}}{\partial x^{j}}\,_{(X)}\Gamma^{\mu}_{\alpha\beta}+\frac{\partial^{2}X^% {\mu}}{\partial x^{i}\partial x^{j}}\end{aligned}
  332. x x
  333. 2 X μ x i x j = X μ x m ( x ) Γ i j m - X α x i X β x j ( X ) Γ α β μ 2 x m X α X β = x m X μ ( X ) Γ α β μ - x i X α x j X β ( x ) Γ i j m \begin{aligned}\displaystyle\frac{\partial^{2}X^{\mu}}{\partial x^{i}\partial x% ^{j}}&\displaystyle=\frac{\partial X^{\mu}}{\partial x^{m}}\,_{(x)}\Gamma^{m}_% {ij}-\frac{\partial X^{\alpha}}{\partial x^{i}}~{}\frac{\partial X^{\beta}}{% \partial x^{j}}\,_{(X)}\Gamma^{\mu}_{\alpha\beta}\\ \displaystyle\frac{\partial^{2}x^{m}}{\partial X^{\alpha}\partial X^{\beta}}&% \displaystyle=\frac{\partial x^{m}}{\partial X^{\mu}}\,_{(X)}\Gamma^{\mu}_{% \alpha\beta}-\frac{\partial x^{i}}{\partial X^{\alpha}}~{}\frac{\partial x^{j}% }{\partial X^{\beta}}\,_{(x)}\Gamma^{m}_{ij}\end{aligned}
  334. s y m b o l F symbol{F}
  335. s y m b o l \timessymbol F = s y m b o l 0 symbol{\nabla}\timessymbol{F}=symbol{0}
  336. s y m b o l C symbol{C}
  337. R α β ρ γ := X ρ [ ( X ) Γ α β γ ] - X β [ ( X ) Γ α ρ γ ] + ( X ) Γ μ ρ γ Γ α β μ ( X ) - ( X ) Γ μ β γ Γ α ρ μ ( X ) = 0 R^{\gamma}_{\alpha\beta\rho}:=\frac{\partial}{\partial X^{\rho}}[\,_{(X)}% \Gamma^{\gamma}_{\alpha\beta}]-\frac{\partial}{\partial X^{\beta}}[\,_{(X)}% \Gamma^{\gamma}_{\alpha\rho}]+\,_{(X)}\Gamma^{\gamma}_{\mu\rho}\,{}_{(X)}% \Gamma^{\mu}_{\alpha\beta}-\,_{(X)}\Gamma^{\gamma}_{\mu\beta}\,{}_{(X)}\Gamma^% {\mu}_{\alpha\rho}=0
  338. s y m b o l C symbol{C}
  339. s y m b o l B symbol{B}

Finite_topology.html

  1. U x 1 , x 2 , , x n = { f Hom ( A , B ) f ( x i ) = 0 for i = 1 , 2 , , n } U_{x_{1},x_{2},\ldots,x_{n}}=\{f\in\operatorname{Hom}(A,B)\mid f(x_{i})=0\mbox% { for }~{}i=1,2,\ldots,n\}

Finite_type_invariant.html

  1. 3 \mathbb{R}^{3}
  2. 3 \mathbb{R}^{3}
  3. V 1 ( K ) = V ( K + ) - V ( K - ) V^{1}(K^{\prime})=V(K_{+})-V(K_{-})
  4. K + K_{+}

First-order_fluid.html

  1. μ eff ( γ ˙ , T ) = μ 0 γ ˙ n - 1 exp ( - b T ) \mu_{\operatorname{eff}}(\dot{\gamma},T)=\mu_{0}{\dot{\gamma}}^{n-1}\exp(-bT)
  2. γ ˙ \dot{\gamma}
  3. μ 0 \mu_{0}
  4. μ eff ( γ ˙ , T ) = exp ( A 0 + A 1 ln ( γ ˙ ) + A 2 T ) \mu_{\operatorname{eff}}(\dot{\gamma},T)=\exp\left(A_{0}+A_{1}\ln(\dot{\gamma}% )+A_{2}T\right)

Five-dimensional_space.html

  1. V = 8 π 2 r 5 15 V=\frac{8\pi^{2}r^{5}}{15}

Five_Equations_That_Changed_the_World.html

  1. F = G M m / r 2 F=GMm/r^{2}

Fixed-point_index.html

  1. g ( x ) = x - f ( x ) || x - f ( x ) || . g(x)=\frac{x-f(x)}{||x-f(x)||}.\,
  2. x Fix ( f ) i ( f , x ) = Λ f , \sum_{x\in\mathrm{Fix}(f)}i(f,x)=\Lambda_{f},

Flack_parameter.html

  1. | F ( h k l ) | 2 |F(hkl)|^{2}
  2. | F ( - h - k - l ) | 2 |F(-h-k-l)|^{2}
  3. I ( h k l ) = ( 1 - x ) | F ( h k l ) | 2 + x | F ( - h - k - l ) | 2 \ I(hkl)=(1-x)|F(hkl)|^{2}+x|F(-h-k-l)|^{2}

Flattening.html

  1. flattening = f = a - b a . \mathrm{flattening}=f=\frac{a-b}{a}.
  2. f f\,\!
  3. a - b a \frac{a-b}{a}\,\!
  4. f f^{\prime}\,\!
  5. a - b b \frac{a-b}{b}\,\!
  6. n ( f ′′ ) n\quad(f^{\prime\prime})\,\!
  7. a - b a + b \frac{a-b}{a+b}\,\!
  8. b \displaystyle b
  9. e e

Flavor-changing_neutral_current.html

  1. S ψ ¯ e ψ τ S\bar{\psi}_{e}\psi_{\tau}
  2. g 2 cos θ W d ¯ L α U α β γ μ d L β Z μ \frac{g}{2\cos\theta_{W}}\overline{d}_{L\alpha}U_{\alpha\beta}\gamma^{\mu}d_{L% \beta}Z_{\mu}

Flexibility_method.html

  1. 𝐪 m = 𝐟 m 𝐐 m + 𝐪 o m ( 1 ) \mathbf{q}^{m}=\mathbf{f}^{m}\mathbf{Q}^{m}+\mathbf{q}^{om}\qquad\qquad\qquad% \mathrm{(1)}
  2. 𝐪 m \mathbf{q}^{m}
  3. 𝐟 m \mathbf{f}^{m}
  4. 𝐐 m \mathbf{Q}^{m}
  5. 𝐪 o m \mathbf{q}^{om}
  6. 𝐐 m = 0 \mathbf{Q}^{m}=0
  7. 𝐪 M × 1 = 𝐟 M × M 𝐐 M × 1 + 𝐪 M × 1 o ( 2 ) \mathbf{q}_{M\times 1}=\mathbf{f}_{M\times M}\mathbf{Q}_{M\times 1}+\mathbf{q}% ^{o}_{M\times 1}\qquad\qquad\qquad\mathrm{(2)}
  8. 𝐐 M × 1 \mathbf{Q}_{M\times 1}
  9. 𝐑 N × 1 = 𝐛 N × M 𝐐 M × 1 + 𝐖 N × 1 ( 3 ) \mathbf{R}_{N\times 1}=\mathbf{b}_{N\times M}\mathbf{Q}_{M\times 1}+\mathbf{W}% _{N\times 1}\qquad\qquad\qquad\mathrm{(3)}
  10. 𝐑 N × 1 \mathbf{R}_{N\times 1}
  11. 𝐛 N × M \mathbf{b}_{N\times M}
  12. 𝐖 N × 1 \mathbf{W}_{N\times 1}
  13. X i = α Q j + β Q k + i = 1 , 2 , I ( 4 ) X_{i}=\alpha Q_{j}+\beta Q_{k}+...\qquad i=1,2,...I\qquad\qquad\mathrm{(4)}
  14. α \alpha
  15. β \beta
  16. X i X_{i}
  17. 𝐐 M × 1 = 𝐁 R 𝐑 N × 1 + 𝐁 X 𝐗 I × 1 + 𝐐 v M × 1 ( 5 ) \mathbf{Q}_{M\times 1}=\mathbf{B}_{R}\mathbf{R}_{N\times 1}+\mathbf{B}_{X}% \mathbf{X}_{I\times 1}+\mathbf{Q}_{v\cdot M\times 1}\qquad\qquad\qquad\mathrm{% (5)}
  18. 𝐪 M × 1 = 𝐟 M × M ( 𝐁 R 𝐑 N × 1 + 𝐁 X 𝐗 I × 1 + 𝐐 v M × 1 ) + 𝐪 M × 1 o ( 6 ) \mathbf{q}_{M\times 1}=\mathbf{f}_{M\times M}\Big(\mathbf{B}_{R}\mathbf{R}_{N% \times 1}+\mathbf{B}_{X}\mathbf{X}_{I\times 1}+\mathbf{Q}_{v\cdot M\times 1}% \Big)+\mathbf{q}^{o}_{M\times 1}\qquad\qquad\qquad\mathrm{(6)}
  19. 𝐗 \mathbf{X}
  20. 𝐗 \mathbf{X}
  21. I I
  22. 𝐗 \mathbf{X}
  23. 𝐫 X \mathbf{r}_{X}
  24. 𝐫 X = 𝐁 X T 𝐪 = 𝐁 X T [ 𝐟 ( 𝐁 R 𝐑 + 𝐁 X 𝐗 + 𝐐 v ) + 𝐪 o ] = 0 ( 7 a ) \mathbf{r}_{X}=\mathbf{B}_{X}^{T}\mathbf{q}=\mathbf{B}_{X}^{T}\Big[\mathbf{f}% \Big(\mathbf{B}_{R}\mathbf{R}+\mathbf{B}_{X}\mathbf{X}+\mathbf{Q}_{v}\Big)+% \mathbf{q}^{o}\Big]=0\qquad\qquad\qquad\mathrm{(7a)}
  25. 𝐫 X = 𝐅 X X 𝐗 + 𝐫 X o = 0 ( 7 b ) \mathbf{r}_{X}=\mathbf{F}_{XX}\mathbf{X}+\mathbf{r}^{o}_{X}=0\qquad\qquad% \qquad\mathrm{(7b)}
  26. 𝐅 X X = 𝐁 X T 𝐟𝐁 X \mathbf{F}_{XX}=\mathbf{B}_{X}^{T}\mathbf{f}\mathbf{B}_{X}
  27. 𝐫 X o = 𝐁 X T [ 𝐟 ( 𝐁 R 𝐑 + 𝐐 v ) + 𝐪 o ] \mathbf{r}^{o}_{X}=\mathbf{B}_{X}^{T}\Big[\mathbf{f}\Big(\mathbf{B}_{R}\mathbf% {R}+\mathbf{Q}_{v}\Big)+\mathbf{q}^{o}\Big]
  28. 𝐫 R = 𝐁 R T 𝐪 = 𝐅 R R 𝐑 + 𝐫 R o \mathbf{r}_{R}=\mathbf{B}_{R}^{T}\mathbf{q}=\mathbf{F}_{RR}\mathbf{R}+\mathbf{% r}^{o}_{R}
  29. 𝐅 R R = 𝐁 R T 𝐟𝐁 R \mathbf{F}_{RR}=\mathbf{B}_{R}^{T}\mathbf{f}\mathbf{B}_{R}
  30. 𝐫 R o = 𝐁 R T [ 𝐟 ( 𝐁 X 𝐗 + 𝐐 v ) + 𝐪 o ] \mathbf{r}^{o}_{R}=\mathbf{B}_{R}^{T}\Big[\mathbf{f}\Big(\mathbf{B}_{X}\mathbf% {X}+\mathbf{Q}_{v}\Big)+\mathbf{q}^{o}\Big]
  31. 𝐫 X o \mathbf{r}^{o}_{X}
  32. 𝐫 R o \mathbf{r}^{o}_{R}

FlexRay.html

  1. { x 0 , x 1 , , x m - 1 } \{x_{0},x_{1},\dots,x_{m-1}\}

Flipped_SO(10).html

  1. 16 10 1 5 ¯ - 3 1 5 16\rightarrow 10_{1}\oplus\bar{5}_{-3}\oplus 1_{5}
  2. 16 1 10 - 2 1 4 16_{1}\oplus 10_{-2}\oplus 1_{4}
  3. χ = - A 4 + 5 B 4 \chi=-{A\over 4}+{5B\over 4}
  4. 16 1 10 1 5 ¯ 2 1 0 10 - 2 5 - 2 5 ¯ - 3 1 4 1 5 \begin{aligned}\displaystyle 16_{1}&\displaystyle\rightarrow 10_{1}\oplus\bar{% 5}_{2}\oplus 1_{0}\\ \displaystyle 10_{-2}&\displaystyle\rightarrow 5_{-2}\oplus\bar{5}_{-3}\\ \displaystyle 1_{4}&\displaystyle\rightarrow 1_{5}\end{aligned}
  5. 5 ¯ - 3 \bar{5}_{-3}
  6. 16 ¯ - 1 H \overline{16}_{-1H}
  7. 5 ¯ 2 \bar{5}_{2}
  8. < 16 ¯ - 1 H > 16 1 ϕ <\overline{16}_{-1H}>16_{1}\phi
  9. < 16 ¯ - 1 H > < 16 ¯ - 1 H > 16 1 16 1 <\overline{16}_{-1H}><\overline{16}_{-1H}>16_{1}16_{1}

Floer_homology.html

  1. [ ω ] , A = λ c 1 , A \langle[\omega],A\rangle=\lambda\langle c_{1},A\rangle
  2. c 1 , A = 0 \langle c_{1},A\rangle=0
  3. c 1 , π 2 ( M ) = N \langle c_{1},\pi_{2}(M)\rangle=N\mathbb{Z}
  4. Q H * ( M ) = H * ( M ) Λ QH_{*}(M)=H_{*}(M)\otimes\Lambda
  5. H F ( L 0 , L 1 ) H F ( L 1 , L 2 ) H F ( L 0 , L 2 ) , HF(L_{0},L_{1})\otimes HF(L_{1},L_{2})\rightarrow HF(L_{0},L_{2}),
  6. Σ \Sigma
  7. Σ \Sigma
  8. Σ \Sigma
  9. Σ \Sigma
  10. Σ \Sigma
  11. Σ \Sigma
  12. X X
  13. A A_{\infty}
  14. A A_{\infty}
  15. X X

Floodgate.html

  1. F = p A \ F=pA
  2. = ρ g h =\rho gh\,

Floquet_theory.html

  1. x ˙ = A ( t ) x , \dot{x}=A(t)x,\,
  2. A ( t ) \displaystyle A(t)
  3. T T
  4. y = Q - 1 ( t ) x \displaystyle y=Q^{-1}(t)x
  5. Q ( t + 2 T ) = Q ( t ) \displaystyle Q(t+2T)=Q(t)
  6. ϕ ( t ) \phi\,(t)
  7. Φ ( t ) \Phi(t)
  8. t 0 t_{0}
  9. Φ ( t 0 ) \Phi(t_{0})
  10. Φ ( t ) = ϕ ( t ) ϕ - 1 ( t 0 ) \Phi(t)=\phi\,(t){\phi\,}^{-1}(t_{0})
  11. x ( 0 ) = x 0 x(0)=x_{0}
  12. x ( t ) = ϕ ( t ) ϕ - 1 ( 0 ) x 0 x(t)=\phi\,(t){\phi\,}^{-1}(0)x_{0}
  13. ϕ ( t ) \phi\,(t)
  14. x ˙ = A ( t ) x \dot{x}=A(t)x
  15. x ( t ) x(t)
  16. n n
  17. A ( t ) A(t)
  18. n × n n\times n
  19. T T
  20. A ( t + T ) = A ( t ) A(t+T)=A(t)
  21. t t
  22. ϕ ( t ) \phi\,(t)
  23. t t\in\mathbb{R}
  24. ϕ ( t + T ) = ϕ ( t ) ϕ - 1 ( 0 ) ϕ ( T ) . \phi(t+T)=\phi(t)\phi^{-1}(0)\phi(T).
  25. ϕ - 1 ( 0 ) ϕ ( T ) \phi^{-1}(0)\phi(T)
  26. B B
  27. e T B = ϕ - 1 ( 0 ) ϕ ( T ) , e^{TB}=\phi^{-1}(0)\phi(T),
  28. T T
  29. t P ( t ) t\mapsto P(t)
  30. ϕ ( t ) = P ( t ) e t B for all t . \phi(t)=P(t)e^{tB}\,\text{ for all }t\in\mathbb{R}.
  31. R R
  32. 2 T 2T
  33. t Q ( t ) t\mapsto Q(t)
  34. ϕ ( t ) = Q ( t ) e t R for all t . \phi(t)=Q(t)e^{tR}\,\text{ for all }t\in\mathbb{R}.
  35. B B
  36. P P
  37. Q Q
  38. R R
  39. n × n n\times n
  40. ϕ ( t ) = Q ( t ) e t R \phi\,(t)=Q(t)e^{tR}
  41. y = Q - 1 ( t ) x y=Q^{-1}(t)x
  42. y ˙ = R y \dot{y}=Ry
  43. Q ( t ) Q(t)
  44. y ( t ) y(t)
  45. x ( t ) x(t)
  46. R R
  47. ϕ ( t ) = P ( t ) e t B \phi\,(t)=P(t)e^{tB}
  48. ϕ ( t ) \phi\,(t)
  49. e T B e^{TB}
  50. x ( t ) x ( t + T ) x(t)\to x(t+T)
  51. μ \mu
  52. e μ T e^{\mu T}
  53. e ( μ + 2 π i k T ) T = e μ T e^{(\mu+\frac{2\pi ik}{T})T}=e^{\mu T}
  54. k k

Flory–Huggins_solution_theory.html

  1. Δ G m \Delta G_{m}
  2. Δ G m = Δ H m - T Δ S m \Delta G_{m}=\Delta H_{m}-T\Delta S_{m}\,
  3. Δ \Delta
  4. Δ H m \Delta H_{m}
  5. Δ S m \Delta S_{m}
  6. Δ G m = R T [ n 1 ln ϕ 1 + n 2 ln ϕ 2 + n 1 ϕ 2 χ 12 ] \Delta G_{m}=RT[\,n_{1}\ln\phi_{1}+n_{2}\ln\phi_{2}+n_{1}\phi_{2}\chi_{12}\,]\,
  7. n 1 n_{1}
  8. ϕ 1 \phi_{1}
  9. 1 1
  10. n 2 n_{2}
  11. ϕ 2 \phi_{2}
  12. 2 2
  13. χ \chi
  14. R R
  15. T T
  16. χ \chi
  17. N = N 1 + x N 2 N=N_{1}+xN_{2}\,
  18. N 1 N_{1}
  19. N 2 N_{2}
  20. x x
  21. Δ S m = - k [ N 1 ln ( N 1 / N ) + N 2 ln ( x N 2 / N ) ] \Delta S_{m}=-k[\,N_{1}\ln(N_{1}/N)+N_{2}\ln(xN_{2}/N)\,]\,
  22. k k
  23. ϕ 1 \phi_{1}
  24. ϕ 2 \phi_{2}
  25. ϕ 1 = N 1 / N \phi_{1}=N_{1}/N\,
  26. ϕ 2 = x N 2 / N \phi_{2}=xN_{2}/N\,
  27. Δ S m = - k [ N 1 ln ϕ 1 + N 2 ln ϕ 2 ] \Delta S_{m}=-k[\,N_{1}\ln\phi_{1}+N_{2}\ln\phi_{2}\,]\,
  28. x x
  29. w 11 w_{11}
  30. w 22 w_{22}
  31. w 12 w_{12}
  32. Δ w = w 12 - 1 2 ( w 22 + w 11 ) \Delta w=w_{12}-\begin{matrix}\frac{1}{2}\end{matrix}(w_{22}+w_{11})\,
  33. x N 2 z ϕ 1 = N 1 ϕ 2 z xN_{2}z\phi_{1}=N_{1}\phi_{2}z\,
  34. z z
  35. x N 2 xN_{2}
  36. x N 2 z xN_{2}z
  37. ϕ 1 \phi_{1}
  38. Δ H m = N 1 ϕ 2 z Δ w \Delta H_{m}=N_{1}\phi_{2}z\Delta w\,
  39. χ 12 = z Δ w / k T \chi_{12}=z\Delta w/kT\,
  40. Δ H m = k T N 1 ϕ 2 χ 12 \Delta H_{m}=kTN_{1}\phi_{2}\chi_{12}\,
  41. Δ G m = R T [ n 1 ln ϕ 1 + n 2 ln ϕ 2 + n 1 ϕ 2 χ 12 ] \Delta G_{m}=RT[\,n_{1}\ln\phi_{1}+n_{2}\ln\phi_{2}+n_{1}\phi_{2}\chi_{12}\,]\,
  42. N 1 N_{1}
  43. N 2 N_{2}
  44. n 1 n_{1}
  45. n 2 n_{2}
  46. N A N_{A}
  47. R = k N A R=kN_{A}
  48. δ a \delta_{a}
  49. δ b \delta_{b}
  50. χ 12 = V s e g ( δ a - δ b ) 2 / R T \chi_{12}=V_{seg}(\delta_{a}-\delta_{b})^{2}/RT\,
  51. V s e g V_{seg}
  52. Δ w \Delta w
  53. χ \chi
  54. x x
  55. P P

Flow_(mathematics).html

  1. X X
  2. X X
  3. φ : X × \R X \varphi:X\times\R\rightarrow X
  4. x x
  5. X X
  6. s s
  7. t t
  8. φ ( x , 0 ) = x ; \varphi(x,0)=x;
  9. φ ( φ ( x , t ) , s ) = φ ( x , s + t ) . \varphi(\varphi(x,t),s)=\varphi(x,s+t).
  10. φ ( x , t ) φ(x,t)
  11. t t
  12. X X
  13. X X
  14. φ φ
  15. X X
  16. φ φ
  17. dom ( φ ) = { ( x , t ) | t [ a x , b x ] , a x < 0 < b x , x X } X × \mathrm{dom}(\varphi)=\{(x,t)\ |\ t\in[a_{x},b_{x}],\ a_{x}<0<b_{x},\ x\in X\}% \subset X\times\mathbb{R}
  18. φ φ
  19. x ( t ) x(t)
  20. x x
  21. t t
  22. V V
  23. X X
  24. Φ V : X × X ; ( x , t ) Φ V t ( x ) . \Phi_{V}:X\times\mathbb{R}\to X;\qquad(x,t)\mapsto\Phi_{V}^{t}(x).
  25. x x
  26. X X
  27. φ ( x , t ) φ(x,t)
  28. t t
  29. x x
  30. φ φ
  31. x x
  32. s y m b o l x ˙ ( t ) = s y m b o l F ( s y m b o l x ( t ) ) , s y m b o l x ( 0 ) = s y m b o l x 0 . \dot{symbol{x}}(t)=symbol{F}(symbol{x}(t)),\qquad symbol{x}(0)=symbol{x}_{0}.
  33. φ φ
  34. s y m b o l x ˙ ( t ) = s y m b o l F ( s y m b o l x ( t ) , t ) , s y m b o l x ( t 0 ) = s y m b o l x 0 . \dot{symbol{x}}(t)=symbol{F}(symbol{x}(t),t),\qquad symbol{x}(t_{0})=symbol{x}% _{0}.
  35. φ : ( n × ) × n × ; φ ( s y m b o l x 0 , t 0 , t ) = ( φ t , t 0 ( s y m b o l x 0 ) , t + t 0 ) \varphi:(\mathbb{R}^{n}\times\mathbb{R})\times\mathbb{R}\to\mathbb{R}^{n}% \times\mathbb{R};\qquad\varphi(symbol{x}_{0},t_{0},t)=(\varphi^{t,t_{0}}(% symbol{x}_{0}),t+t_{0})
  36. φ ( φ ( s y m b o l x 0 , t 0 , t ) , s ) = φ ( φ t , t 0 ( s y m b o l x 0 ) , t + t 0 , s ) = ( φ s , t + t 0 ( s y m b o l x 0 ) , s + t + t 0 ) = φ ( s y m b o l x 0 , t 0 , s + t ) . \varphi(\varphi(symbol{x}_{0},t_{0},t),s)=\varphi(\varphi^{t,t_{0}}(symbol{x}_% {0}),t+t_{0},s)=(\varphi^{s,t+t_{0}}(symbol{x}_{0}),s+t+t_{0})=\varphi(symbol{% x}_{0},t_{0},s+t).
  37. s y m b o l G ( s y m b o l x , t ) := ( s y m b o l F ( s y m b o l x , t ) , 1 ) , s y m b o l y ( t ) := ( s y m b o l x ( t ) , t + t 0 ) . symbol{G}(symbol{x},t):=(symbol{F}(symbol{x},t),1),\qquad symbol{y}(t):=(% symbol{x}(t),t+t_{0}).
  38. s y m b o l y ˙ ( s ) = s y m b o l G ( s y m b o l y ( s ) ) , s y m b o l y ( 0 ) = ( s y m b o l x 0 , t 0 ) \dot{symbol{y}}(s)=symbol{G}(symbol{y}(s)),\qquad symbol{y}(0)=(symbol{x}_{0},% t_{0})
  39. 𝐱 ( t ) \mathbf{x}(t)
  40. φ φ
  41. Ω Ω
  42. n n
  43. Γ Γ
  44. Ω Ω
  45. T T
  46. T T
  47. u t - Δ u = 0 in Ω × ( 0 , T ) , u = 0 on Γ × ( 0 , T ) , \begin{array}[]{rcll}u_{t}-\Delta u&=&0&\mbox{ in }~{}\Omega\times(0,T),\\ u&=&0&\mbox{ on }~{}\Gamma\times(0,T),\end{array}
  48. Ω Ω
  49. u u
  50. Γ × ( 0 , T ) Γ×(0,T)
  51. L 2 ( Ω ) L^{2}(\Omega)
  52. D ( Δ D ) = H 2 ( Ω ) H 0 1 ( Ω ) D(\Delta_{D})=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)
  53. H k ( Ω ) = W k , 2 ( Ω ) H^{k}(\Omega)=W^{k,2}(\Omega)
  54. H 0 1 ( Ω ) = C 0 ( Ω ) ¯ H 1 ( Ω ) H_{0}^{1}(\Omega)={\overline{C_{0}^{\infty}(\Omega)}}^{H^{1}(\Omega)}
  55. Ω Ω
  56. H 1 ( Ω ) - H^{1}(\Omega)-
  57. v D ( Δ D ) v\in D(\Delta_{D})
  58. Δ D v = Δ v = i = 1 n 2 x i 2 v . \Delta_{D}v=\Delta v=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}v~{}.
  59. u ( t ) = Δ D u ( t ) u^{\prime}(t)=\Delta_{D}u(t)
  60. φ ( u 0 , t ) = e u 0 t Δ D \varphi(u^{0},t)=\mbox{e}~{}^{t\Delta_{D}}u^{0}
  61. Ω Ω
  62. n n
  63. Γ Γ
  64. Ω × ( 0 , T ) \Omega\times(0,T)
  65. T T
  66. u t t - Δ u = 0 in Ω × ( 0 , T ) , u = 0 on Γ × ( 0 , T ) , \begin{array}[]{rcll}u_{tt}-\Delta u&=&0&\mbox{ in }~{}\Omega\times(0,T),\\ u&=&0&\mbox{ on }~{}\Gamma\times(0,T),\end{array}
  67. Ω \Omega
  68. u t ( 0 ) = u 2 , 0 in Ω u_{t}(0)=u^{2,0}\mbox{ in }~{}\Omega
  69. 𝒜 = ( 0 I d Δ D 0 ) \mathcal{A}=\left(\begin{array}[]{cc}0&Id\\ \Delta_{D}&0\end{array}\right)
  70. D ( 𝒜 ) = H 2 ( Ω ) H 0 1 ( Ω ) × H 0 1 ( Ω ) D(\mathcal{A})=H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)
  71. H = H 0 1 ( Ω ) × L 2 ( Ω ) H=H^{1}_{0}(\Omega)\times L^{2}(\Omega)
  72. Δ D \Delta_{D}
  73. U = ( u 1 u 2 ) U=\left(\begin{array}[]{c}u^{1}\\ u^{2}\end{array}\right)
  74. u 1 = u u^{1}=u
  75. u 2 = u t u^{2}=u_{t}
  76. U 0 = ( u 1 , 0 u 2 , 0 ) U^{0}=\left(\begin{array}[]{c}u^{1,0}\\ u^{2,0}\end{array}\right)
  77. U ( t ) = 𝒜 U ( t ) U^{\prime}(t)=\mathcal{A}U(t)
  78. U ( 0 ) = U 0 U(0)=U^{0}
  79. φ ( U 0 , t ) = e U 0 t 𝒜 \varphi(U^{0},t)=\mbox{e}~{}^{t\mathcal{A}}U^{0}
  80. e t 𝒜 \mbox{e}~{}^{t\mathcal{A}}
  81. 𝒜 \mathcal{A}
  82. H H
  83. φ ( x , t ) φ(x,t)
  84. t t
  85. φ ( x , 1 ) φ(x,1)
  86. ψ ( x , t ) ψ(x,t)
  87. ψ ( x , t ) = φ ( x , t ) ψ(x,t)=φ(x,t)
  88. c c

Floyd–Steinberg_dithering.html

  1. [ * 7 16 3 16 5 16 1 16 ] \begin{bmatrix}&&&&\\ &&*&\frac{\displaystyle 7}{\displaystyle 16}&...\\ ...&\frac{\displaystyle 3}{\displaystyle 16}&\frac{\displaystyle 5}{% \displaystyle 16}&\frac{\displaystyle 1}{\displaystyle 16}&...\\ \end{bmatrix}

Fluent_(artificial_intelligence).html

  1. On ( box , table ) \mathrm{On}(\mathrm{box},\mathrm{table})
  2. On \mathrm{On}
  3. On ( box , table , t ) \mathrm{On}(\mathrm{box},\mathrm{table},t)
  4. t t
  5. o n ( b o x , t a b l e ) on(box,table)
  6. o n on
  7. H o l d s A t ( o n ( b o x , t a b l e ) , t ) HoldsAt(on(box,table),t)
  8. t t
  9. H o l d s A t HoldsAt
  10. o n ( b o x , t ) on(box,t)
  11. t t
  12. o n ( b o x , t ) = t a b l e on(box,t)=table

Fluent_calculus.html

  1. \circ
  2. s s
  3. t . s = o n ( b o x , t a b l e ) t \exists t.s=on(box,table)\circ t
  4. S t a t e ( D o ( m o v e ( b o x , t a b l e , f l o o r ) , s ) ) o n ( b o x , t a b l e ) = S t a t e ( s ) o n ( b o x , f l o o r ) State(Do(move(box,table,floor),s))\circ on(box,table)=State(s)\circ on(box,floor)
  5. o n ( b o x , f l o o r ) on(box,floor)
  6. o n ( b o x , t a b l e ) on(box,table)
  7. \circ

Fluid_mechanics.html

  1. ρ D 𝐮 D t = \cdotsymbol σ + ρ 𝐟 \rho\frac{D\mathbf{u}}{Dt}=\nabla\cdotsymbol{\sigma}+\rho\mathbf{f}
  2. ρ \rho
  3. D D t \frac{D}{Dt}
  4. 𝐮 \mathbf{u}
  5. 𝐟 \mathbf{f}
  6. s y m b o l σ symbol{\sigma}
  7. s y m b o l σ symbol{\sigma}
  8. s y m b o l σ i j = - p δ i j + τ i j symbol\sigma_{ij}=-p\delta_{ij}+\tau_{ij}
  9. - p δ i j -p\delta_{ij}
  10. τ i j \tau_{ij}
  11. p p
  12. τ i j \tau_{ij}
  13. τ \mathbf{\tau}
  14. ρ D 𝐮 D t = - p + τ + ρ 𝐟 \rho\frac{D\mathbf{u}}{Dt}=-\nabla p+\nabla\cdot\mathbf{\tau}+\rho\mathbf{f}
  15. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  16. p p
  17. τ = - μ d v d y \tau=-\mu\frac{dv}{dy}
  18. τ \tau
  19. μ \mu
  20. d v d y \frac{dv}{dy}
  21. τ i j = μ ( v i x j + v j x i ) \tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}% {\partial x_{i}}\right)
  22. τ i j \tau_{ij}
  23. i t h i^{th}
  24. j t h j^{th}
  25. v i v_{i}
  26. i t h i^{th}
  27. x j x_{j}
  28. j t h j^{th}
  29. τ i j = μ ( v i x j + v j x i - 2 3 δ i j 𝐯 ) + κ δ i j 𝐯 \tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}% {\partial x_{i}}-\frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{v}\right)+\kappa% \delta_{ij}\nabla\cdot\mathbf{v}
  30. κ \kappa
  31. τ \mathbf{\tau}

Fluid_solution.html

  1. T a b = μ u a u b + p h a b + ( u a q b + q a u b ) + π a b T^{ab}=\mu\,u^{a}\,u^{b}+p\,h^{ab}+\left(u^{a}\,q^{b}+q^{a}\,u^{b}\right)+\pi^% {ab}
  2. u a u^{a}
  3. h a b = g a b + u a u b h_{ab}=g_{ab}+u_{a}\,u_{b}
  4. u a u^{a}
  5. μ \mu
  6. p p
  7. q a q^{a}
  8. π a b \pi^{ab}
  9. q a u a = 0 , π a b u b = 0 q_{a}\,u^{a}=0,\;\;\pi_{ab}\,u^{b}=0
  10. T a b = ( μ + p ) u a u b + p g a b T^{ab}=(\mu+p)\,u^{a}\,u^{b}+p\,g^{ab}
  11. T a b = μ u a u b T^{ab}=\mu\,u^{a}\,u^{b}
  12. μ = 3 p \mu=3p
  13. T a b = p ( 4 u a u b + g a b ) T^{ab}=p\,\left(4\,u^{a}\,u^{b}+\,g^{ab}\right)
  14. r = r [ 0 ] r=r[0]
  15. r 0 r_{0}
  16. e 0 , e 1 , e 2 , e 3 \vec{e}_{0},\;\vec{e}_{1},\;\vec{e}_{2},\;\vec{e}_{3}
  17. G a ^ b ^ = 8 π [ μ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ] G^{\hat{a}\hat{b}}=8\pi\,\left[\begin{matrix}\mu&0&0&0\\ 0&p&0&0\\ 0&0&p&0\\ 0&0&0&p\end{matrix}\right]
  18. μ \mu
  19. p p
  20. e 0 \vec{e}_{0}
  21. u = e 0 \vec{u}=\vec{e}_{0}
  22. χ ( λ ) = ( λ - 8 π μ ) ( λ - 8 π p ) 3 \chi(\lambda)=\left(\lambda-8\pi\mu\right)\,\left(\lambda-8\pi p\right)^{3}
  23. μ , p \mu,\,p
  24. 12 a 4 + a 2 2 - 3 a 1 a 3 = 0 12a_{4}+a_{2}^{2}-3a_{1}a_{3}=0
  25. a 1 a 2 a 3 - 9 a 3 2 - 9 a 1 2 a 4 + 32 a 2 a 4 = 0 a_{1}a_{2}a_{3}-9a_{3}^{2}-9a_{1}^{2}a_{4}+32a_{2}a_{4}=0
  26. G a a = t 1 = a 1 {G^{a}}_{a}=t_{1}=a_{1}
  27. G a b G b a = t 2 = a 1 2 - 2 a 2 {G^{a}}_{b}\,{G^{b}}_{a}=t_{2}=a_{1}^{2}-2a_{2}
  28. G a b G b c G c a = t 3 = a 1 3 - 3 a 1 a 2 + 3 a 3 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=t_{3}=a_{1}^{3}-3a_{1}a_{2}+3a_{3}
  29. G a b G b c G c d G d a = t 4 = a 1 4 - 4 a 1 2 a 2 + 4 a 1 a 3 + 2 a 2 2 - a 4 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=t_{4}=a_{1}^{4}-4a_{1}^{2}a% _{2}+4a_{1}a_{3}+2a_{2}^{2}-a_{4}
  30. t 2 3 + 4 t 3 2 + t 1 2 t 4 - 4 t 2 t 4 - 2 t 1 t 2 t 3 = 0 t_{2}^{3}+4t_{3}^{2}+t_{1}^{2}t_{4}-4t_{2}t_{4}-2t_{1}t_{2}t_{3}=0
  31. t 1 4 + 7 t 2 2 - 8 t 1 2 t 2 + 12 t 1 t 3 - 12 t 4 = 0 t_{1}^{4}+7t_{2}^{2}-8t_{1}^{2}t_{2}+12t_{1}t_{3}-12t_{4}=0
  32. a 2 = a 3 = a 4 = 0 a_{2}\,=a_{3}=a_{4}=0
  33. t 2 = t 1 2 , t 3 = t 1 3 , t 4 = t 1 4 t_{2}=t_{1}^{2},\;\;t_{3}=t_{1}^{3},\;\;t_{4}=t_{1}^{4}
  34. G a a = - R {G^{a}}_{a}=-R
  35. G a b G b a = R 2 {G^{a}}_{b}\,{G^{b}}_{a}=R^{2}
  36. G a b G b c G c a = - R 3 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=-R^{3}
  37. G a b G b c G c d G d a = - R 4 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=-R^{4}
  38. a 1 = 0 , 27 a 3 2 + 8 a 2 3 = 0 , 12 a 4 + a 2 2 = 0 a_{1}=0,\;27\,a_{3}^{2}+8a_{2}^{3}=0,\;12\,a_{4}+a_{2}^{2}=0
  39. t 1 = 0 , 7 t 3 2 - t 2 t 4 = 0 , 12 t 4 - 7 t 2 2 = 0 t_{1}=0,7\,t_{3}^{2}-t_{2}\,t_{4}=0,\;12\,t_{4}-7\,t_{2}^{2}=0

Fluidized_bed.html

  1. u m f u_{mf}
  2. u u m f u\geq u_{mf}
  3. u > u m f u>u_{mf}
  4. Δ p w = H w ( 1 - ϵ w ) ( ρ s - ρ f ) g = [ M s g / A ] [ ( ρ s - ρ f ) / ρ s ] \Delta p_{w}=H_{w}(1-\epsilon_{w})(\rho_{s}-\rho_{f})g=[M_{s}g/A][(\rho_{s}-% \rho_{f})/\rho_{s}]
  5. Δ p w \Delta p_{w}
  6. H w H_{w}
  7. ϵ w \epsilon_{w}
  8. ρ s \rho_{s}
  9. ρ f \rho_{f}
  10. g g
  11. M s M_{s}
  12. A A

Fluorescence_correlation_spectroscopy.html

  1. P S F ( r , z ) = I 0 e - 2 r 2 / ω x y 2 e - 2 z 2 / ω z 2 PSF(r,z)=I_{0}e^{-2r^{2}/\omega_{xy}^{2}}e^{-2z^{2}/\omega_{z}^{2}}
  2. I 0 I_{0}
  3. ω x y \omega_{xy}
  4. ω z \omega_{z}
  5. ω z > ω x y \omega_{z}>\omega_{xy}
  6. ω x y \omega_{xy}
  7. ω z \omega_{z}
  8. τ \tau
  9. τ \tau
  10. G ( τ ) = δ I ( t ) δ I ( t + τ ) I ( t ) 2 = I ( t ) I ( t + τ ) I ( t ) 2 - 1 G(\tau)=\frac{\langle\delta I(t)\delta I(t+\tau)\rangle}{\langle I(t)\rangle^{% 2}}=\frac{\langle I(t)I(t+\tau)\rangle}{\langle I(t)\rangle^{2}}-1
  11. δ I ( t ) = I ( t ) - I ( t ) \delta I(t)=I(t)-\langle I(t)\rangle
  12. τ = 0 \tau=0
  13. P S F ( r , z ) PSF(r,z)
  14. G ( τ ) = 1 N exp ( - Δ X ( τ ) 2 + Δ Y ( τ ) 2 w x y 2 - Δ Z ( τ ) 2 w z 2 ) , G(\tau)=\frac{1}{\langle N\rangle}\left\langle\exp\left(-\frac{\Delta X(\tau)^% {2}+\Delta Y(\tau)^{2}}{w_{xy}^{2}}-\frac{\Delta Z(\tau)^{2}}{w_{z}^{2}}\right% )\right\rangle,
  15. Δ R ( τ ) = ( Δ X ( τ ) , Δ Y ( τ ) , Δ Z ( τ ) ) \Delta\vec{R}(\tau)=(\Delta X(\tau),\Delta Y(\tau),\Delta Z(\tau))
  16. τ \tau
  17. N \langle N\rangle
  18. G ( τ ) G(\tau)
  19. w x y , w z w_{xy},w_{z}
  20. | Δ R ( τ ) | 2 |\Delta\vec{R}(\tau)|^{2}
  21. w x y , w z w_{xy},w_{z}
  22. G ( τ ) = G ( 0 ) 1 ( 1 + ( τ / τ D ) ) ( 1 + a - 2 ( τ / τ D ) ) 1 / 2 + G ( ) \ G(\tau)=G(0)\frac{1}{(1+(\tau/\tau_{D}))(1+a^{-2}(\tau/\tau_{D}))^{1/2}}+G(\infty)
  23. a = ω z / ω x y a=\omega_{z}/\omega_{xy}
  24. e - 2 e^{-2}
  25. τ D \tau_{D}
  26. G ( ) G(\infty)
  27. τ D \tau_{D}
  28. G ( 0 ) = 1 N = 1 V eff C , \ G(0)=\frac{1}{\langle N\rangle}=\frac{1}{V\text{eff}\langle C\rangle},
  29. V eff = π 3 / 2 ω x y 2 ω z . \ V\text{eff}=\pi^{3/2}\omega_{xy}^{2}\omega_{z}.\,
  30. τ D \tau_{D}
  31. D = ω x y 2 / 4 τ D . \ D=\omega_{xy}^{2}/{4\tau_{D}}.
  32. M S D = 6 D a t α \ MSD=6D_{a}t^{\alpha}\,
  33. D a D_{a}
  34. G ( τ ) = G ( 0 ) 1 ( 1 + ( τ / τ D ) α ) ( 1 + a - 2 ( τ / τ D ) α ) 1 / 2 + G ( ) , G(\tau)=G(0)\frac{1}{(1+(\tau/\tau_{D})^{\alpha})(1+a^{-2}(\tau/\tau_{D})^{% \alpha})^{1/2}}+G(\infty),
  35. α \alpha
  36. G ( τ ) = G ( 0 ) i α i ( 1 + ( τ / τ D , i ) ) ( 1 + a - 2 ( τ / τ D , i ) ) 1 / 2 + G ( ) \ G(\tau)=G(0)\sum_{i}\frac{\alpha_{i}}{(1+(\tau/\tau_{D,i}))(1+a^{-2}(\tau/% \tau_{D,i}))^{1/2}}+G(\infty)
  37. α i \alpha_{i}
  38. τ D , i \tau_{D,i}
  39. v v
  40. G ( τ ) = G ( 0 ) 1 ( 1 + ( τ / τ D ) ) ( 1 + a - 2 ( τ / τ D ) ) 1 / 2 × exp [ - ( τ / τ v ) 2 × 1 1 + τ / τ D ] + G ( ) \ G(\tau)=G(0)\frac{1}{(1+(\tau/\tau_{D}))(1+a^{-2}(\tau/\tau_{D}))^{1/2}}% \times\exp[-(\tau/\tau_{v})^{2}\times\frac{1}{1+\tau/\tau_{D}}]+G(\infty)
  41. τ v = ω x y / v \tau_{v}=\omega_{xy}/v
  42. G ( τ ) = G ( 0 ) exp ( - τ / τ B ) + G ( ) \ G(\tau)=G(0)\exp(-\tau/\tau_{B})+G(\infty)
  43. τ B = ( k on + k off ) - 1 \ \tau_{B}=(k\text{on}+k\text{off})^{-1}
  44. G ( 0 ) = 1 N k o n k o f f = 1 N K G(0)=\frac{1}{\langle N\rangle}\frac{k_{on}}{k_{off}}=\frac{1}{\langle N% \rangle}K
  45. τ F \tau_{F}
  46. τ F \tau_{F}
  47. τ D \tau_{D}
  48. G ( τ ) = G ( 0 ) ( 1 - F + F e - τ / τ F ) ( 1 - F ) 1 ( 1 + ( τ / τ D , i ) ) ( 1 + a - 2 ( τ / τ D , i ) ) 1 / 2 + G ( ) \ G(\tau)=G(0)\frac{(1-F+Fe^{-\tau/\tau_{F}})}{(1-F)}\frac{1}{(1+(\tau/\tau_{D% ,i}))(1+a^{-2}(\tau/\tau_{D,i}))^{1/2}}+G(\infty)
  49. F \ F
  50. τ F \ \tau_{F}
  51. D \ D
  52. ω x y 2 = 4 D τ D + t 0 \ \omega_{xy}^{2}=4D\tau_{D}+t_{0}
  53. t 0 t_{0}
  54. t 0 = 0 t_{0}=0
  55. t 0 > 0 t_{0}>0
  56. t 0 < 0 t_{0}<0
  57. τ D = 3 π ω x y 2 η 2 k T ( M ) 1 / 3 \ \tau_{D}=\frac{3\pi\omega_{xy}^{2}\eta}{2kT}(M)^{1/3}
  58. η \ \eta
  59. M \ M
  60. ϵ \langle\epsilon\rangle
  61. σ 2 \sigma^{2}
  62. I \langle I\rangle
  63. ε = σ 2 - I I = i f i ε i \ \langle\varepsilon\rangle=\frac{\sigma^{2}-\langle I\rangle}{\langle I% \rangle}=\sum_{i}f_{i}\varepsilon_{i}
  64. f i f_{i}
  65. ϵ i \epsilon_{i}
  66. i i

Fluxon.html

  1. B c 1 B_{c_{1}}
  2. B c 2 B_{c_{2}}
  3. λ L \lambda_{L}
  4. Φ 0 \Phi_{0}
  5. λ L \lambda_{L}

Folium_of_Descartes.html

  1. x 3 + y 3 - 3 a x y = 0 x^{3}+y^{3}-3axy=0\,
  2. x + y + a = 0 x+y+a=0\,
  3. y = x y=x
  4. r = 3 a sin θ cos θ sin 3 θ + cos 3 θ . r=\frac{3a\sin\theta\cos\theta}{\sin^{3}\theta+\cos^{3}\theta}.
  5. x = 3 a p 1 + p 3 , y = 3 a p 2 1 + p 3 x={{3ap}\over{1+p^{3}}},\,y={{3ap^{2}}\over{1+p^{3}}}
  6. x = u + v 2 , y = u - v 2 x={{u+v}\over{\sqrt{2}}},\,y={{u-v}\over{\sqrt{2}}}
  7. v = ± u 3 a 2 - 2 u 6 u + 3 a 2 v=\pm u\sqrt{\frac{3a\sqrt{2}-2u}{6u+3a\sqrt{2}}}
  8. x 3 + y 3 = 3 a x y x^{3}+y^{3}=3axy\,
  9. x = X + Y 2 , y = X - Y 2 x={{X+Y}\over\sqrt{2}},y={{X-Y}\over\sqrt{2}}
  10. X , Y X,Y
  11. 2 X ( X 2 + 3 Y 2 ) = 3 2 a ( X 2 - Y 2 ) 2X(X^{2}+3Y^{2})=3\sqrt{2}a(X^{2}-Y^{2})
  12. Y Y
  13. 3 \sqrt{3}
  14. 2 X ( X 2 + Y 2 ) = a 2 ( 3 X 2 - Y 2 ) 2X(X^{2}+Y^{2})=a\sqrt{2}(3X^{2}-Y^{2})

Folk_theorem_(game_theory).html

  1. ( 1 - δ ) t 0 δ t u i ( h t ) , (1-\delta)\sum_{t\geq 0}\delta^{t}u_{i}(h_{t}),
  2. 1 1 - δ ϵ . \frac{1}{1-\delta}\epsilon.

Force-free_magnetic_field.html

  1. 0 = - p + 𝐣 × 𝐁 . 0=-\nabla p+\mathbf{j}\times\mathbf{B}.
  2. p p
  3. p B 2 / 2 μ p\ll B^{2}/2\mu
  4. 𝐣 × 𝐁 = 0 \mathbf{j}\times\mathbf{B}=0
  5. μ 0 𝐣 = α 𝐁 \mu_{0}\mathbf{j}=\alpha\mathbf{B}
  6. α \alpha
  7. × 𝐁 = μ 0 𝐣 \nabla\times\mathbf{B}=\mu_{0}\mathbf{j}
  8. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  9. ( × 𝐁 ) = 0 \nabla\cdot(\nabla\times\mathbf{B})=0
  10. α \alpha
  11. 𝐁 \mathbf{B}
  12. 𝐁 α = 0 \mathbf{B}\cdot\nabla\alpha=0
  13. × 𝐁 = α 𝐁 \nabla\times\mathbf{B}=\alpha\mathbf{B}
  14. 𝐣 = 0 \mathbf{j}=0
  15. × 𝐁 = 0 \nabla\times\mathbf{B}=0
  16. 𝐁 = ϕ \mathbf{B}=\nabla\phi
  17. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  18. 2 ϕ = 0 \nabla^{2}\phi=0
  19. μ 𝐣 = α 𝐁 \mu\mathbf{j}=\alpha\mathbf{B}
  20. × 𝐁 = α 𝐁 \nabla\times\mathbf{B}=\alpha\mathbf{B}
  21. α \alpha
  22. 𝐁 α = 0 \mathbf{B}\cdot\nabla\alpha=0
  23. × 𝐁 = α 𝐁 \nabla\times\mathbf{B}=\alpha\mathbf{B}
  24. × ( × 𝐁 ) = × ( α 𝐁 ) \nabla\times(\nabla\times\mathbf{B})=\nabla\times(\alpha\mathbf{B})
  25. × ( α 𝐁 ) = α ( × 𝐁 ) = α 2 𝐁 \nabla\times(\alpha\mathbf{B})=\alpha(\nabla\times\mathbf{B})=\alpha^{2}% \mathbf{B}
  26. × ( × 𝐁 ) = ( 𝐁 ) - 2 𝐁 = - 2 𝐁 \nabla\times(\nabla\times\mathbf{B})=\nabla(\nabla\cdot\mathbf{B})-\nabla^{2}% \mathbf{B}=-\nabla^{2}\mathbf{B}
  27. - 2 𝐁 = α 2 𝐁 -\nabla^{2}\mathbf{B}=\alpha^{2}\mathbf{B}
  28. × ( α 𝐁 ) = α ( × 𝐁 ) + α × 𝐁 = α 2 𝐁 + α × 𝐁 \nabla\times(\alpha\mathbf{B})=\alpha(\nabla\times\mathbf{B})+\nabla\alpha% \times\mathbf{B}=\alpha^{2}\mathbf{B}+\nabla\alpha\times\mathbf{B}
  29. 2 𝐁 + α 2 𝐁 = 𝐁 × α \nabla^{2}\mathbf{B}+\alpha^{2}\mathbf{B}=\mathbf{B}\times\nabla\alpha
  30. 𝐁 α = 0 \mathbf{B}\cdot\nabla\alpha=0

Force_field_(chemistry).html

  1. E total = E bonded + E nonbonded \ E_{\,\text{total}}=E_{\,\text{bonded}}+E_{\,\text{nonbonded}}
  2. E bonded = E bond + E angle + E dihedral \ E_{\,\text{bonded}}=E_{\,\text{bond}}+E_{\,\text{angle}}+E_{\,\text{dihedral}}
  3. E nonbonded = E electrostatic + E van der Waals \ E_{\,\text{nonbonded}}=E_{\,\text{electrostatic}}+E_{\,\text{van der Waals}}

Force_field_(physics).html

  1. F ( x ) \vec{F}(\vec{x})
  2. F \vec{F}
  3. x \vec{x}
  4. g = - G M r 2 r ^ \vec{g}=\frac{-GM}{r^{2}}\hat{r}
  5. r ^ \hat{r}
  6. F = m g \vec{F}=m\vec{g}
  7. E \vec{E}
  8. F = q E \vec{F}=q\vec{E}
  9. W = C F d r W=\int_{C}\vec{F}\cdot d\vec{r}
  10. C F d r = 0 \oint_{C}\vec{F}\cdot d\vec{r}=0
  11. F = ϕ \vec{F}=\nabla\phi
  12. W = ϕ ( b ) - ϕ ( a ) W=\phi(b)-\phi(a)

Foreign_exchange_swap.html

  1. F = S ( 1 + r d T 1 + r f T ) , F=S\left(\frac{1+r_{d}T}{1+r_{f}T}\right),
  2. F - S = S ( 1 + r d T 1 + r f T - 1 ) = S ( r d - r f ) T 1 + r f T S ( r d - r f ) T , F-S=S\left(\frac{1+r_{d}T}{1+r_{f}T}-1\right)=\frac{S(r_{d}-r_{f})T}{1+r_{f}T}% \approx S\left(r_{d}-r_{f}\right)T,
  3. r f T r_{f}T

Forest-fire_model.html

  1. f p T smax f\ll p\ll T_{\mathrm{smax}}\,
  2. f p f\ll p
  3. p T smax p\ll T_{\mathrm{smax}}

Formic_acid_(data_page).html

  1. log 10 P m m H g = 6.94459 - 1295.26 218.0 + T \scriptstyle\log_{10}P_{mmHg}=6.94459-\frac{1295.26}{218.0+T}

Formula_game.html

  1. Φ \Phi
  2. Φ \Phi

Forward_scattering_alignment.html

  1. S F S A = [ 1 0 0 - 1 ] S B S A S_{FSA}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}S_{BSA}

Fouling.html

  1. 𝖢𝖺 ( 𝖧𝖢𝖮 𝟥 ) 𝟤 ( a q u e o u s ) 𝖢𝖺𝖢𝖮 𝟥 + 𝖢𝖮 𝟤 + 𝖧 𝟤 𝖮 \mathsf{Ca(HCO_{3})_{2}}(aqueous)\longrightarrow\mathsf{CaCO_{3}}\downarrow+% \mathsf{CO_{2}}\uparrow+\mathsf{H_{2}O}
  2. d m d t = k t ( C b - C i ) \frac{dm}{dt}=k_{t}(C_{b}-C_{i})
  3. d m d t = k r ( C i - C e ) n 1 \frac{dm}{dt}={k_{r}}(C_{i}-C_{e})^{n1}
  4. d m d t = k d ( C b - C e ) n 2 \frac{dm}{dt}=k_{d}(C_{b}-C_{e})^{n2}
  5. k d = P k t k_{d}=Pk_{t}
  6. d m d t = k a C i \frac{dm}{dt}={k_{a}}C_{i}
  7. k d = ( 1 k a + 1 k t ) - 1 k_{d}=\left(\frac{1}{k_{a}}+\frac{1}{k_{t}}\right)^{-1}
  8. d m d t = k d C b \frac{dm}{dt}={k_{d}}C_{b}
  9. [ rate of deposit accumulation ] = [ rate of deposition ] - [ rate of re-entrainment of unconsolidated deposit ] \left[\begin{array}[]{c}\,\text{rate of}\\ \,\text{deposit}\\ \,\text{accumulation}\end{array}\right]=\left[\begin{array}[]{c}\,\text{rate % of}\\ \,\text{deposition}\end{array}\right]-\left[\begin{array}[]{c}\,\text{rate of}% \\ \,\text{re-entrainment of}\\ \,\text{unconsolidated deposit}\end{array}\right]
  10. [ rate of accumulation of unconsolidated deposit ] = [ rate of deposition ] - [ rate of re-entrainment of unconsolidated deposit ] - [ rate of consolidation of unconsolidated deposit ] \left[\begin{array}[]{c}\,\text{rate of}\\ \,\text{accumulation of}\\ \,\text{unconsolidated deposit}\end{array}\right]=\left[\begin{array}[]{c}\,% \text{rate of}\\ \,\text{deposition}\end{array}\right]-\left[\begin{array}[]{c}\,\text{rate of}% \\ \,\text{re-entrainment of}\\ \,\text{unconsolidated deposit}\end{array}\right]-\left[\begin{array}[]{c}\,% \text{rate of}\\ \,\text{consolidation of}\\ \,\text{unconsolidated deposit}\end{array}\right]
  11. { d m / d t = k d C m ρ - λ r m r ( t ) d m r / d t = k d C m ρ - λ r m r ( t ) - λ c m r ( t ) \left\{\begin{array}[]{c}{dm/dt}=k_{d}C_{m}\rho-\lambda_{r}m_{r}(t)\\ {dm_{r}/dt}=k_{d}C_{m}\rho-\lambda_{r}m_{r}(t)-\lambda_{c}\cdot m_{r}(t)\end{% array}\right.
  12. m ( t ) = k d C m ρ λ ( t λ c + λ r λ ( 1 - e - λ t ) ) m(t)={{k_{d}C_{m}\rho}\over{\lambda}}\left(t\lambda_{c}+{{\lambda_{r}}\over{% \lambda}}\left(1-e^{-\lambda t}\right)\right)
  13. m ( t ) = m * ( 1 - e - λ r t ) m(t)=m^{*}\left(1-e^{-\lambda_{r}t}\right)

Four-tensor.html

  1. A ν 1 , ν 2 , , ν m μ 1 , μ 2 , , μ n A^{\mu_{1},\mu_{2},...,\mu_{n}}_{\;\nu_{1},\nu_{2},...,\nu_{m}}
  2. x μ = ( x 0 , x 1 , x 2 , x 3 ) x^{\mu}=\left(x^{0},x^{1},x^{2},x^{3}\right)
  3. x 0 = c t x^{0}=ct
  4. c c
  5. x 0 x^{0}
  6. 𝐱 \mathbf{x}
  7. p μ = ( E / c , p x , p y , p z ) p^{\mu}=\left(E/c,p_{x},p_{y},p_{z}\right)
  8. p 0 = E / c p^{0}=E/c
  9. p 1 p^{1}
  10. p 2 p^{2}
  11. p 3 p^{3}
  12. 𝐩 \mathbf{p}
  13. F μ ν = ( 0 - E x / c - E y / c - E z / c E x / c 0 - B z B y E y / c B z 0 - B x E z / c - B y B x 0 ) F^{\mu\nu}=\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\ E_{x}/c&0&-B_{z}&B_{y}\\ E_{y}/c&B_{z}&0&-B_{x}\\ E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}

Fourier_number.html

  1. F o = diffusive transport rate storage rate Fo=\dfrac{\mbox{diffusive transport rate}~{}}{\mbox{storage rate}~{}}
  2. 𝐹𝑜 h = α t L 2 \mathit{Fo}_{h}=\frac{\alpha t}{L^{2}}
  3. 𝐹𝑜 m = D t L 2 \mathit{Fo}_{m}=\frac{Dt}{L^{2}}
  4. t = ρ c p V h A ln T 0 - T T - T t=\frac{\rho{{c}_{p}}V}{hA}\ln\frac{{{T}_{0}}-{{T}_{\infty}}}{T-{{T}_{\infty}}}

Fourier_transform_ion_cyclotron_resonance.html

  1. f = q B 2 π m f=\frac{qB}{2\pi m}
  2. ω c = q B m \omega_{c}=\frac{qB}{m}
  3. ω c \omega_{c}
  4. f = ω 2 π f=\frac{\omega}{2\pi}
  5. ω t = q α m \omega_{t}=\sqrt{{\frac{q\alpha}{m}}}
  6. α \alpha
  7. ω ± = ω c 2 ± ( ω c 2 ) 2 - ω t 2 2 \omega_{\pm}=\frac{\omega_{c}}{2}\pm\sqrt{\left({\frac{\omega_{c}}{2}}\right)^% {2}-{\frac{\omega_{t}^{2}}{2}}}
  8. ω t \omega_{t}
  9. ω + \omega_{+}
  10. ω - \omega_{-}
  11. ω + \omega_{+}
  12. ω t \omega_{t}
  13. ω c / 2 \omega_{c}/2
  14. ω + \omega_{+}
  15. ω c \omega_{c}
  16. ω - \omega_{-}
  17. ω c / 2 \omega_{c}/2
  18. ω c / 2 \omega_{c}/2
  19. ω c - ω + \omega_{c}-\omega_{+}

Frame_fields_in_general_relativity.html

  1. e 0 \vec{e}_{0}
  2. e 1 , e 2 , e 3 \vec{e}_{1},\vec{e}_{2},\,\vec{e}_{3}
  3. X = X j x j . \vec{X}=X^{j}\,\partial_{x^{j}}.
  4. X j X^{j}
  5. e a = e a j x j . \vec{e}_{a}={e_{a}}^{j}\,\partial_{x^{j}}.
  6. g = - σ 0 σ 0 + i = 1 3 σ i σ i , g=-\sigma^{0}\otimes\sigma^{0}+\sum_{i=1}^{3}\sigma^{i}\otimes\sigma^{i},
  7. \otimes
  8. e a μ e^{\mu}_{\ a}
  9. μ \mu\,
  10. a a\,
  11. g μ ν g^{\mu\nu}\,
  12. g μ ν = e a μ e b ν η a b g^{\mu\nu}=e^{\mu}_{\ a}e^{\nu}_{\ b}\eta^{ab}\,
  13. η a b \eta^{ab}\,
  14. T a = η a b T b . T^{a}=\eta^{ab}T_{b}.
  15. T a = e a μ T μ . T_{a}=e^{\mu}_{\ a}T_{\mu}.
  16. e a ν = e a μ e μ ν e^{\nu}_{\ a}=e^{\mu}_{\ a}e^{\nu}_{\ \mu}\,
  17. e μ ν = δ μ ν . e^{\nu}_{\ \mu}=\delta^{\nu}_{\mu}.
  18. T a = e μ a T μ . T^{a}=e_{\mu}^{\ a}T^{\mu}.
  19. T μ a = e ν a T μ ν . T^{\mu a}=e_{\nu}^{\ a}T^{\mu\nu}.
  20. T μ a = x μ x ν T ν a T^{\prime\mu a}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}T^{\nu a}
  21. T μ a = Λ ( x ) b a T μ b . T^{\prime\mu a}=\Lambda(x)^{a}_{\ b}T^{\mu b}.
  22. e 0 e 0 = 0 \nabla_{\vec{e}_{0}}\,\vec{e}_{0}=0
  23. e 0 e j = 0 , j = 0 3 \nabla_{\vec{e}_{0}}\,\vec{e}_{j}=0,\;\;j=0\dots 3
  24. e 0 e 0 0 \nabla_{\vec{e}_{0}}\,\vec{e}_{0}\neq 0
  25. e 0 e j , j = 1 3 \nabla_{\vec{e}_{0}}\,\vec{e}_{j},\;j=1\dots 3
  26. d s 2 = - ( 1 - 2 m / r ) d t 2 + d r 2 1 - 2 m / r + r 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) ds^{2}=-(1-2m/r)\,dt^{2}+\frac{dr^{2}}{1-2m/r}+r^{2}\,\left(d\theta^{2}+\sin(% \theta)^{2}\,d\phi^{2}\right)
  27. - < t < , 2 m < r < , 0 < θ < π , - π < ϕ < π -\infty<t<\infty,\;2m<r<\infty,\;0<\theta<\pi,\;-\pi<\phi<\pi
  28. g = - ( 1 - 2 m / r ) d t d t + 1 1 - 2 m / r d r d r + r 2 d θ d θ + r 2 sin ( θ ) 2 d ϕ d ϕ g=-(1-2m/r)\,dt\otimes dt+\frac{1}{1-2m/r}\,dr\otimes dr+r^{2}\,d\theta\otimes d% \theta+r^{2}\sin(\theta)^{2}\,d\phi\otimes d\phi
  29. σ 0 = - 1 - 2 m / r d t , σ 1 = d r 1 - 2 m / r , σ 2 = r d θ , σ 3 = r sin ( θ ) d ϕ \sigma^{0}=-\sqrt{1-2m/r}\,dt,\;\sigma^{1}=\frac{dr}{\sqrt{1-2m/r}},\;\sigma^{% 2}=rd\theta,\;\sigma^{3}=r\sin(\theta)d\phi
  30. g = - σ 0 σ 0 + σ 1 σ 1 + σ 2 σ 2 + σ 3 σ 3 g=-\sigma^{0}\otimes\sigma^{0}+\sigma^{1}\otimes\sigma^{1}+\sigma^{2}\otimes% \sigma^{2}+\sigma^{3}\otimes\sigma^{3}
  31. e 0 = 1 1 - 2 m / r t , e 1 = 1 - 2 m / r r , e 2 = 1 r θ , e 3 = 1 r sin ( θ ) ϕ \vec{e}_{0}=\frac{1}{\sqrt{1-2m/r}}\partial_{t},\;\vec{e}_{1}=\sqrt{1-2m/r}% \partial_{r},\;\vec{e}_{2}=\frac{1}{r}\partial_{\theta},\;\vec{e}_{3}=\frac{1}% {r\sin(\theta)}\partial_{\phi}
  32. σ 0 \sigma^{0}
  33. e 0 \vec{e}_{0}
  34. e 0 e 0 = m / r 2 1 - 2 m / r e 1 \nabla_{\vec{e}_{0}}\vec{e}_{0}=\frac{m/r^{2}}{\sqrt{1-2m/r}}\,\vec{e}_{1}
  35. e 0 \vec{e}_{0}
  36. E [ X ] a b = R a m b n X m X n E[X]_{ab}=R_{ambn}\,X^{m}\,X^{n}
  37. X = e 0 \vec{X}=\vec{e}_{0}
  38. E [ X ] 11 = - 2 m / r 3 , E [ X ] 22 = E [ X ] 33 = m / r 3 E[X]_{11}=-2m/r^{3},\;E[X]_{22}=E[X]_{33}=m/r^{3}
  39. E [ X ] r r = - 2 m / r 3 / ( 1 - 2 m / r ) , E [ X ] θ θ = m / r , E [ X ] ϕ ϕ = m sin ( θ ) 2 / r E[X]_{rr}=-2m/r^{3}/(1-2m/r),\;E[X]_{\theta\theta}=m/r,\;E[X]_{\phi\phi}=m\sin% (\theta)^{2}/r
  40. t , r , θ , ϕ t,r,\theta,\phi
  41. S a b = 36 m / r S_{ab}=36m/r
  42. Φ \Phi
  43. U U
  44. Φ i j = U , i j - 1 3 U , k , k η i j \Phi_{ij}=U_{,ij}-\frac{1}{3}{U^{,k}}_{,k}\,\eta_{ij}
  45. m / ( r + h ) 2 - m / r 2 = - 2 m / r 3 h + 3 m / r 4 h 2 + O ( h 3 ) m/(r+h)^{2}-m/r^{2}=-2m/r^{3}\,h+3m/r^{4}\,h^{2}+O(h^{3})
  46. Φ 11 = - 2 m / r 3 \Phi_{11}=-2m/r^{3}
  47. r = r 0 r=r_{0}
  48. m r 0 2 sin ( θ ) m r 0 2 h r 0 = m r 0 3 h \frac{m}{r_{0}^{2}}\,\sin(\theta)\approx\frac{m}{r_{0}^{2}}\,\frac{h}{r_{0}}=% \frac{m}{r_{0}^{3}}\,h
  49. O ( h 2 ) O(h^{2})
  50. Φ 22 = Φ 33 = m / r 3 \Phi_{22}=\Phi_{33}=m/r^{3}
  51. ϵ 1 = r , ϵ 2 = 1 r θ , ϵ 3 = 1 r sin θ ϕ \vec{\epsilon}_{1}=\partial_{r},\;\vec{\epsilon}_{2}=\frac{1}{r}\,\partial_{% \theta},\;\vec{\epsilon}_{3}=\frac{1}{r\sin\theta}\,\partial_{\phi}
  52. E [ X ] θ θ , E [ X ] ϕ ϕ E[X]_{\theta\theta},\,E[X]_{\phi\phi}
  53. e 1 \vec{e}_{1}
  54. f 0 = 1 1 - 2 m / r t - 2 m / r r \vec{f}_{0}=\frac{1}{1-2m/r}\,\partial_{t}-\sqrt{2m/r}\,\partial_{r}
  55. f 1 = r - 2 m / r 1 - 2 m / r t \vec{f}_{1}=\partial_{r}-\frac{\sqrt{2m/r}}{1-2m/r}\,\partial_{t}
  56. f 2 = 1 r θ \vec{f}_{2}=\frac{1}{r}\,\partial_{\theta}
  57. f 3 = 1 r sin ( θ ) ϕ \vec{f}_{3}=\frac{1}{r\sin(\theta)}\,\partial_{\phi}
  58. e 0 f 0 , e 1 f 1 \vec{e}_{0}\neq\vec{f}_{0},\;\vec{e}_{1}\neq\vec{f}_{1}
  59. e 0 \vec{e}_{0}
  60. e 0 \vec{e}_{0}
  61. r = 2 m r=2m
  62. T ( t , r ) = t - 2 m / r 1 - 2 m / r d r = t + 2 2 m r + 2 m log ( r - 2 m r + 2 m ) T(t,r)=t-\int\frac{\sqrt{2m/r}}{1-2m/r}\,dr=t+2\sqrt{2mr}+2m\log\left(\frac{% \sqrt{r}-\sqrt{2m}}{\sqrt{r}+\sqrt{2m}}\right)
  63. d s 2 = - d T 2 + ( d r + 2 m / r d T ) 2 + r 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) ds^{2}=-dT^{2}+\left(dr+\sqrt{2m/r}\,dT\right)^{2}+r^{2}\left(d\theta^{2}+\sin% (\theta)^{2}\,d\phi^{2}\right)
  64. - < T < , 0 < r < , 0 < θ < π , - π < ϕ < π -\infty<T<\infty,\;0<r<\infty,\;0<\theta<\pi,\;-\pi<\phi<\pi
  65. f 0 = T - 2 m / r r \vec{f}_{0}=\partial_{T}-\sqrt{2m/r}\,\partial_{r}
  66. f 1 = r \vec{f}_{1}=\partial_{r}
  67. f 2 = 1 r θ \vec{f}_{2}=\frac{1}{r}\,\partial_{\theta}
  68. f 3 = 1 r sin ( θ ) ϕ \vec{f}_{3}=\frac{1}{r\sin(\theta)}\,\partial_{\phi}
  69. T = T 0 T=T_{0}
  70. E [ Y ] a b = R a m b n Y m Y n E[Y]_{ab}=R_{ambn}\,Y^{m}\,Y^{n}
  71. Y = f 0 Y=\vec{f}_{0}
  72. E [ Y ] 11 = - 2 m / r 3 , E [ Y ] 22 = E [ Y ] 33 = m / r 3 E[Y]_{11}=-2m/r^{3},\,E[Y]_{22}=E[Y]_{33}=m/r^{3}
  73. e 3 \vec{e}_{3}
  74. θ = π / 2 \theta=\pi/2
  75. h 0 = 1 1 - 3 m / r t + m / r 3 1 - 3 m / r sin ( θ ) ϕ \vec{h}_{0}=\frac{1}{\sqrt{1-3m/r}}\,\partial_{t}+\frac{\sqrt{m/r^{3}}}{\sqrt{% 1-3m/r}\,\sin(\theta)}\,\partial_{\phi}
  76. h 1 = 1 - 2 m / r r \vec{h}_{1}=\sqrt{1-2m/r}\,\partial_{r}
  77. h 2 = 1 r θ \vec{h}_{2}=\frac{1}{r}\,\partial_{\theta}
  78. h 3 = 1 - 2 m / r 1 - 3 m / r sin ( θ ) ϕ - m / r 3 1 - 2 m / r 1 - 3 m / r t \vec{h}_{3}=\frac{\sqrt{1-2m/r}}{\sqrt{1-3m/r}\,\sin(\theta)}\,\partial_{\phi}% -\frac{\sqrt{m/r^{3}}}{\sqrt{1-2m/r}\,\sqrt{1-3m/r}}\,\partial_{t}
  79. h 0 = 1 1 - 3 m / r t + m / r 3 1 - 3 m / r ϕ \vec{h}_{0}=\frac{1}{\sqrt{1-3m/r}}\,\partial_{t}+\frac{\sqrt{m/r^{3}}}{\sqrt{% 1-3m/r}}\,\partial_{\phi}
  80. h 1 = 1 - 2 m / r r \vec{h}_{1}=\sqrt{1-2m/r}\,\partial_{r}
  81. h 2 = 1 r θ \vec{h}_{2}=\frac{1}{r}\,\partial_{\theta}
  82. h 3 = 1 - 2 m / r 1 - 3 m / r ϕ - m / r 3 1 - 2 m / r 1 - 3 m / r t \vec{h}_{3}=\frac{\sqrt{1-2m/r}}{\sqrt{1-3m/r}}\,\partial_{\phi}-\frac{\sqrt{m% /r^{3}}}{\sqrt{1-2m/r}\,\sqrt{1-3m/r}}\,\partial_{t}
  83. E [ Z ] a b E[Z]_{ab}
  84. Z = h 0 \vec{Z}=\vec{h}_{0}
  85. E [ Z ] 11 = - m r 3 2 - 3 m / r 1 - 2 m / r = - 2 m r 3 - m 2 r 4 + O ( 1 / r 5 ) E[Z]_{11}=-\frac{m}{r^{3}}\,\frac{2-3m/r}{1-2m/r}=-\frac{2m}{r^{3}}-\frac{m^{2% }}{r^{4}}+O(1/r^{5})
  86. E [ Z ] 22 = m r 3 1 1 - 3 m / r = - m r 3 + 3 m 2 r 4 + O ( 1 / r 5 ) E[Z]_{22}=\frac{m}{r^{3}}\,\frac{1}{1-3m/r}=-\frac{m}{r^{3}}+\frac{3m^{2}}{r^{% 4}}+O(1/r^{5})
  87. E [ Z ] 33 = m r 3 E[Z]_{33}=\frac{m}{r^{3}}
  88. r > 3 m r>3m
  89. r > 6 m r>6m
  90. h 1 , h 3 \vec{h}_{1},\;\vec{h}_{3}
  91. h 2 \vec{h}_{2}
  92. V V
  93. V V

Frank–Wolfe_algorithm.html

  1. 𝒟 \mathcal{D}
  2. f : 𝒟 f\colon\mathcal{D}\to\mathbb{R}
  3. f ( 𝐱 ) f(\mathbf{x})
  4. 𝐱 𝒟 \mathbf{x}\in\mathcal{D}
  5. k 0 k\leftarrow 0
  6. 𝐱 0 \mathbf{x}_{0}\!
  7. 𝒟 \mathcal{D}
  8. 𝐬 k \mathbf{s}_{k}
  9. 𝐬 T f ( 𝐱 k ) \mathbf{s}^{T}\nabla f(\mathbf{x}_{k})
  10. 𝐬 𝒟 \mathbf{s}\in\mathcal{D}
  11. f f
  12. 𝐱 k \mathbf{x}_{k}\!
  13. γ 2 k + 2 \gamma\leftarrow\frac{2}{k+2}
  14. γ \gamma
  15. f ( 𝐱 k + γ ( 𝐬 k - 𝐱 k ) ) f(\mathbf{x}_{k}+\gamma(\mathbf{s}_{k}-\mathbf{x}_{k}))
  16. 0 γ 1 0\leq\gamma\leq 1
  17. 𝐱 k + 1 𝐱 k + γ ( 𝐬 k - 𝐱 k ) \mathbf{x}_{k+1}\leftarrow\mathbf{x}_{k}+\gamma(\mathbf{s}_{k}-\mathbf{x}_{k})
  18. k k + 1 k\leftarrow k+1
  19. O ( 1 / k ) O(1/k)
  20. O ( 1 / k ) O(1/k)
  21. f f
  22. f ( 𝐲 ) f(\mathbf{y})
  23. f f
  24. 𝐱 𝒟 \mathbf{x}\in\mathcal{D}
  25. f ( 𝐲 ) f ( 𝐱 ) + ( 𝐲 - 𝐱 ) T f ( 𝐱 ) f(\mathbf{y})\geq f(\mathbf{x})+(\mathbf{y}-\mathbf{x})^{T}\nabla f(\mathbf{x})
  26. 𝐱 * \mathbf{x}^{*}
  27. 𝐱 \mathbf{x}
  28. f ( 𝐱 * ) min 𝐲 D f ( 𝐱 ) + ( 𝐲 - 𝐱 ) T f ( 𝐱 ) = f ( 𝐱 ) - 𝐱 T f ( 𝐱 ) + min 𝐲 D 𝐲 T f ( 𝐱 ) f(\mathbf{x}^{*})\geq\min_{\mathbf{y}\in D}f(\mathbf{x})+(\mathbf{y}-\mathbf{x% })^{T}\nabla f(\mathbf{x})=f(\mathbf{x})-\mathbf{x}^{T}\nabla f(\mathbf{x})+% \min_{\mathbf{y}\in D}\mathbf{y}^{T}\nabla f(\mathbf{x})
  29. 𝐬 k \mathbf{s}_{k}
  30. k k
  31. l k l_{k}
  32. l 0 = - l_{0}=-\infty
  33. l k := max ( l k - 1 , f ( 𝐱 k ) + ( 𝐬 k - 𝐱 k ) T f ( 𝐱 k ) ) l_{k}:=\max(l_{k-1},f(\mathbf{x}_{k})+(\mathbf{s}_{k}-\mathbf{x}_{k})^{T}% \nabla f(\mathbf{x}_{k}))
  34. l k f ( 𝐱 * ) f ( 𝐱 k ) l_{k}\leq f(\mathbf{x}^{*})\leq f(\mathbf{x}_{k})
  35. f ( 𝐱 k ) f(\mathbf{x}_{k})
  36. l k l_{k}
  37. f ( 𝐱 k ) - l k = O ( 1 / k ) . f(\mathbf{x}_{k})-l_{k}=O(1/k).

Fredholm_alternative.html

  1. T : V V T:V\to V
  2. T ( u ) = v T(u)=v
  3. dim ( ker ( T ) ) > 0 \dim(\ker(T))>0
  4. ( 𝐛 Im ( A ) ) (\mathbf{b}\in\operatorname{Im}(A))
  5. ( 𝐛 ker ( A T ) ) (\mathbf{b}\in\ker(A^{T})^{\bot})
  6. K ( x , y ) K(x,y)
  7. λ φ ( x ) - a b K ( x , y ) φ ( y ) d y = 0 \lambda\varphi(x)-\int_{a}^{b}K(x,y)\varphi(y)\,dy=0
  8. λ φ ( x ) - a b K ( x , y ) φ ( y ) d y = f ( x ) . \lambda\varphi(x)-\int_{a}^{b}K(x,y)\varphi(y)\,dy=f(x).
  9. λ \lambda\in\mathbb{C}
  10. f ( x ) f(x)
  11. K ( x , y ) K(x,y)
  12. [ a , b ] × [ a , b ] [a,b]\times[a,b]
  13. T = λ - K T=\lambda-K
  14. T ( x , y ) = λ δ ( x - y ) - K ( x , y ) T(x,y)=\lambda\;\delta(x-y)-K(x,y)
  15. δ ( x - y ) \delta(x-y)
  16. ϕ ( x ) \phi(x)
  17. T : V V T:V\to V
  18. ϕ ψ \phi\mapsto\psi
  19. ψ \psi
  20. ψ ( x ) = a b T ( x , y ) ϕ ( y ) d y = λ ϕ ( x ) - a b K ( x , y ) ϕ ( y ) d y . \psi(x)=\int_{a}^{b}T(x,y)\phi(y)\,dy=\lambda\;\phi(x)-\int_{a}^{b}K(x,y)\phi(% y)\,dy.
  21. λ \lambda
  22. R ( λ ; K ) = ( K - λ Id ) - 1 . R(\lambda;K)=(K-\lambda\operatorname{Id})^{-1}.
  23. L u = f , u d o m ( L ) X , Lu=f,\qquad u\in dom(L)\subseteq X,
  24. ( * ) L u := - Δ u + h ( x ) u = f in Ω , (*)\qquad Lu:=-\Delta u+h(x)u=f\qquad\,\text{in }\Omega,
  25. ( * * ) u = 0 on Ω , (**)\qquad u=0\qquad\,\text{on }\partial\Omega,

Fredholm_determinant.html

  1. ( I + T ) - 1 - I = - T ( I + T ) - 1 , (I+T)^{-1}-I=-T(I+T)^{-1},
  2. ( , ) (\cdot,\cdot)
  3. Λ k H \Lambda^{k}H
  4. ( v 1 v 2 v k , w 1 w 2 w k ) = det ( v i , w j ) . (v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k},w_{1}\wedge w_{2}\wedge\cdots\wedge w% _{k})={\rm det}\,(v_{i},w_{j}).
  5. e i 1 e i 2 e i k , ( i 1 < i 2 < < i k ) e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{k}},\qquad(i_{1}<i_{2}<% \cdots<i_{k})
  6. Λ k H \Lambda^{k}H
  7. Λ k ( A ) \Lambda^{k}(A)
  8. Λ k H \Lambda^{k}H
  9. Λ k ( A ) v 1 v 2 v k = A v 1 A v 2 A v k . \Lambda^{k}(A)v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k}=Av_{1}\wedge Av_{2}% \wedge\cdots\wedge Av_{k}.
  10. Λ k ( A ) \Lambda^{k}(A)
  11. Λ k ( A ) 1 A 1 k / k ! . \|\Lambda^{k}(A)\|_{1}\leq\|A\|_{1}^{k}/k!.
  12. det ( I + A ) = k = 0 Tr Λ k ( A ) {\rm det}\,(I+A)=\sum_{k=0}^{\infty}{\rm Tr}\Lambda^{k}(A)
  13. det ( I + z A ) = k = 0 z k Tr Λ k ( A ) {\rm det}\,(I+zA)=\sum_{k=0}^{\infty}z^{k}{\rm Tr}\Lambda^{k}(A)
  14. | det ( I + z A ) | exp ( | z | A 1 ) . |{\rm det}\,(I+zA)|\leq\exp(|z|\cdot\|A\|_{1}).
  15. | det ( I + A ) - det ( I + B ) | A - B 1 exp ( A 1 + B 1 + 1 ) . |{\rm det}(I+A)-{\rm det}(I+B)|\leq\|A-B\|_{1}\exp(\|A\|_{1}+\|B\|_{1}+1).
  16. | det ( I + A ) - det ( I + B ) | A - B 1 exp ( max ( A 1 , B 1 ) + 1 ) . |{\rm det}(I+A)-{\rm det}(I+B)|\leq\|A-B\|_{1}\exp(\max(\|A\|_{1},\|B\|_{1})+1).
  17. det ( I + A ) det ( I + B ) = det ( I + A ) ( I + B ) . {\rm det}(I+A)\cdot{\rm det}(I+B)={\rm det}(I+A)(I+B).
  18. det X T X - 1 = det T . {\rm det}\,XTX^{-1}={\rm det}\,T.
  19. det e A = exp Tr ( A ) . {\rm det}\,e^{A}=\exp\,{\rm Tr}(A).
  20. log det ( I + z A ) = Tr ( log ( I + z A ) ) = k = 1 ( - 1 ) k + 1 Tr A k k z k \log{\rm det}\,(I+zA)={\rm Tr}(\log{(I+zA)})=\sum_{k=1}^{\infty}(-1)^{k+1}% \frac{{\rm Tr}A^{k}}{k}z^{k}
  21. F ˙ ( t ) = lim h 0 F ( t + h ) - F ( t ) h \dot{F}(t)=\lim_{h\rightarrow 0}{F(t+h)-F(t)\over h}
  22. F - 1 F ˙ = id - exp - ad g ( t ) ad g ( t ) g ˙ ( t ) , F^{-1}\dot{F}={{\rm id}-\exp-{\rm ad}g(t)\over{\rm ad}g(t)}\cdot\dot{g}(t),
  23. ad ( X ) Y = X Y - Y X . {\rm ad}(X)\cdot Y=XY-YX.
  24. f - 1 f ˙ = Tr F - 1 F ˙ . f^{-1}\dot{f}={\rm Tr}F^{-1}\dot{F}.
  25. det e A e B e - A e - B = exp Tr ( A B - B A ) . {\rm det}\,e^{A}e^{B}e^{-A}e^{-B}=\exp{\rm Tr}(AB-BA).
  26. T ( f ) = P m ( f ) P , T(f)=Pm(f)P,
  27. T ( f ) T ( g ) - T ( g ) T ( f ) T(f)T(g)-T(g)T(f)
  28. tr ( T ( f ) T ( g ) - T ( g ) T ( f ) ) = 1 2 π i 0 2 π f d g . {\rm tr}(T(f)T(g)-T(g)T(f))={1\over 2\pi i}\int_{0}^{2\pi}fdg.
  29. T ( e f + g ) T ( e - f ) T ( e - g ) T(e^{f+g})T(e^{-f})T(e^{-g})
  30. det T ( e f ) T ( e - f ) = exp n > 0 n a n a - n , {\rm det}\,T(e^{f})T(e^{-f})=\exp\sum_{n>0}na_{n}a_{-n},
  31. f ( z ) = a n z n . f(z)=\sum a_{n}z^{n}.
  32. lim N det P N m ( e f ) P N = exp n > 0 n a n a - n , \lim_{N\rightarrow\infty}{\rm det}P_{N}m(e^{f})P_{N}=\exp\sum_{n>0}na_{n}a_{-n},
  33. det ( I - λ T ) = n = 0 ( - λ ) n Tr Λ n ( T ) = exp ( - n = 1 Tr ( T n ) n λ n ) \det(I-\lambda T)=\sum_{n=0}^{\infty}(-\lambda)^{n}\operatorname{Tr}\Lambda^{n% }(T)=\exp{\left(-\sum_{n=1}^{\infty}\frac{\operatorname{Tr}(T^{n})}{n}\lambda^% {n}\right)}
  34. Tr T = K ( x , x ) d x \operatorname{Tr}T=\int K(x,x)\,dx
  35. Tr Λ 2 ( T ) = 1 2 ! K ( x , x ) K ( y , y ) - K ( x , y ) K ( y , x ) d x d y \operatorname{Tr}\Lambda^{2}(T)=\frac{1}{2!}\iint K(x,x)K(y,y)-K(x,y)K(y,x)\,dxdy
  36. Tr Λ n ( T ) = 1 n ! det K ( x i , x j ) | 1 i , j n d x 1 d x n \operatorname{Tr}\Lambda^{n}(T)=\frac{1}{n!}\int\cdots\int\det K(x_{i},x_{j})|% _{1\leq i,j\leq n}\,dx_{1}\cdots dx_{n}

Fredholm_kernel.html

  1. B * B B^{*}\otimes B
  2. X π = inf { i } e i * e i \|X\|_{\pi}=\inf\sum_{\{i\}}\|e^{*}_{i}\|\|e_{i}\|
  3. X = { i } e i * e i B * B X=\sum_{\{i\}}e^{*}_{i}\otimes e_{i}\in B^{*}\otimes B
  4. B * ^ π B B^{*}\widehat{\,\otimes\,}_{\pi}B
  5. X = { i } λ i e i * e i X=\sum_{\{i\}}\lambda_{i}e^{*}_{i}\otimes e_{i}
  6. e i B e_{i}\in B
  7. e i * B * e^{*}_{i}\in B^{*}
  8. e i = e i * = 1 \|e_{i}\|=\|e^{*}_{i}\|=1
  9. { i } | λ i | < . \sum_{\{i\}}|\lambda_{i}|<\infty.\,
  10. X : B B \mathcal{L}_{X}:B\to B
  11. X f = { i } λ i e i * ( f ) e i . \mathcal{L}_{X}f=\sum_{\{i\}}\lambda_{i}e^{*}_{i}(f)e_{i}.\,
  12. tr X = { i } λ i e i * ( e i ) . \mbox{tr}~{}X=\sum_{\{i\}}\lambda_{i}e^{*}_{i}(e_{i}).\,
  13. { i } | λ i | p < \sum_{\{i\}}|\lambda_{i}|^{p}<\infty
  14. 0 < p 1 0<p\leq 1
  15. : B B \mathcal{L}:B\to B
  16. X B * ^ π B X\in B^{*}\widehat{\,\otimes\,}_{\pi}B
  17. = X \mathcal{L}=\mathcal{L}_{X}
  18. q 2 / 3 q\leq 2/3
  19. : B B \mathcal{L}:B\to B
  20. q 2 / 3 q\leq 2/3
  21. Tr = { i } ρ i \mbox{Tr}~{}\mathcal{L}=\sum_{\{i\}}\rho_{i}
  22. ρ i \rho_{i}
  23. \mathcal{L}
  24. det ( 1 - z ) = i ( 1 - ρ i z ) \det\left(1-z\mathcal{L}\right)=\prod_{i}\left(1-\rho_{i}z\right)
  25. det ( 1 - z ) = exp Tr log ( 1 - z ) \det\left(1-z\mathcal{L}\right)=\exp\mbox{Tr}~{}\log\left(1-z\mathcal{L}\right)
  26. \mathcal{L}
  27. = w \mathcal{L}=\mathcal{L}_{w}
  28. det ( 1 - z w ) \det\left(1-z\mathcal{L}_{w}\right)
  29. D k D\subset\mathbb{C}^{k}

Free-radical_reaction.html

  1. rate = k obs [ I ] 3 / 2 \ \,\text{rate}=k_{\,\text{obs}}[I]^{3/2}\,

Free_Boolean_algebra.html

  1. 2 2 n 2^{2^{n}}
  2. 0 \aleph_{0}
  3. 2 2 n 2^{2^{n}}
  4. 0 \aleph_{0}
  5. 0 \aleph_{0}

Free_Lie_algebra.html

  1. N k = 1 k d | k μ ( d ) m k / d , N_{k}=\frac{1}{k}\sum_{d|k}\mu(d)\cdot m^{k/d},
  2. μ \mu

Freiman's_theorem.html

  1. A + A A+A\,
  2. | A + A | < c | A | |A+A|<c|A|\,
  3. c c
  4. c | A | c^{\prime}|A|\,
  5. | A + A | 2 | A | - 1 |A+A|\geq 2|A|-1

Frequency-resolved_optical_gating.html

  1. ω \omega
  2. τ \tau
  3. I FROG ( ω , τ ) = | E sig ( ω , τ ) | 2 = | F T [ E sig ( t , τ ) ] | 2 = | - E s i g ( t , τ ) e - i ω t d t | 2 . I\text{FROG}(\omega,\tau)=\left|E\text{sig}(\omega,\tau)\right|^{2}=\left|FT[E% \text{sig}(t,\tau)]\right|^{2}=\left|\int_{-\infty}^{\infty}E_{sig}(t,\tau)e^{% -i\omega t}\,dt\right|^{2}.
  4. E sig ( t , τ ) E\text{sig}(t,\tau)
  5. E ( t ) E(t)
  6. E gate ( t - τ ) E\text{gate}(t-\tau)
  7. E sig ( t , τ ) = E ( t ) E gate ( t - τ ) E\text{sig}(t,\tau)=E(t)E\text{gate}(t-\tau)
  8. E gate ( t - τ ) = E ( t - τ ) E\text{gate}(t-\tau)=E(t-\tau)
  9. I SHG FROG ( ω , τ ) = | - E ( t ) E ( t - τ ) e - i ω t d t | 2 . I\text{SHG FROG}(\omega,\tau)=\left|\int_{-\infty}^{\infty}E(t)E(t-\tau)e^{-i% \omega t}\,dt\right|^{2}.
  10. E gate ( t - τ ) E\text{gate}(t-\tau)
  11. E sig ( t , τ ) = E ( t ) E ( t - τ ) E\text{sig}(t,\tau)=E(t)E(t-\tau)
  12. E sig ( t , τ ) E\text{sig}(t,\tau)
  13. E sig ( ω , τ ) E\text{sig}(\omega,\tau)
  14. E sig ( ω , τ ) E\text{sig}(\omega,\tau)
  15. E sig ( ω , τ ) E\text{sig}(\omega,\tau)
  16. E ( t ) E(t)
  17. E sig ( t , τ ) E\text{sig}(t,\tau)
  18. τ \tau

Fresnel_diffraction.html

  1. F F
  2. F 1 F\gg 1
  3. F θ 2 / 4 1 F\theta^{2}/4\ll 1
  4. θ \theta
  5. θ a / L \theta\approx a/L
  6. a a
  7. L L
  8. E ( x , y , z ) = z i λ - + E ( x , y , 0 ) e i k r r 2 d x d y E(x,y,z)={z\over{i\lambda}}\iint_{-\infty}^{+\infty}{E(x^{\prime},y^{\prime},0% )\frac{e^{ikr}}{r^{2}}}dx^{\prime}dy^{\prime}
  9. E ( x , y , 0 ) E(x^{\prime},y^{\prime},0)
  10. r = ( x - x ) 2 + ( y - y ) 2 + z 2 r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2}}
  11. i i\,
  12. ρ 2 = ( x - x ) 2 + ( y - y ) 2 \rho^{2}=(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\,
  13. r = ρ 2 + z 2 = z 1 + ρ 2 z 2 r=\sqrt{\rho^{2}+z^{2}}=z\sqrt{1+\frac{\rho^{2}}{z^{2}}}
  14. 1 + u = ( 1 + u ) 1 / 2 = 1 + u 2 - u 2 8 + \sqrt{1+u}=(1+u)^{1/2}=1+\frac{u}{2}-\frac{u^{2}}{8}+\cdots
  15. r = z 1 + ρ 2 z 2 r=z\sqrt{1+\frac{\rho^{2}}{z^{2}}}
  16. = z [ 1 + ρ 2 2 z 2 - 1 8 ( ρ 2 z 2 ) 2 + ] =z\left[1+\frac{\rho^{2}}{2z^{2}}-\frac{1}{8}\left(\frac{\rho^{2}}{z^{2}}% \right)^{2}+\cdots\right]
  17. = z + ρ 2 2 z - ρ 4 8 z 3 + =z+\frac{\rho^{2}}{2z}-\frac{\rho^{4}}{8z^{3}}+\cdots
  18. 2 π 2\pi
  19. k ρ 4 8 z 3 2 π k\frac{\rho^{4}}{8z^{3}}\ll 2\pi
  20. k = 2 π λ k={2\pi\over\lambda}\,
  21. ρ 4 z 3 λ 8 \frac{\rho^{4}}{z^{3}\lambda}\ll 8
  22. z 3 / λ 3 z^{3}/\lambda^{3}
  23. ρ 4 λ 4 8 z 3 λ 3 \frac{\rho^{4}}{\lambda^{4}}\ll 8{z^{3}\over\lambda^{3}}
  24. [ ( x - x ) 2 + ( y - y ) 2 ] 2 λ 4 8 z 3 λ 3 \frac{[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]^{2}}{\lambda^{4}}\ll 8{z^{3}% \over\lambda^{3}}
  25. λ z \lambda\ll z
  26. λ ρ \lambda\ll\rho
  27. ρ z \rho\ll z
  28. r z + ρ 2 2 z = z + ( x - x ) 2 + ( y - y ) 2 2 z r\approx z+\frac{\rho^{2}}{2z}=z+\frac{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}{% 2z}
  29. F = a 2 L λ 1 F=\frac{a^{2}}{L\lambda}\geq 1
  30. F = a 2 L λ 1 F=\frac{a^{2}}{L\lambda}\ll 1
  31. a a
  32. λ \lambda
  33. L L
  34. r z r\approx z
  35. L λ L\gg\lambda
  36. E ( x , y , z ) = e i k z i λ z - + E ( x , y , 0 ) e i k 2 z [ ( x - x ) 2 + ( y - y ) 2 ] d x d y E(x,y,z)=\frac{e^{ikz}}{i\lambda z}\iint_{-\infty}^{+\infty}E(x^{\prime},y^{% \prime},0)e^{{ik\over 2z}[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]}dx^{\prime}dy% ^{\prime}
  37. h ( x , y , z ) = e i k z i λ z e i k 2 z ( x 2 + y 2 ) h(x,y,z)=\frac{e^{ikz}}{i\lambda z}e^{i\frac{k}{2z}(x^{2}+y^{2})}
  38. E ( x , y , z ) = E ( x , y , 0 ) * h ( x , y , z ) E(x,y,z)=E(x,y,0)*h(x,y,z)\,
  39. k = 2 π λ k={2\pi\over\lambda}
  40. ( x - x ) 2 = x 2 + x 2 - 2 x x (x-x^{\prime})^{2}=x^{2}+x^{\prime 2}-2xx^{\prime}\,
  41. ( y - y ) 2 = y 2 + y 2 - 2 y y (y-y^{\prime})^{2}=y^{2}+y^{\prime 2}-2yy^{\prime}\,
  42. G ( p , q ) = { g ( x , y ) } - g ( x , y ) e - i 2 π ( p x + q y ) d x d y G(p,q)=\mathcal{F}\left\{g(x,y)\right\}\equiv\iint_{-\infty}^{\infty}g(x,y)e^{% -i2\pi(px+qy)}dxdy
  43. E ( x , y , z ) = e i k z i λ z e i π λ z ( x 2 + y 2 ) { E ( x , y , 0 ) e i π λ z ( x 2 + y 2 ) } | p = x λ z ; q = y λ z E(x,y,z)=\frac{e^{ikz}}{i\lambda z}e^{i\frac{\pi}{\lambda z}(x^{2}+y^{2})}% \mathcal{F}\left.\left\{E(x^{\prime},y^{\prime},0)e^{i\frac{\pi}{\lambda z}(x^% {\prime 2}+y^{\prime 2})}\right\}\right|_{p=\frac{x}{\lambda z};q=\frac{y}{% \lambda z}}
  44. = H ( p , q ) G ( p , q ) | p = x λ z , q = y λ z =H(p,q)\cdot G(p,q)|_{p={x\over\lambda z},q={y\over\lambda z}}
  45. H ( p , q ) = { h ( x , y ) } H(p,q)=\mathcal{F}\left\{h(x,y)\right\}
  46. ( x λ z , y λ z ) \left(\frac{x}{\lambda z},\frac{y}{\lambda z}\right)
  47. e i k r / r e^{ikr}/r

Fresnel_number.html

  1. F = a 2 L λ F=\frac{a^{2}}{L\lambda}
  2. a a\!
  3. L L\!
  4. λ \lambda\!
  5. π \pi
  6. F 1 \ F\gg 1
  7. F 1 \ F\sim 1
  8. F 1 \ F\ll 1
  9. 1 \leq 1
  10. > 1 >1

Fréchet_derivative.html

  1. U V U\subset V
  2. x U x\in U
  3. A : V W A:V\to W
  4. lim h 0 f ( x + h ) - f ( x ) - A h W h V = 0. \lim_{h\to 0}\frac{\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}=0.
  5. h n n = 1 \langle h_{n}\rangle_{n=1}^{\infty}
  6. h n 0. h_{n}{\rightarrow}0.
  7. f ( x + h ) = f ( x ) + A h + o ( h ) . f(x+h)=f(x)+Ah+o(h).
  8. D f ( x ) = A Df(x)=A
  9. D f : U B ( V , W ) ; x D f ( x ) Df:U\to B(V,W);x\mapsto Df(x)
  10. D f ( x ) : V W Df(x):V\to W
  11. x x
  12. t t f ( x ) t\mapsto tf^{\prime}(x)
  13. D ( g f ) ( x ) = D g ( f ( x ) ) D f ( x ) . D(g\circ f)(x)=Dg(f(x))\circ Df(x).
  14. D f ( a ) : 𝐑 n 𝐑 m with D f ( a ) ( v ) = J f ( a ) v , Df(a):\mathbf{R}^{n}\to\mathbf{R}^{m}\quad\mbox{with}~{}\quad Df(a)(v)=J_{f}(a% )\,v,
  15. f x i ( a ) = D f ( a ) ( e i ) = J f ( a ) e i , \frac{\partial f}{\partial x_{i}}(a)=Df(a)(e_{i})=J_{f}(a)\,e_{i},
  16. D f ( a ) ( h ) = i = 1 n h i f x i ( a ) . Df(a)(h)=\sum_{i=1}^{n}h_{i}\frac{\partial f}{\partial x_{i}}(a).
  17. f ( x , y ) = { ( x 2 + y 2 ) sin ( 1 x 2 + y 2 ) if ( x , y ) ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f(x,y)=\begin{cases}(x^{2}+y^{2})\sin(\frac{1}{\sqrt{x^{2}+y^{2}}})&\mbox{ if % }~{}(x,y)\neq(0,0)\\ 0&\mbox{ if }~{}(x,y)=(0,0)\end{cases}
  18. ( 0 , 0 ) (0,0)
  19. g ( h ) = lim t 0 f ( x + t h ) - f ( x ) t g(h)=\lim_{t\to 0}\frac{f(x+th)-f(x)}{t}
  20. f ( x , y ) = { x 3 x 2 + y 2 if ( x , y ) ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f(x,y)=\begin{cases}\frac{x^{3}}{x^{2}+y^{2}}&\mbox{ if }~{}(x,y)\neq(0,0)\\ 0&\mbox{ if }~{}(x,y)=(0,0)\end{cases}
  21. g ( a , b ) = { a 3 a 2 + b 2 if ( a , b ) ( 0 , 0 ) 0 if ( a , b ) = ( 0 , 0 ) . g(a,b)=\begin{cases}\frac{a^{3}}{a^{2}+b^{2}}&\mbox{ if }~{}(a,b)\neq(0,0)\\ 0&\mbox{ if }~{}(a,b)=(0,0).\end{cases}
  22. f ( x , y ) = g ( r ) h ( ϕ ) f(x,y)=g(r)h(\phi)
  23. h ( ϕ + π ) = - h ( ϕ ) h(\phi+\pi)=-h(\phi)
  24. f ( x , y ) = { x 3 y x 6 + y 2 if ( x , y ) ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f(x,y)=\begin{cases}\frac{x^{3}y}{x^{6}+y^{2}}&\mbox{ if }~{}(x,y)\neq(0,0)\\ 0&\mbox{ if }~{}(x,y)=(0,0)\end{cases}
  25. f ( x , y ) = { x 2 y x 4 + y 2 x 2 + y 2 if ( x , y ) ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) f(x,y)=\begin{cases}\frac{x^{2}y}{x^{4}+y^{2}}\sqrt{x^{2}+y^{2}}&\mbox{ if }~{% }(x,y)\neq(0,0)\\ 0&\mbox{ if }~{}(x,y)=(0,0)\end{cases}
  26. lim ( x , y ) ( 0 , 0 ) | x 2 y x 4 + y 2 | \lim_{(x,y)\to(0,0)}\left|\frac{x^{2}y}{x^{4}+y^{2}}\right|
  27. f ( x ) = x φ ( x ) . f(x)=\|x\|\varphi(x).\,
  28. lim x 0 φ ( x ) \lim_{x\to 0}\varphi(x)
  29. D f : U L ( V , W ) Df:U\to L(V,W)\,
  30. D 2 f : U L ( V , L ( V , W ) ) . D^{2}f:U\to L\big(V,L(V,W)\big).
  31. φ ( x ) ( y ) = ψ ( x , y ) \varphi(x)(y)=\psi(x,y)\,
  32. D 2 f : U L 2 ( V × V , W ) D^{2}f:U\to L^{2}(V\times V,W)\,
  33. D n f : U L n ( V × V × × V , W ) , D^{n}f:U\to L^{n}(V\times V\times\cdots\times V,W),
  34. lim h n + 1 0 D n f ( x + h n + 1 ) ( h 1 , h 2 , , h n ) - D n f ( x ) ( h 1 , h 2 , , h n ) - A ( h 1 , h 2 , , h n , h n + 1 ) h n + 1 = 0 \lim_{h_{n+1}\to 0}\frac{\|D^{n}f(x+h_{n+1})(h_{1},h_{2},\dots,h_{n})-D^{n}f(x% )(h_{1},h_{2},\dots,h_{n})-A(h_{1},h_{2},\dots,h_{n},h_{n+1})\|}{\|h_{n+1}\|}=0
  35. L n ( V × V × × V , W ) L^{n}(V\times V\times\cdots\times V,W)
  36. L ( j = 1 n V j , W ) L(\bigotimes_{j=1}^{n}V_{j},W)
  37. f ( x 1 , x 2 , , x n ) = f ( x 1 x 2 x n ) f(x_{1},x_{2},\ldots,x_{n})=f(x_{1}\otimes x_{2}\otimes\cdots\otimes x_{n})
  38. f : U Y f:U\to Y
  39. f ( 0 ) = 0 f(0)=0
  40. W \sub Y W\sub Y
  41. V \sub X V\sub X
  42. o : o:\mathbb{R}\to\mathbb{R}
  43. lim t 0 o ( t ) t = 0 , \lim_{t\to 0}\frac{o(t)}{t}=0,\,
  44. f ( t V ) \sub o ( t ) W f(tV)\sub o(t)W
  45. f ( 0 ) = 0 f(0)=0
  46. x 0 U x_{0}\in U
  47. λ : X Y \lambda:X\to Y
  48. f ( x 0 + h ) - f ( x 0 ) - λ h f(x_{0}+h)-f(x_{0})-\lambda h
  49. v X v\in X
  50. lim τ 0 f ( x 0 + τ v ) - f ( x 0 ) τ = f ( x 0 ) v \lim_{\tau\to 0}\frac{f(x_{0}+\tau v)-f(x_{0})}{\tau}=f^{\prime}(x_{0})v
  51. f ( x 0 ) f^{\prime}(x_{0})

Friedman_test.html

  1. { x i j } n × k \{x_{ij}\}_{n\times k}
  2. n n
  3. k k
  4. { r i j } n × k \{r_{ij}\}_{n\times k}
  5. r i j r_{ij}
  6. x i j x_{ij}
  7. i i
  8. r ¯ j = 1 n i = 1 n r i j \bar{r}_{\cdot j}=\frac{1}{n}\sum_{i=1}^{n}{r_{ij}}
  9. r ¯ = 1 n k i = 1 n j = 1 k r i j \bar{r}=\frac{1}{nk}\sum_{i=1}^{n}\sum_{j=1}^{k}r_{ij}
  10. S S t = n j = 1 k ( r ¯ j - r ¯ ) 2 SS_{t}=n\sum_{j=1}^{k}(\bar{r}_{\cdot j}-\bar{r})^{2}
  11. S S e = 1 n ( k - 1 ) i = 1 n j = 1 k ( r i j - r ¯ ) 2 SS_{e}=\frac{1}{n(k-1)}\sum_{i=1}^{n}\sum_{j=1}^{k}(r_{ij}-\bar{r})^{2}
  12. Q = S S t S S e Q=\frac{SS_{t}}{SS_{e}}
  13. 𝐏 ( χ k - 1 2 Q ) \mathbf{P}(\chi^{2}_{k-1}\geq Q)

Friis_transmission_equation.html

  1. P r P_{r}
  2. P t P_{t}
  3. P r P t = G t G r ( λ 4 π R ) 2 \frac{P_{r}}{P_{t}}=G_{t}G_{r}\left(\frac{\lambda}{4\pi R}\right)^{2}
  4. G t G_{t}
  5. G r G_{r}
  6. λ \lambda
  7. R R
  8. P r = P t + G t + G r + 20 log 10 ( λ 4 π R ) P_{r}=P_{t}+G_{t}+G_{r}+20\log_{10}\left(\frac{\lambda}{4\pi R}\right)
  9. R λ R\gg\lambda
  10. R R
  11. λ \lambda
  12. R < λ R<\lambda
  13. P r P_{r}
  14. P t P_{t}
  15. P r P t = G t ( θ t , ϕ t ) G r ( θ r , ϕ r ) ( λ 4 π R ) 2 ( 1 - | Γ t | 2 ) ( 1 - | Γ r | 2 ) | 𝐚 t 𝐚 r * | 2 e - α R \frac{P_{r}}{P_{t}}=G_{t}(\theta_{t},\phi_{t})G_{r}(\theta_{r},\phi_{r})\left(% \frac{\lambda}{4\pi R}\right)^{2}(1-|\Gamma_{t}|^{2})(1-|\Gamma_{r}|^{2})|% \mathbf{a}_{t}\cdot\mathbf{a}_{r}^{*}|^{2}e^{-\alpha R}
  16. G t ( θ t , ϕ t ) G_{t}(\theta_{t},\phi_{t})
  17. ( θ t , ϕ t ) (\theta_{t},\phi_{t})
  18. G r ( θ r , ϕ r ) G_{r}(\theta_{r},\phi_{r})
  19. ( θ r , ϕ r ) (\theta_{r},\phi_{r})
  20. Γ t \Gamma_{t}
  21. Γ r \Gamma_{r}
  22. 𝐚 t \mathbf{a}_{t}
  23. 𝐚 r \mathbf{a}_{r}
  24. α \alpha
  25. P r P t G t G r ( λ R ) n \frac{P_{r}}{P_{t}}\propto G_{t}G_{r}\left(\frac{\lambda}{R}\right)^{n}
  26. n n
  27. G t G_{t}
  28. G r G_{r}

Frisch–Waugh–Lovell_theorem.html

  1. Y = X 1 β 1 + X 2 β 2 + u Y=X_{1}\beta_{1}+X_{2}\beta_{2}+u\!
  2. X 1 X_{1}
  3. X 2 X_{2}
  4. n × k 1 n\times k_{1}
  5. n × k 2 n\times k_{2}
  6. β 1 \beta_{1}
  7. β 2 \beta_{2}
  8. β 2 \beta_{2}
  9. M X 1 Y = M X 1 X 2 β 2 + M X 1 u , M_{X_{1}}Y=M_{X_{1}}X_{2}\beta_{2}+M_{X_{1}}u\!,
  10. M X 1 M_{X_{1}}
  11. X 1 ( X 1 X 1 ) - 1 X 1 X_{1}(X_{1}^{\prime}X_{1})^{-1}X_{1}^{\prime}
  12. M X 1 = I - X 1 ( X 1 X 1 ) - 1 X 1 . M_{X_{1}}=I-X_{1}(X_{1}^{\prime}X_{1})^{-1}X_{1}^{\prime}.\!

Frobenius_pseudoprime.html

  1. x 2 - P x + Q \scriptstyle x^{2}-Px+Q
  2. D = P 2 - 4 Q \scriptstyle D=P^{2}-4Q
  3. gcd ( n , 2 Q D ) = 1 \scriptstyle\gcd(n,2QD)=1
  4. ( D n ) \scriptstyle\left(\tfrac{D}{n}\right)
  5. ( 1 ) U n - k 0 ( mod n ) \,\text{ }(1)\,\text{ }U_{n-k}\equiv 0\;\;(\mathop{{\rm mod}}n)
  6. ( 2 ) V n - k { 2 Q if k = - 1 2 if k = 1 . \,\text{ }(2)\,\text{ }V_{n-k}\equiv\begin{cases}2Q&\mbox{if }~{}k=-1\\ 2&\mbox{if }~{}k=1\mbox{.}\end{cases}
  7. x 2 - x - 1 \scriptstyle x^{2}-x-1
  8. x 2 - x - 1 \scriptstyle x^{2}-x-1
  9. ( 5 n ) = - 1 \left(\tfrac{5}{n}\right)=-1
  10. x 2 - 3 x - 1 \scriptstyle x^{2}-3x-1
  11. x 2 - 3 x - 5 \scriptstyle x^{2}-3x-5
  12. ( P , Q ) = ( 3 , - 5 ) \scriptstyle(P,Q)=(3,-5)
  13. ( D n ) = - 1 \scriptstyle\left(\tfrac{D}{n}\right)=-1
  14. 2 64 \scriptstyle 2^{64}
  15. x 2 - P x + Q \scriptstyle x^{2}-Px+Q
  16. V k \scriptstyle V_{k}
  17. x 3 - r x 2 + s x - 1 x^{3}-rx^{2}+sx-1
  18. ( c n ) = - 1 \left(\tfrac{c}{n}\right)=-1
  19. ( 1 + c ) n ( 1 - c ) ( mod n ) (1+\sqrt{c})^{n}\equiv(1-\sqrt{c})\;\;(\mathop{{\rm mod}}n)
  20. ( 13 1763 ) \left(\tfrac{13}{1763}\right)
  21. 1 7710 \tfrac{1}{7710}
  22. 1 131040 t \tfrac{1}{131040^{t}}
  23. 256 331776 t \tfrac{256}{{331776}^{t}}
  24. 1 4096 t \tfrac{1}{{4096}^{t}}
  25. 16 336442 t \tfrac{16}{336442^{t}}

Front_velocity.html

  1. f ( t ) = u ( t ) sin ω t , f(t)=u(t)\sin\omega t\ ,

Frölicher_space.html

  1. S : F B × C A × C ( 𝐑 , 𝐑 ) Mor ( C ( A , B ) , 𝐑 ) : ( f , c , λ ) S ( f , c , λ ) , S ( f , c , λ ) ( m ) := λ ( f m c ) S:F_{B}\times C_{A}\times\mathrm{C}^{\infty}(\mathbf{R},\mathbf{R})^{\prime}% \to\mathrm{Mor}(\mathrm{C}^{\infty}(A,B),\mathbf{R}):(f,c,\lambda)\mapsto S(f,% c,\lambda),\quad S(f,c,\lambda)(m):=\lambda(f\circ m\circ c)

Fubini–Study_metric.html

  1. 𝐂𝐏 n = { 𝐙 = [ Z 0 , Z 1 , , Z n ] 𝐂 n + 1 { 0 } } / { 𝐙 c 𝐙 , c 𝐂 * } . \mathbf{CP}^{n}=\left\{\mathbf{Z}=[Z_{0},Z_{1},\ldots,Z_{n}]\in{\mathbf{C}}^{n% +1}\setminus\{0\}\,\right\}/\{\mathbf{Z}\sim c\mathbf{Z},c\in\mathbf{C}^{*}\}.
  2. θ \theta
  3. 𝐂 n + 1 { 0 } ( a ) S 2 n + 1 ( b ) 𝐂𝐏 n \mathbf{C}^{n+1}\setminus\{0\}\stackrel{(a)}{\longrightarrow}S^{2n+1}\stackrel% {(b)}{\longrightarrow}\mathbf{CP}^{n}
  4. S 2 n + 1 S^{2n+1}
  5. g g
  6. X , Y X,Y
  7. d s 2 = d 𝐙 d 𝐙 ¯ = d Z 0 d Z 0 ¯ + + d Z n d Z n ¯ ds^{2}=d\mathbf{Z}\otimes d\overline{\mathbf{Z}}=dZ_{0}\otimes d\overline{Z_{0% }}+\cdots+dZ_{n}\otimes d\overline{Z_{n}}
  8. S 2 n + 1 S^{2n+1}
  9. [ Z 0 , , Z n ] [ 1 , z 1 , , z n ] , [Z_{0},\dots,Z_{n}]{\sim}[1,z_{1},\dots,z_{n}],
  10. { 1 , , n } \{\partial_{1},\ldots,\partial_{n}\}
  11. h i j ¯ = h ( i , ¯ j ) = ( 1 + | 𝐳 | 2 ) δ i j ¯ - z ¯ i z j ( 1 + | 𝐳 | 2 ) 2 . h_{i\bar{j}}=h(\partial_{i},\bar{\partial}_{j})=\frac{(1+|\mathbf{z}|^{2})% \delta_{i\bar{j}}-\bar{z}_{i}z_{j}}{(1+|\mathbf{z}|^{2})^{2}}.
  12. ( h i j ¯ ) = 1 ( 1 + | 𝐳 | 2 ) 2 [ 1 + | 𝐳 | 2 - | z 1 | 2 - z ¯ 1 z 2 - z ¯ 1 z n - z ¯ 2 z 1 1 + | 𝐳 | 2 - | z 2 | 2 - z ¯ 2 z n - z ¯ n z 1 - z ¯ n z 2 1 + | 𝐳 | 2 - | z n | 2 ] \bigl(h_{i\bar{j}}\bigr)=\frac{1}{(1+|\mathbf{z}|^{2})^{2}}\left[\begin{array}% []{cccc}1+|\mathbf{z}|^{2}-|z_{1}|^{2}&-\bar{z}_{1}z_{2}&\cdots&-\bar{z}_{1}z_% {n}\\ -\bar{z}_{2}z_{1}&1+|\mathbf{z}|^{2}-|z_{2}|^{2}&\cdots&-\bar{z}_{2}z_{n}\\ \vdots&\vdots&\ddots&\vdots\\ -\bar{z}_{n}z_{1}&-\bar{z}_{n}z_{2}&\cdots&1+|\mathbf{z}|^{2}-|z_{n}|^{2}\end{% array}\right]
  13. 𝐳 e i θ 𝐳 \mathbf{z}\mapsto e^{i\theta}\mathbf{z}
  14. d s 2 = ( 1 + | 𝐳 | 2 ) | d 𝐳 | 2 - ( 𝐳 ¯ d 𝐳 ) ( 𝐳 d 𝐳 ¯ ) ( 1 + | 𝐳 | 2 ) 2 = ( 1 + z i z ¯ i ) d z j d z ¯ j - z ¯ j z i d z j d z ¯ i ( 1 + z i z ¯ i ) 2 . \begin{aligned}\displaystyle ds^{2}&\displaystyle=\frac{(1+|\mathbf{z}|^{2})|d% \mathbf{z}|^{2}-(\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{% \mathbf{z}})}{(1+|\mathbf{z}|^{2})^{2}}\\ &\displaystyle=\frac{(1+z_{i}\bar{z}^{i})dz_{j}d\bar{z}^{j}-\bar{z}^{j}z_{i}dz% _{j}d\bar{z}^{i}}{(1+z_{i}\bar{z}^{i})^{2}}.\end{aligned}
  15. K = ln ( 1 + δ i j * z i z ¯ j * ) K=\ln(1+\delta_{ij^{*}}z^{i}\bar{z}^{j^{*}})
  16. g i j * = K i j * = 2 z i z ¯ j * K g_{ij^{*}}=K_{ij^{*}}=\frac{\partial^{2}}{\partial z^{i}\partial\bar{z}^{j^{*}% }}K
  17. d s 2 = | 𝐙 | 2 | d 𝐙 | 2 - ( 𝐙 ¯ d 𝐙 ) ( 𝐙 d 𝐙 ¯ ) | 𝐙 | 4 = Z α Z ¯ α d Z β d Z ¯ β - Z ¯ α Z β d Z α d Z ¯ β ( Z α Z ¯ α ) 2 = 2 Z [ α d Z β ] Z ¯ [ α d Z ¯ β ] ( Z α Z ¯ α ) 2 . \begin{aligned}\displaystyle ds^{2}&\displaystyle=\frac{|\mathbf{Z}|^{2}|d% \mathbf{Z}|^{2}-(\bar{\mathbf{Z}}\cdot d\mathbf{Z})(\mathbf{Z}\cdot d\bar{% \mathbf{Z}})}{|\mathbf{Z}|^{4}}\\ &\displaystyle=\frac{Z_{\alpha}\bar{Z}^{\alpha}dZ_{\beta}d\bar{Z}^{\beta}-\bar% {Z}^{\alpha}Z_{\beta}dZ_{\alpha}d\bar{Z}^{\beta}}{(Z_{\alpha}\bar{Z}^{\alpha})% ^{2}}\\ &\displaystyle=\frac{2Z_{[\alpha}dZ_{\beta]}\overline{Z}^{[\alpha}\overline{dZ% }^{\beta]}}{\left(Z_{\alpha}\overline{Z}^{\alpha}\right)^{2}}.\end{aligned}
  18. Z [ α W β ] = 1 2 ( Z α W β - Z β W α ) . Z_{[\alpha}W_{\beta]}=\frac{1}{2}\left(Z_{\alpha}W_{\beta}-Z_{\beta}W_{\alpha}% \right).
  19. ω = i ¯ log | 𝐙 | 2 \omega=i\partial\overline{\partial}\log|\mathbf{Z}|^{2}
  20. S 2 1 S^{2}\cong\mathbb{CP}^{1}
  21. d s 2 = Re ( d z d z ¯ ) ( 1 + | z | 2 ) 2 = d x 2 + d y 2 ( 1 + r 2 ) 2 = 1 4 ( d ϕ 2 + sin 2 ϕ d θ 2 ) = 1 4 d s u s 2 ds^{2}=\frac{\operatorname{Re}(dz\otimes d\overline{z})}{\left(1+|z|^{2}\right% )^{2}}=\frac{dx^{2}+dy^{2}}{\left(1+r^{2}\right)^{2}}=\frac{1}{4}(d\phi^{2}+% \sin^{2}\phi\,d\theta^{2})=\frac{1}{4}ds^{2}_{us}
  22. d s u s 2 ds^{2}_{us}
  23. 1 / R 2 1/R^{2}
  24. K ( σ ) = 1 + 3 J X , Y 2 K(\sigma)=1+3\langle JX,Y\rangle^{2}
  25. { X , Y } T p 𝐂𝐏 n \{X,Y\}\in T_{p}\mathbf{CP}^{n}
  26. , \langle\cdot,\cdot\rangle
  27. 1 K ( σ ) 4 1\leq K(\sigma)\leq 4
  28. σ \sigma
  29. R i c i j = λ g i j Ric_{ij}=\lambda g_{ij}
  30. | ψ = k = 0 n Z k | e k = [ Z 0 : Z 1 : : Z n ] |\psi\rangle=\sum_{k=0}^{n}Z_{k}|e_{k}\rangle=[Z_{0}:Z_{1}:\ldots:Z_{n}]
  31. { | e k } \{|e_{k}\rangle\}
  32. Z k Z_{k}
  33. Z α = [ Z 0 : Z 1 : : Z n ] Z_{\alpha}=[Z_{0}:Z_{1}:\ldots:Z_{n}]
  34. P n \mathbb{C}P^{n}
  35. | ψ = Z α |\psi\rangle=Z_{\alpha}
  36. | ϕ = W α |\phi\rangle=W_{\alpha}
  37. γ ( ψ , ϕ ) = arccos ψ | ϕ ϕ | ψ ψ | ψ ϕ | ϕ \gamma(\psi,\phi)=\arccos\sqrt{\frac{\langle\psi|\phi\rangle\;\langle\phi|\psi% \rangle}{\langle\psi|\psi\rangle\;\langle\phi|\phi\rangle}}
  38. γ ( ψ , ϕ ) = γ ( Z , W ) = arccos Z α W ¯ α W β Z ¯ β Z α Z ¯ α W β W ¯ β . \gamma(\psi,\phi)=\gamma(Z,W)=\arccos\sqrt{\frac{Z_{\alpha}\overline{W}^{% \alpha}\;W_{\beta}\overline{Z}^{\beta}}{Z_{\alpha}\overline{Z}^{\alpha}\;W_{% \beta}\overline{W}^{\beta}}}.
  39. Z ¯ α \overline{Z}^{\alpha}
  40. Z α Z_{\alpha}
  41. ψ | ψ \langle\psi|\psi\rangle
  42. | ψ |\psi\rangle
  43. | ϕ |\phi\rangle
  44. π / 2 \pi/2
  45. ϕ = ψ + δ ψ \phi=\psi+\delta\psi
  46. W α = Z α + d Z α W_{\alpha}=Z_{\alpha}+dZ_{\alpha}
  47. d s 2 = δ ψ | δ ψ ψ | ψ - δ ψ | ψ ψ | δ ψ ψ | ψ 2 . ds^{2}=\frac{\langle\delta\psi|\delta\psi\rangle}{\langle\psi|\psi\rangle}-% \frac{\langle\delta\psi|\psi\rangle\;\langle\psi|\delta\psi\rangle}{{\langle% \psi|\psi\rangle}^{2}}.
  48. | ψ |\psi\rangle
  49. | ψ = | ψ A | ψ B |\psi\rangle=|\psi_{A}\rangle\otimes|\psi_{B}\rangle
  50. d s 2 = d s A 2 + d s B 2 ds^{2}={ds_{A}}^{2}+{ds_{B}}^{2}
  51. d s A 2 {ds_{A}}^{2}
  52. d s B 2 {ds_{B}}^{2}

Fujikawa_method.html

  1. 𝔤 . \mathfrak{g}\,.
  2. D / = def / + i A / D\!\!\!\!/\ \stackrel{\mathrm{def}}{=}\ \partial\!\!\!/+iA\!\!\!/
  3. d d x ψ ¯ i D / ψ \int d^{d}x\,\overline{\psi}iD\!\!\!\!/\psi
  4. Z [ A ] = 𝒟 ψ ¯ 𝒟 ψ e - d d x ψ ¯ i D / ψ . Z[A]=\int\mathcal{D}\overline{\psi}\mathcal{D}\psi e^{-\int d^{d}x\overline{% \psi}iD\!\!\!\!/\psi}.
  5. ψ e i γ d + 1 α ( x ) ψ \psi\to e^{i\gamma_{d+1}\alpha(x)}\psi\,
  6. ψ ¯ ψ ¯ e i γ d + 1 α ( x ) \overline{\psi}\to\overline{\psi}e^{i\gamma_{d+1}\alpha(x)}
  7. S S + d d x α ( x ) μ ( ψ ¯ γ μ γ 5 ψ ) S\to S+\int d^{d}x\,\alpha(x)\partial_{\mu}\left(\overline{\psi}\gamma^{\mu}% \gamma^{5}\psi\right)
  8. j d + 1 μ ψ ¯ γ μ γ 5 ψ j_{d+1}^{\mu}\equiv\overline{\psi}\gamma^{\mu}\gamma^{5}\psi
  9. 0 = μ j d + 1 μ 0=\partial_{\mu}j_{d+1}^{\mu}
  10. ψ = i ψ i a i , \psi=\sum\limits_{i}\psi_{i}a^{i},
  11. ψ ¯ = i ψ i b i , \overline{\psi}=\sum\limits_{i}\psi_{i}b^{i},
  12. { a i , b i } \{a^{i},b^{i}\}
  13. { ψ i } \{\psi_{i}\}
  14. D / ψ i = - λ i ψ i . D\!\!\!\!/\psi_{i}=-\lambda_{i}\psi_{i}.
  15. δ i j = d d x ( 2 π ) d ψ j ( x ) ψ i ( x ) . \delta_{i}^{j}=\int\frac{d^{d}x}{(2\pi)^{d}}\psi^{\dagger j}(x)\psi_{i}(x).
  16. 𝒟 ψ 𝒟 ψ ¯ = i d a i d b i \mathcal{D}\psi\mathcal{D}\overline{\psi}=\prod\limits_{i}da^{i}db^{i}
  17. ψ ψ = ( 1 + i α γ d + 1 ) ψ = i ψ i a i , \psi\to\psi^{\prime}=(1+i\alpha\gamma_{d+1})\psi=\sum\limits_{i}\psi_{i}a^{% \prime i},
  18. ψ ¯ ψ ¯ = ψ ¯ ( 1 + i α γ d + 1 ) = i ψ i b i . \overline{\psi}\to\overline{\psi}^{\prime}=\overline{\psi}(1+i\alpha\gamma_{d+% 1})=\sum\limits_{i}\psi_{i}b^{\prime i}.
  19. C j i ( δ a δ a ) j i = d d x ψ i ( x ) [ 1 - i α ( x ) γ d + 1 ] ψ j ( x ) = δ j i - i d d x α ( x ) ψ i ( x ) γ d + 1 ψ j ( x ) . C^{i}_{j}\equiv\left(\frac{\delta a}{\delta a^{\prime}}\right)^{i}_{j}=\int d^% {d}x\,\psi^{\dagger i}(x)[1-i\alpha(x)\gamma_{d+1}]\psi_{j}(x)=\delta^{i}_{j}% \,-i\int d^{d}x\,\alpha(x)\psi^{\dagger i}(x)\gamma_{d+1}\psi_{j}(x).
  20. { b i } \{b_{i}\}
  21. 𝒟 ψ 𝒟 ψ ¯ = i d a i d b i = i d a i d b i det - 2 ( C j i ) , \mathcal{D}\psi\mathcal{D}\overline{\psi}=\prod\limits_{i}da^{i}db^{i}=\prod% \limits_{i}da^{\prime i}db^{\prime i}{\det}^{-2}(C^{i}_{j}),
  22. det - 2 ( C j i ) \displaystyle{\det}^{-2}(C^{i}_{j})
  23. - 2 t r ln C j i = 2 i lim M α d d x ψ i ( x ) γ d + 1 e - λ i 2 / M 2 ψ i ( x ) = 2 i lim M α d d x ψ i ( x ) γ d + 1 e D / 2 / M 2 ψ i ( x ) \begin{aligned}\displaystyle-2{\rm tr}\ln C^{i}_{j}&\displaystyle=2i\lim% \limits_{M\to\infty}\alpha\int d^{d}x\,\psi^{\dagger i}(x)\gamma_{d+1}e^{-% \lambda_{i}^{2}/M^{2}}\psi_{i}(x)\\ &\displaystyle=2i\lim\limits_{M\to\infty}\alpha\int d^{d}x\,\psi^{\dagger i}(x% )\gamma_{d+1}e^{{D\!\!\!\!/}^{2}/M^{2}}\psi_{i}(x)\end{aligned}
  24. D / 2 {D\!\!\!\!/}^{2}
  25. D 2 + 1 4 [ γ μ , γ ν ] F μ ν D^{2}+\tfrac{1}{4}[\gamma^{\mu},\gamma^{\nu}]F_{\mu\nu}
  26. = 2 i lim M α d d x d d k ( 2 π ) d d d k ( 2 π ) d ψ i ( k ) e i k x γ d + 1 e - k 2 / M 2 + 1 / ( 4 M 2 ) [ γ μ , γ ν ] F μ ν e - i k x ψ i ( k ) =2i\lim\limits_{M\to\infty}\alpha\int d^{d}x\int\frac{d^{d}k}{(2\pi)^{d}}\int% \frac{d^{d}k^{\prime}}{(2\pi)^{d}}\psi^{\dagger i}(k^{\prime})e^{ik^{\prime}x}% \gamma_{d+1}e^{-k^{2}/M^{2}+1/(4M^{2})[\gamma^{\mu},\gamma^{\nu}]F_{\mu\nu}}e^% {-ikx}\psi_{i}(k)
  27. = - - 2 α ( 2 π ) d / 2 ( d 2 ) ! ( 1 2 F ) d / 2 , =-\frac{-2\alpha}{(2\pi)^{d/2}(\frac{d}{2})!}(\tfrac{1}{2}F)^{d/2},
  28. F F μ ν d x μ d x ν . F\equiv F_{\mu\nu}\,dx^{\mu}\wedge dx^{\nu}\,.
  29. ( d 2 ) th (\tfrac{d}{2})^{\rm th}
  30. 𝔤 \mathfrak{g}

Full_Domain_Hash.html

  1. ( t , ϵ ) (t^{\prime},\epsilon^{\prime})
  2. ( t , ϵ ) (t,\epsilon)
  3. t = t - ( q h a s h + q s i g + 1 ) 𝒪 ( k 3 ) t=t^{\prime}-(q_{hash}+q_{sig}+1)\cdot\mathcal{O}(k^{3})
  4. ϵ = ( 1 + 1 q s i g ) q s i g + 1 q s i g ϵ \epsilon=\left(1+\frac{1}{q_{sig}}\right)^{q_{sig}+1}\cdot q_{sig}\cdot% \epsilon^{\prime}
  5. q s i g q_{sig}
  6. ϵ e x p ( 1 ) q s i g ϵ \epsilon\sim exp(1)\cdot q_{sig}\cdot\epsilon^{\prime}
  7. q h a s h q_{hash}
  8. q s i g q_{sig}
  9. ϵ \epsilon
  10. ϵ \epsilon^{\prime}
  11. t t^{\prime}

Full_reptend_prime.html

  1. b p - 1 - 1 p \frac{b^{p-1}-1}{p}
  2. 1 / p 1/p
  3. a / p a/p
  4. 1 / p 1/p
  5. a ( i ) = 2 i mod p mod 2 a(i)=2^{i}~{}\bmod p~{}\bmod 2

Function_field_of_an_algebraic_variety.html

  1. y 2 = x 5 + 1 y^{2}=x^{5}+1
  2. y 2 = x 5 + 1 y^{2}=x^{5}+1

Functional_determinant.html

  1. ζ S ( a ) = tr S - a , \zeta_{S}(a)=\operatorname{tr}\,S^{-a}\,,
  2. det S = e - ζ S ( 0 ) , \det S=e^{-\zeta_{S}^{\prime}(0)}\,,
  3. det S ( V 𝒟 ϕ e - ϕ , S ϕ ) - 2 . \det S\propto\left(\int_{V}\mathcal{D}\phi\;e^{-\langle\phi,S\phi\rangle}% \right)^{-2}\,.
  4. 1 det S = V e - π x , S x d x \frac{1}{\sqrt{\det S}}=\int_{V}e^{-\pi\langle x,Sx\rangle}\,dx
  5. V e - π ϕ , S ϕ 𝒟 ϕ \int_{V}e^{-\pi\langle\phi,S\phi\rangle}\,\mathcal{D}\phi
  6. - , - \langle-,-\rangle
  7. 𝒟 ϕ \mathcal{D}\phi
  8. | ϕ = i c i | f i with c i = f i | ϕ . |\phi\rangle=\sum_{i}c_{i}|f_{i}\rangle\quad\,\text{with }c_{i}=\langle f_{i}|% \phi\rangle.\,
  9. ϕ | S | ϕ = i , j c i * c j f i | S | f j = i , j c i * c j δ i j λ i = i | c i | 2 λ i . \langle\phi|S|\phi\rangle=\sum_{i,j}c_{i}^{*}c_{j}\langle f_{i}|S|f_{j}\rangle% =\sum_{i,j}c_{i}^{*}c_{j}\delta_{ij}\lambda_{i}=\sum_{i}|c_{i}|^{2}\lambda_{i}.
  10. 𝒟 ϕ = i d c i 2 π . \mathcal{D}\phi=\prod_{i}\frac{dc_{i}}{2\pi}.
  11. V 𝒟 ϕ e - ϕ | S | ϕ = i - + d c i 2 π e - λ i c i 2 . \int_{V}\mathcal{D}\phi\;e^{-\langle\phi|S|\phi\rangle}=\prod_{i}\int_{-\infty% }^{+\infty}\frac{dc_{i}}{2\pi}e^{-\lambda_{i}c_{i}^{2}}.
  12. V 𝒟 ϕ e - ϕ | S | ϕ = i 1 2 π λ i = N i λ i \int_{V}\mathcal{D}\phi\;e^{-\langle\phi|S|\phi\rangle}=\prod_{i}\frac{1}{2% \sqrt{\pi\lambda_{i}}}=\frac{N}{\sqrt{\prod_{i}\lambda_{i}}}
  13. V 𝒟 ϕ e - ϕ | S | ϕ 1 det S . \int_{V}\mathcal{D}\phi\;e^{-\langle\phi|S|\phi\rangle}\propto\frac{1}{\sqrt{% \det S}}.
  14. ϕ , S ϕ c ϕ , ϕ \langle\phi,S\phi\rangle\geq c\langle\phi,\phi\rangle
  15. 0 < λ 1 λ 2 , λ n . 0<\lambda_{1}\leq\lambda_{2}\leq\cdots,\qquad\lambda_{n}\to\infty.
  16. ζ S ( s ) = n = 1 1 λ n s . \zeta_{S}(s)=\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}^{s}}.
  17. s = 0 s=0
  18. ζ S ( s ) = n = 1 - log λ n λ n s , \zeta_{S}^{\prime}(s)=\sum_{n=1}^{\infty}\frac{-\log\lambda_{n}}{\lambda_{n}^{% s}},
  19. det S = exp ( - ζ S ( 0 ) ) . \det S=\exp\left(-\zeta_{S}^{\prime}(0)\right).
  20. n = 0 1 ( n + a ) \sum_{n=0}^{\infty}\frac{1}{(n+a)}
  21. n = 0 log ( n + a ) \sum_{n=0}^{\infty}\log(n+a)
  22. - s ζ H ( 0 , a ) -\partial_{s}\zeta_{H}(0,a)
  23. ζ H ( s , a ) \zeta_{H}(s,a)
  24. det ( - d 2 d x 2 + A ) ( x [ 0 , L ] ) , \det\left(-\frac{d^{2}}{dx^{2}}+A\right)\qquad(x\in[0,L]),
  25. λ n = n 2 π 2 L 2 + A ( n 0 ) . \lambda_{n}=\frac{n^{2}\pi^{2}}{L^{2}}+A\qquad(n\in\mathbb{N}_{0}).
  26. det ( - d 2 d x 2 + A ) det ( - d 2 d x 2 ) = n = 1 + n 2 π 2 L 2 + A n 2 π 2 L 2 = n = 1 + ( 1 + L 2 A n 2 π 2 ) . \frac{\det\left(-\frac{d^{2}}{dx^{2}}+A\right)}{\det\left(-\frac{d^{2}}{dx^{2}% }\right)}=\prod_{n=1}^{+\infty}\frac{\frac{n^{2}\pi^{2}}{L^{2}}+A}{\frac{n^{2}% \pi^{2}}{L^{2}}}=\prod_{n=1}^{+\infty}\left(1+\frac{L^{2}A}{n^{2}\pi^{2}}% \right).
  27. sin z = z n = 1 ( 1 - z 2 n 2 π 2 ) \sin z=z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right)
  28. sinh z = - i sin i z = z n = 1 ( 1 + z 2 n 2 π 2 ) . \sinh z=-i\sin iz=z\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}\pi^{2}}% \right).
  29. det ( - d 2 d x 2 + A ) det ( - d 2 d x 2 ) = n = 1 + ( 1 + L 2 A n 2 π 2 ) = sinh L A L A . \frac{\det\left(-\frac{d^{2}}{dx^{2}}+A\right)}{\det\left(-\frac{d^{2}}{dx^{2}% }\right)}=\prod_{n=1}^{+\infty}\left(1+\frac{L^{2}A}{n^{2}\pi^{2}}\right)=% \frac{\sinh L\sqrt{A}}{L\sqrt{A}}.
  30. det ( - d 2 d x 2 + V 1 ( x ) - m ) det ( - d 2 d x 2 + V 2 ( x ) - m ) \frac{\det\left(-\frac{d^{2}}{dx^{2}}+V_{1}(x)-m\right)}{\det\left(-\frac{d^{2% }}{dx^{2}}+V_{2}(x)-m\right)}
  31. ( - d 2 d x 2 + V i ( x ) - m ) ψ i m ( x ) = 0 \left(-\frac{d^{2}}{dx^{2}}+V_{i}(x)-m\right)\psi_{i}^{m}(x)=0
  32. ψ i m ( 0 ) = 0 , d ψ i m d x ( 0 ) = 1. \psi_{i}^{m}(0)=0,\quad\qquad\frac{d\psi_{i}^{m}}{dx}(0)=1.
  33. Δ ( m ) = ψ 1 m ( L ) ψ 2 m ( L ) , \Delta(m)=\frac{\psi_{1}^{m}(L)}{\psi_{2}^{m}(L)},
  34. det ( - d 2 d x 2 + V 1 ( x ) - m ) det ( - d 2 d x 2 + V 2 ( x ) - m ) = ψ 1 m ( L ) ψ 2 m ( L ) \frac{\det\left(-\frac{d^{2}}{dx^{2}}+V_{1}(x)-m\right)}{\det\left(-\frac{d^{2% }}{dx^{2}}+V_{2}(x)-m\right)}=\frac{\psi_{1}^{m}(L)}{\psi_{2}^{m}(L)}
  35. det ( - d 2 d x 2 + V 1 ( x ) ) det ( - d 2 d x 2 + V 2 ( x ) ) = ψ 1 0 ( L ) ψ 2 0 ( L ) . \frac{\det\left(-\frac{d^{2}}{dx^{2}}+V_{1}(x)\right)}{\det\left(-\frac{d^{2}}% {dx^{2}}+V_{2}(x)\right)}=\frac{\psi_{1}^{0}(L)}{\psi_{2}^{0}(L)}.
  36. ( - d 2 d x 2 + A ) ψ 1 0 = 0 , ψ 1 0 ( 0 ) = 0 , d ψ 1 0 d x ( 0 ) = 1 , - d 2 d x 2 ψ 2 0 = 0 , ψ 2 0 ( 0 ) = 0 , d ψ 2 0 d x ( 0 ) = 1 , \begin{aligned}&\displaystyle\left(-\frac{d^{2}}{dx^{2}}+A\right)\psi_{1}^{0}=% 0,\qquad\psi_{1}^{0}(0)=0\quad,\qquad\frac{d\psi_{1}^{0}}{dx}(0)=1,\\ &\displaystyle-\frac{d^{2}}{dx^{2}}\psi_{2}^{0}=0,\qquad\psi_{2}^{0}(0)=0,% \qquad\frac{d\psi_{2}^{0}}{dx}(0)=1,\end{aligned}
  37. ψ 1 0 ( x ) = 1 A sinh x A , \displaystyle\psi_{1}^{0}(x)=\frac{1}{\sqrt{A}}\sinh x\sqrt{A},
  38. det ( - d 2 d x 2 + A ) det ( - d 2 d x 2 ) = sinh L A L A . \frac{\det\left(-\frac{d^{2}}{dx^{2}}+A\right)}{\det\left(-\frac{d^{2}}{dx^{2}% }\right)}=\frac{\sinh L\sqrt{A}}{L\sqrt{A}}.

Functional_near-infrared_spectroscopy.html

  1. O D = L o g 10 ( I / I 0 ) = ϵ * [ X ] * l * D P F + G OD=Log_{10}(I/I_{0})=\epsilon\ *[X]*l*DPF+G
  2. O D OD
  3. I 0 I_{0}
  4. I I
  5. ϵ \epsilon
  6. [ X ] [X]
  7. l l
  8. D P F DPF
  9. G G
  10. ϵ \epsilon
  11. Δ [ X ] = Δ O D / ( ϵ * d ) \Delta[X]=\Delta OD/(\epsilon*d)
  12. d d
  13. | Δ O D λ 1 Δ O D λ 2 | = | ϵ λ 1 H b d ϵ λ 1 H b O 2 d ϵ λ 2 H b d ϵ λ 2 H b O 2 d | | Δ [ X ] H b Δ [ X ] H b O 2 | \begin{vmatrix}\Delta OD_{\lambda 1}\\ \Delta OD_{\lambda 2}\end{vmatrix}=\begin{vmatrix}\epsilon^{Hb}_{\lambda 1}d&% \epsilon^{HbO_{2}}_{\lambda 1}d\\ \epsilon^{Hb}_{\lambda 2}d&\epsilon^{HbO_{2}}_{\lambda 2}d\end{vmatrix}\begin{% vmatrix}\Delta[X]^{Hb}\\ \Delta[X]^{HbO_{2}}\end{vmatrix}

Fundamental_lemma_of_calculus_of_variations.html

  1. δ f δf
  2. f f
  3. δ f δf
  4. δ f δf
  5. f f
  6. a b f ( x ) h ( x ) d x = 0 \int_{a}^{b}f(x)\,h(x)\,\mathrm{d}x=0
  7. a b ( f ( x ) h ( x ) + g ( x ) h ( x ) ) d x = 0 \int_{a}^{b}(f(x)\,h(x)+g(x)\,h^{\prime}(x))\,\mathrm{d}x=0
  8. a b g ( x ) h ( x ) d x = 0 \int_{a}^{b}g(x)\,h^{\prime}(x)\,\mathrm{d}x=0
  9. f 0 , f 1 , , f n f_{0},f_{1},\dots,f_{n}
  10. a b ( f 0 ( x ) h ( x ) + f 1 ( x ) h ( x ) + + f n ( x ) h ( n ) ( x ) ) d x = 0 \int_{a}^{b}(f_{0}(x)\,h(x)+f_{1}(x)\,h^{\prime}(x)+\dots+f_{n}(x)\,h^{(n)}(x)% )\,\mathrm{d}x=0
  11. u 0 , u 1 , , u n - 1 u_{0},u_{1},\dots,u_{n-1}
  12. f 0 = u 0 , f 1 = u 0 + u 1 , , f n - 1 = u n - 2 + u n - 1 , f n = u n - 1 f_{0}=u^{\prime}_{0},\;f_{1}=u_{0}+u^{\prime}_{1},\;\dots,\;f_{n-1}=u_{n-2}+u^% {\prime}_{n-1},\;f_{n}=u_{n-1}
  13. ( u 0 h ) + ( u 1 h ) + + ( u n - 1 h ( n - 1 ) ) . (u_{0}h)^{\prime}+(u_{1}h^{\prime})^{\prime}+\dots+(u_{n-1}h^{(n-1)})^{\prime}.
  14. f = f 0 = u 0 f=f_{0}=u^{\prime}_{0}
  15. f 1 = u 0 , f_{1}=u_{0},
  16. f 0 - f 1 = 0. f_{0}-f^{\prime}_{1}=0.
  17. f 0 - f 1 + f 2 ′′ = 0 , f_{0}-f^{\prime}_{1}+f^{\prime\prime}_{2}=0,
  18. f 2 = u 1 f_{2}=u_{1}
  19. f 0 - f 1 + f 2 ′′ = 0 f_{0}-f^{\prime}_{1}+f^{\prime\prime}_{2}=0
  20. f 0 - ( f 1 - f 2 ) = 0 f_{0}-(f_{1}-f^{\prime}_{2})^{\prime}=0
  21. f 0 - ( f 1 - ( f 2 - f 3 ) ) = 0 f_{0}-(f_{1}-(f_{2}-f^{\prime}_{3})^{\prime})^{\prime}=0
  22. ( a , b ) d (a,b)\to\mathbb{R}^{d}
  23. Ω d \Omega\subset\mathbb{R}^{d}
  24. Ω f ( x ) h ( x ) d x = 0 \int_{\Omega}f(x)\,h(x)\,\mathrm{d}x=0
  25. Ω d \Omega\subset\mathbb{R}^{d}
  26. f L 2 ( Ω ) f\in L^{2}(\Omega)
  27. Ω f ( x ) h ( x ) d x = 0 \int_{\Omega}f(x)\,h(x)\,\mathrm{d}x=0
  28. J [ y ] = x 0 x 1 L ( t , y ( t ) , y ˙ ( t ) ) d t J[y]=\int_{x_{0}}^{x_{1}}L(t,y(t),\dot{y}(t))\,\mathrm{d}t
  29. y : [ x 0 , x 1 ] V y:[x_{0},x_{1}]\to V
  30. V V
  31. L ( t , y ( t ) , y ˙ ( t ) ) y = d d t L ( t , y ( t ) , y ˙ ( t ) ) y ˙ . {\partial L(t,y(t),\dot{y}(t))\over\partial y}={\mathrm{d}\over\mathrm{d}t}{% \partial L(t,y(t),\dot{y}(t))\over\partial\dot{y}}.

Fuzzy_clustering.html

  1. n n
  2. X = { 𝐱 1 , , 𝐱 n } X=\{\mathbf{x}_{1},...,\mathbf{x}_{n}\}
  3. c c
  4. C = { 𝐜 1 , , 𝐜 c } C=\{\mathbf{c}_{1},...,\mathbf{c}_{c}\}
  5. W = w i , j [ 0 , 1 ] , i = 1 , , n , j = 1 , , c W=w_{i,j}\in[0,1],\;i=1,...,n,\;j=1,...,c
  6. w i j w_{ij}
  7. 𝐱 i \mathbf{x}_{i}
  8. 𝐜 j \mathbf{c}_{j}
  9. arg min 𝐶 i = 1 n j = 1 c w i j m 𝐱 i - 𝐜 j 2 , \underset{C}{\operatorname{arg\,min}}\sum_{i=1}^{n}\sum_{j=1}^{c}w_{ij}^{m}% \left\|\mathbf{x}_{i}-\mathbf{c}_{j}\right\|^{2},
  10. w i j = 1 k = 1 c ( 𝐱 i - 𝐜 j 𝐱 i - 𝐜 k ) 2 m - 1 . w_{ij}=\frac{1}{\sum_{k=1}^{c}\left(\frac{\left\|\mathbf{x}_{i}-\mathbf{c}_{j}% \right\|}{\left\|\mathbf{x}_{i}-\mathbf{c}_{k}\right\|}\right)^{\frac{2}{m-1}}}.
  11. w i j w_{ij}
  12. m R m\in R
  13. m 1 m\geq 1
  14. m m
  15. m m
  16. w i j w_{ij}
  17. m = 1 m=1
  18. w i j w_{ij}
  19. m m
  20. c k = x w k ( x ) m x x w k ( x ) m . c_{k}={{\sum_{x}{w_{k}(x)}^{m}x}\over{\sum_{x}{w_{k}(x)}^{m}}}.
  21. ε \varepsilon

Fuzzy_sphere.html

  1. j 2 j^{2}
  2. J a , a = 1 , 2 , 3 J_{a},~{}a=1,2,3
  3. [ J a , J b ] = i ϵ a b c J c [J_{a},J_{b}]=i\epsilon_{abc}J_{c}
  4. ϵ a b c \epsilon_{abc}
  5. ϵ 123 = 1 \epsilon_{123}=1
  6. M j M_{j}
  7. J 1 2 + J 2 2 + J 3 2 = 1 4 ( j 2 - 1 ) I J_{1}^{2}+J_{2}^{2}+J_{3}^{2}=\frac{1}{4}(j^{2}-1)I
  8. x a = k r - 1 J a x_{a}=kr^{-1}J_{a}
  9. 4 r 4 = k 2 ( j 2 - 1 ) 4r^{4}=k^{2}(j^{2}-1)
  10. x 1 2 + x 2 2 + x 3 2 = r 2 x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=r^{2}
  11. S 2 f d Ω := 2 π k Tr ( F ) \int_{S^{2}}fd\Omega:=2\pi k\,\,\text{Tr}(F)
  12. 2 π k Tr ( I ) = 2 π k j = 4 π r 2 j j 2 - 1 2\pi k\,\,\text{Tr}(I)=2\pi kj=4\pi r^{2}\frac{j}{\sqrt{j^{2}-1}}

Gafftopsail_catfish.html

  1. W = c L b W=cL^{b}\!\,

Galerkin_method.html

  1. V V
  2. u V u\in V
  3. v V , a ( u , v ) = f ( v ) v\in V,a(u,v)=f(v)
  4. a ( , ) a(\cdot,\cdot)
  5. a ( , ) a(\cdot,\cdot)
  6. f f
  7. V V
  8. V n V V_{n}\subset V
  9. u n V n u_{n}\in V_{n}
  10. v n V n , a ( u n , v n ) = f ( v n ) v_{n}\in V_{n},a(u_{n},v_{n})=f(v_{n})
  11. u n u_{n}
  12. V n V_{n}
  13. V n V V_{n}\subset V
  14. v n v_{n}
  15. ϵ n = u - u n \epsilon_{n}=u-u_{n}
  16. u u
  17. u n u_{n}
  18. a ( ϵ n , v n ) = a ( u , v n ) - a ( u n , v n ) = f ( v n ) - f ( v n ) = 0. a(\epsilon_{n},v_{n})=a(u,v_{n})-a(u_{n},v_{n})=f(v_{n})-f(v_{n})=0.
  19. e 1 , e 2 , , e n e_{1},e_{2},\ldots,e_{n}
  20. V n V_{n}
  21. u n V n u_{n}\in V_{n}
  22. a ( u n , e i ) = f ( e i ) i = 1 , , n . a(u_{n},e_{i})=f(e_{i})\quad i=1,\ldots,n.
  23. u n u_{n}
  24. u n = j = 1 n u j e j u_{n}=\sum_{j=1}^{n}u_{j}e_{j}
  25. a ( j = 1 n u j e j , e i ) = j = 1 n u j a ( e j , e i ) = f ( e i ) i = 1 , , n . a\left(\sum_{j=1}^{n}u_{j}e_{j},e_{i}\right)=\sum_{j=1}^{n}u_{j}a(e_{j},e_{i})% =f(e_{i})\quad i=1,\ldots,n.
  26. A u = f Au=f
  27. A i j = a ( e j , e i ) , f i = f ( e i ) . A_{ij}=a(e_{j},e_{i}),\quad f_{i}=f(e_{i}).
  28. a ( , ) a(\cdot,\cdot)
  29. a ( u , v ) = a ( v , u ) . a(u,v)=a(v,u).
  30. u n u_{n}
  31. u , v V u,v\in V
  32. a ( u , v ) C u v a(u,v)\leq C\|u\|\,\|v\|
  33. C > 0 C>0
  34. u V u\in V
  35. a ( u , u ) c u 2 a(u,u)\geq c\|u\|^{2}
  36. c > 0. c>0.
  37. V n V V_{n}\subset V
  38. V n V_{n}
  39. u - u n u-u_{n}
  40. u - u n C c inf v n V n u - v n . \|u-u_{n}\|\leq\frac{C}{c}\inf_{v_{n}\in V_{n}}\|u-v_{n}\|.
  41. C / c C/c
  42. u n u_{n}
  43. u u
  44. V n V_{n}
  45. V n V_{n}
  46. v n V n v_{n}\in V_{n}
  47. c u - u n 2 a ( u - u n , u - u n ) = a ( u - u n , u - v n ) C u - u n u - v n . c\|u-u_{n}\|^{2}\leq a(u-u_{n},u-u_{n})=a(u-u_{n},u-v_{n})\leq C\|u-u_{n}\|\,% \|u-v_{n}\|.
  48. c u - u n c\|u-u_{n}\|
  49. v n v_{n}

Games_behind.html

  1. Games Behind = ( TeamA’s wins - TeamB’s wins ) + ( TeamB’s losses - TeamA’s losses ) 2 \,\text{Games Behind}=\frac{(\,\text{TeamA's wins - TeamB's wins})+(\,\text{% TeamB's losses - TeamA's losses})}{2}

Gamma_matrices.html

  1. { γ 0 , γ 1 , γ 2 , γ 3 } \{\gamma^{0},\gamma^{1},\gamma^{2},\gamma^{3}\}
  2. γ 0 = ( 1 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 - 1 ) , γ 1 = ( 0 0 0 1 0 0 1 0 0 - 1 0 0 - 1 0 0 0 ) \gamma^{0}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix},\quad\gamma^{1}=\begin{pmatrix}0&0&0&1\\ 0&0&1&0\\ 0&-1&0&0\\ -1&0&0&0\end{pmatrix}
  3. γ 2 = ( 0 0 0 - i 0 0 i 0 0 i 0 0 - i 0 0 0 ) , γ 3 = ( 0 0 1 0 0 0 0 - 1 - 1 0 0 0 0 1 0 0 ) . \gamma^{2}=\begin{pmatrix}0&0&0&-i\\ 0&0&i&0\\ 0&i&0&0\\ -i&0&0&0\end{pmatrix},\quad\gamma^{3}=\begin{pmatrix}0&0&1&0\\ 0&0&0&-1\\ -1&0&0&0\\ 0&1&0&0\end{pmatrix}.
  4. { γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2 η μ ν I 4 \displaystyle\{\gamma^{\mu},\gamma^{\nu}\}=\gamma^{\mu}\gamma^{\nu}+\gamma^{% \nu}\gamma^{\mu}=2\eta^{\mu\nu}I_{4}
  5. { , } \{,\}
  6. η μ ν \eta^{\mu\nu}
  7. I 4 I_{4}
  8. γ μ = η μ ν γ ν = { γ 0 , - γ 1 , - γ 2 , - γ 3 } , \displaystyle\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\left\{\gamma^{0},-\gamma^% {1},-\gamma^{2},-\gamma^{3}\right\},
  9. { γ μ , γ ν } = - 2 η μ ν I 4 \displaystyle\{\gamma^{\mu},\gamma^{\nu}\}=-2\eta^{\mu\nu}I_{4}
  10. i i
  11. γ μ = η μ ν γ ν = { - γ 0 , + γ 1 , + γ 2 , + γ 3 } \displaystyle\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\left\{-\gamma^{0},+\gamma% ^{1},+\gamma^{2},+\gamma^{3}\right\}
  12. V V
  13. V V
  14. E n d ( V ) End(V)
  15. V V
  16. 4 × 4 4 × 4
  17. Ψ Ψ
  18. x x
  19. Ψ ( x ) Ψ(x)
  20. x x
  21. S S
  22. E E
  23. S S
  24. S ( Λ ) S(Λ)
  25. Λ Λ
  26. V V
  27. γ μ S ( Λ ) γ μ S ( Λ ) - 1 = ( Λ - 1 ) μ ν γ ν := Λ ν μ γ ν , \gamma^{\mu}\mapsto S(\Lambda)\gamma^{\mu}S(\Lambda)^{-1}={{({\Lambda}^{-1})}^% {\mu}}_{\nu}\gamma^{\nu}:={\Lambda_{\nu}}^{\mu}\gamma^{\nu},
  28. a / := a μ γ μ a\!\!\!/:=a_{\mu}\gamma^{\mu}
  29. γ γ
  30. V V
  31. a / μ Λ μ ν a / ν . {a\!\!\!/}^{\mu}\mapsto{\Lambda^{\mu}}_{\nu}{a\!\!\!/}^{\nu}.
  32. S ( Λ ) S(Λ)
  33. ( i γ μ μ - m ) ψ = 0 (i\gamma^{\mu}\partial_{\mu}-m)\psi=0
  34. ψ \psi
  35. ( i / - m ) ψ = 0. (i\partial\!\!\!/-m)\psi=0.
  36. γ 5 := i γ 0 γ 1 γ 2 γ 3 = ( 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ) \gamma^{5}:=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\begin{pmatrix}0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\end{pmatrix}
  37. γ 5 \gamma^{5}
  38. γ 0 \gamma^{0}
  39. γ 4 \gamma^{4}
  40. γ 5 \gamma^{5}
  41. γ 5 = i 4 ! ε μ ν α β γ μ γ ν γ α γ β \gamma^{5}=\frac{i}{4!}\varepsilon_{\mu\nu\alpha\beta}\gamma^{\mu}\gamma^{\nu}% \gamma^{\alpha}\gamma^{\beta}
  42. γ 0 γ 1 γ 2 γ 3 = γ [ 0 γ 1 γ 2 γ 3 ] = 1 4 ! δ μ ν ϱ σ 0123 γ μ γ ν γ ϱ γ σ \gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\gamma^{[0}\gamma^{1}\gamma^{2}\gamma% ^{3]}=\frac{1}{4!}\delta^{0123}_{\mu\nu\varrho\sigma}\gamma^{\mu}\gamma^{\nu}% \gamma^{\varrho}\gamma^{\sigma}
  43. δ μ ν ϱ σ α β γ δ \delta^{\alpha\beta\gamma\delta}_{\mu\nu\varrho\sigma}
  44. ε α β \varepsilon_{\alpha\dots\beta}
  45. δ μ ν ϱ σ α β γ δ = ε α β γ δ ε μ ν ϱ σ \delta^{\alpha\beta\gamma\delta}_{\mu\nu\varrho\sigma}=\varepsilon^{\alpha% \beta\gamma\delta}\varepsilon_{\mu\nu\varrho\sigma}
  46. γ 5 = i γ 0 γ 1 γ 2 γ 3 = i 4 ! ε 0123 ε μ ν ϱ σ γ μ γ ν γ ϱ γ σ = i 4 ! ε μ ν ϱ σ γ μ γ ν γ ϱ γ σ \gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\frac{i}{4!}\varepsilon^{% 0123}\varepsilon_{\mu\nu\varrho\sigma}\,\gamma^{\mu}\gamma^{\nu}\gamma^{% \varrho}\gamma^{\sigma}=\frac{i}{4!}\varepsilon_{\mu\nu\varrho\sigma}\,\gamma^% {\mu}\gamma^{\nu}\gamma^{\varrho}\gamma^{\sigma}
  47. ψ L = 1 - γ 5 2 ψ , ψ R = 1 + γ 5 2 ψ \psi_{L}=\frac{1-\gamma^{5}}{2}\psi,\qquad\psi_{R}=\frac{1+\gamma^{5}}{2}\psi
  48. ( γ 5 ) = γ 5 . (\gamma^{5})^{\dagger}=\gamma^{5}.\,
  49. ( γ 5 ) 2 = I 4 . (\gamma^{5})^{2}=I_{4}.\,
  50. { γ 5 , γ μ } = γ 5 γ μ + γ μ γ 5 = 0. \left\{\gamma^{5},\gamma^{\mu}\right\}=\gamma^{5}\gamma^{\mu}+\gamma^{\mu}% \gamma^{5}=0.\,
  51. 5 5
  52. ( 1 , 4 ) (1,4)
  53. ( 4 , 1 ) (4,1)
  54. ( 3 , 1 ) (3,1)
  55. 2 n 2n
  56. 2 n + 1 2n+1
  57. n 1 n≥1
  58. γ 5 \gamma^{5}
  59. γ μ γ μ = 4 I 4 \displaystyle\gamma^{\mu}\gamma_{\mu}=4I_{4}
  60. γ μ γ ν γ μ = - 2 γ ν \displaystyle\gamma^{\mu}\gamma^{\nu}\gamma_{\mu}=-2\gamma^{\nu}
  61. γ μ γ ν γ ρ γ μ = 4 η ν ρ I 4 \displaystyle\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}=4\eta^{\nu\rho}% I_{4}
  62. γ μ γ ν γ ρ γ σ γ μ = - 2 γ σ γ ρ γ ν \displaystyle\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu}=% -2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}
  63. γ μ γ ν γ λ = η μ ν γ λ + η ν λ γ μ - η μ λ γ ν - i ϵ σ μ ν λ γ σ γ 5 \displaystyle\gamma^{\mu}\gamma^{\nu}\gamma^{\lambda}=\eta^{\mu\nu}\gamma^{% \lambda}+\eta^{\nu\lambda}\gamma^{\mu}-\eta^{\mu\lambda}\gamma^{\nu}-i\epsilon% ^{\sigma\mu\nu\lambda}\gamma_{\sigma}\gamma^{5}
  64. γ μ γ μ = 4 I 4 \displaystyle\gamma^{\mu}\gamma_{\mu}=4I_{4}
  65. { γ μ , γ ν } = γ μ γ ν + γ ν γ μ = 2 η μ ν I 4 . \displaystyle\{\gamma^{\mu},\gamma^{\nu}\}=\gamma^{\mu}\gamma^{\nu}+\gamma^{% \nu}\gamma^{\mu}=2\eta^{\mu\nu}I_{4}.
  66. η \eta
  67. γ μ γ μ = γ μ η μ ν γ ν = η μ ν γ μ γ ν \gamma^{\mu}\gamma_{\mu}\,=\gamma^{\mu}\eta_{\mu\nu}\gamma^{\nu}=\eta_{\mu\nu}% \gamma^{\mu}\gamma^{\nu}
  68. = 1 2 ( η μ ν + η ν μ ) γ μ γ ν =\frac{1}{2}(\eta_{\mu\nu}+\eta_{\nu\mu})\gamma^{\mu}\gamma^{\nu}
  69. η \eta
  70. = 1 2 ( η μ ν γ μ γ ν + η ν μ γ μ γ ν ) =\frac{1}{2}(\eta_{\mu\nu}\gamma^{\mu}\gamma^{\nu}+\eta_{\nu\mu}\gamma^{\mu}% \gamma^{\nu})
  71. = 1 2 ( η μ ν γ μ γ ν + η μ ν γ ν γ μ ) =\frac{1}{2}(\eta_{\mu\nu}\gamma^{\mu}\gamma^{\nu}+\eta_{\mu\nu}\gamma^{\nu}% \gamma^{\mu})
  72. = 1 2 η μ ν { γ μ , γ ν } =\frac{1}{2}\eta_{\mu\nu}\{\gamma^{\mu},\gamma^{\nu}\}\,
  73. = 1 2 η μ ν ( 2 η μ ν I 4 ) = η μ ν η μ ν I 4 = 4 I 4 . =\frac{1}{2}\eta_{\mu\nu}\left(2\eta^{\mu\nu}I_{4}\right)=\eta_{\mu\nu}\eta^{% \mu\nu}I_{4}=4I_{4}.\,
  74. γ μ γ ν γ μ = - 2 γ ν . \gamma^{\mu}\gamma^{\nu}\gamma_{\mu}=-2\gamma^{\nu}.\,
  75. γ μ γ ν γ μ \gamma^{\mu}\gamma^{\nu}\gamma_{\mu}\,
  76. = γ μ ( 2 η μ ν I 4 - γ μ γ ν ) =\gamma^{\mu}\left(2\eta_{\mu}^{\nu}I_{4}-\gamma_{\mu}\gamma^{\nu}\right)\,
  77. = 2 γ μ η μ ν - γ μ γ μ γ ν =2\gamma^{\mu}\eta_{\mu}^{\nu}-\gamma^{\mu}\gamma_{\mu}\gamma^{\nu}\,
  78. = 2 γ ν - 4 γ ν = - 2 γ ν . =2\gamma^{\nu}-4\gamma^{\nu}=-2\gamma^{\nu}.\,
  79. γ μ γ ν γ ρ γ μ = 4 η ν ρ I 4 . \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}=4\eta^{\nu\rho}I_{4}.\,
  80. γ μ \gamma^{\mu}
  81. γ μ γ ν γ ρ γ μ \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}\,
  82. = { γ μ , γ ν } γ ρ γ μ - γ ν γ μ γ ρ γ μ =\{\gamma^{\mu},\gamma^{\nu}\}\gamma^{\rho}\gamma_{\mu}-\gamma^{\nu}\gamma^{% \mu}\gamma^{\rho}\gamma_{\mu}\,
  83. = 2 η μ ν γ ρ γ μ - γ ν { γ μ , γ ρ } γ μ + γ ν γ ρ γ μ γ μ . =2\ \eta^{\mu\nu}\gamma^{\rho}\gamma_{\mu}-\gamma^{\nu}\{\gamma^{\mu},\gamma^{% \rho}\}\gamma_{\mu}+\gamma^{\nu}\gamma^{\rho}\gamma^{\mu}\gamma_{\mu}.\,
  84. γ μ γ μ = 4 I \gamma^{\mu}\gamma_{\mu}=4I
  85. γ μ γ ν γ ρ γ μ \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}\,
  86. = 2 γ ρ γ ν - γ ν 2 η μ ρ γ μ + 4 γ ν γ ρ =2\ \gamma^{\rho}\gamma^{\nu}-\gamma^{\nu}2\eta^{\mu\rho}\gamma_{\mu}+4\ % \gamma^{\nu}\gamma^{\rho}\,
  87. = 2 γ ρ γ ν - 2 γ ν γ ρ + 4 γ ν γ ρ =2\ \gamma^{\rho}\gamma^{\nu}-2\ \gamma^{\nu}\gamma^{\rho}+4\ \gamma^{\nu}% \gamma^{\rho}\,
  88. = 2 ( γ ρ γ ν + γ ν γ ρ ) =2\ (\gamma^{\rho}\gamma^{\nu}+\gamma^{\nu}\gamma^{\rho})\,
  89. = 2 { γ ν , γ ρ } . =2\ \{\gamma^{\nu},\gamma^{\rho}\}.\,
  90. γ μ γ ν γ ρ γ μ = 4 η ν ρ I 4 . \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}=4\ \eta^{\nu\rho}I_{4}.\,
  91. γ μ γ ν γ ρ γ σ γ μ \gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu}\,
  92. = ( 2 η μ ν - γ ν γ μ ) γ ρ γ σ γ μ =(2\eta^{\mu\nu}-\gamma^{\nu}\gamma^{\mu})\gamma^{\rho}\gamma^{\sigma}\gamma_{% \mu}\,\quad
  93. = 2 η μ ν γ ρ γ σ γ μ - 4 γ ν η ρ σ =2\eta^{\mu\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu}-4\gamma^{\nu}\eta^{% \rho\sigma}\,\quad
  94. = 2 γ ρ γ σ γ ν - 4 γ ν η ρ σ =2\gamma^{\rho}\gamma^{\sigma}\gamma^{\nu}-4\gamma^{\nu}\eta^{\rho\sigma}\,
  95. = 2 ( 2 η ρ σ - γ σ γ ρ ) γ ν - 4 γ ν η ρ σ =2(2\eta^{\rho\sigma}-\gamma^{\sigma}\gamma^{\rho})\gamma^{\nu}-4\gamma^{\nu}% \eta^{\rho\sigma}\,
  96. = - 2 γ σ γ ρ γ ν =-2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}\,
  97. μ = ν = ρ \mu=\nu=\rho
  98. ϵ σ μ ν ρ = 0 \epsilon^{\sigma\mu\nu\rho}=0
  99. μ = ν ρ \mu=\nu\neq\rho
  100. μ = ρ ν \mu=\rho\neq\nu
  101. ν = ρ μ \nu=\rho\neq\mu
  102. η μ ν = 0 \eta^{\mu\nu}=0
  103. η μ ρ = 0 \eta^{\mu\rho}=0
  104. η ν ρ = 0 \eta^{\nu\rho}=0
  105. γ \gamma
  106. ϵ σ μ ν ρ \epsilon_{\sigma\mu\nu\rho}
  107. γ 0 γ 1 γ 2 \gamma^{0}\gamma^{1}\gamma^{2}
  108. γ 0 γ 1 γ 3 \gamma^{0}\gamma^{1}\gamma^{3}
  109. γ 0 γ 2 γ 3 \gamma^{0}\gamma^{2}\gamma^{3}
  110. γ 1 γ 2 γ 3 \gamma^{1}\gamma^{2}\gamma^{3}
  111. - i ϵ σ 012 γ σ γ 5 = - i ϵ 3012 ( - γ 3 ) ( i γ 0 γ 1 γ 2 γ 3 ) = - ϵ 3012 γ 0 γ 1 γ 2 = ϵ 0123 γ 0 γ 1 γ 2 -i\epsilon^{\sigma 012}\gamma_{\sigma}\gamma^{5}=-i\epsilon^{3012}(-\gamma^{3}% )(i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})=-\epsilon^{3012}\gamma^{0}\gamma^% {1}\gamma^{2}=\epsilon^{0123}\gamma^{0}\gamma^{1}\gamma^{2}
  112. - i ϵ σ 013 γ σ γ 5 = - i ϵ 2013 ( - γ 2 ) ( i γ 0 γ 1 γ 2 γ 3 ) = ϵ 2013 γ 0 γ 1 γ 3 = ϵ 0123 γ 0 γ 1 γ 3 -i\epsilon^{\sigma 013}\gamma_{\sigma}\gamma^{5}=-i\epsilon^{2013}(-\gamma^{2}% )(i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})=\epsilon^{2013}\gamma^{0}\gamma^{% 1}\gamma^{3}=\epsilon^{0123}\gamma^{0}\gamma^{1}\gamma^{3}
  113. - i ϵ σ 023 γ σ γ 5 = - i ϵ 1023 ( - γ 1 ) ( i γ 0 γ 1 γ 2 γ 3 ) = - ϵ 1023 γ 0 γ 2 γ 3 = ϵ 0123 γ 0 γ 2 γ 3 -i\epsilon^{\sigma 023}\gamma_{\sigma}\gamma^{5}=-i\epsilon^{1023}(-\gamma^{1}% )(i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})=-\epsilon^{1023}\gamma^{0}\gamma^% {2}\gamma^{3}=\epsilon^{0123}\gamma^{0}\gamma^{2}\gamma^{3}
  114. - i ϵ σ 123 γ σ γ 5 = - i ϵ 0123 ( γ 0 ) ( i γ 0 γ 1 γ 2 γ 3 ) = ϵ 0123 γ 1 γ 2 γ 3 -i\epsilon^{\sigma 123}\gamma_{\sigma}\gamma^{5}=-i\epsilon^{0123}(\gamma^{0})% (i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})=\epsilon^{0123}\gamma^{1}\gamma^{2% }\gamma^{3}
  115. tr ( γ μ ) = 0 \operatorname{tr}(\gamma^{\mu})=0
  116. γ μ \gamma^{\mu}
  117. γ 5 \gamma^{5}
  118. γ μ \gamma^{\mu}
  119. tr ( γ μ γ ν ) = 4 η μ ν \operatorname{tr}(\gamma^{\mu}\gamma^{\nu})=4\eta^{\mu\nu}
  120. tr ( γ μ γ ν γ ρ γ σ ) = 4 ( η μ ν η ρ σ - η μ ρ η ν σ + η μ σ η ν ρ ) \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=4(\eta% ^{\mu\nu}\eta^{\rho\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta% ^{\nu\rho})
  121. tr ( γ 5 ) = tr ( γ μ γ ν γ 5 ) = 0 \operatorname{tr}(\gamma^{5})=\operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma% ^{5})=0
  122. tr ( γ μ γ ν γ ρ γ σ γ 5 ) = - 4 i ϵ μ ν ρ σ \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{% 5})=-4i\epsilon^{\mu\nu\rho\sigma}
  123. tr ( γ μ 1 γ μ n ) = tr ( γ μ n γ μ 1 ) \operatorname{tr}(\gamma^{\mu 1}\dots\gamma^{\mu n})=\operatorname{tr}(\gamma^% {\mu n}\dots\gamma^{\mu 1})
  124. γ μ γ ν + γ ν γ μ = 2 η μ ν \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2\eta^{\mu\nu}\,
  125. γ μ γ μ = η μ μ \gamma^{\mu}\gamma^{\mu}=\eta^{\mu\mu}\,
  126. γ μ γ μ η μ μ = I \frac{\gamma^{\mu}\gamma^{\mu}}{\eta^{\mu\mu}}=I\,
  127. η μ μ \eta^{\mu\mu}
  128. γ μ γ μ \gamma^{\mu}\gamma^{\mu}
  129. tr ( γ ν ) = 1 η μ μ tr ( γ ν γ μ γ μ ) \operatorname{tr}(\gamma^{\nu})=\frac{1}{\eta^{\mu\mu}}\operatorname{tr}(% \gamma^{\nu}\gamma^{\mu}\gamma^{\mu})
  130. = - 1 η μ μ tr ( γ μ γ ν γ μ ) =-\frac{1}{\eta^{\mu\mu}}\operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\mu})
  131. μ ν \mu\neq\nu
  132. = - 1 η μ μ tr ( γ ν γ μ γ μ ) =-\frac{1}{\eta^{\mu\mu}}\operatorname{tr}(\gamma^{\nu}\gamma^{\mu}\gamma^{\mu})
  133. = - tr ( γ ν ) =-\operatorname{tr}(\gamma^{\nu})
  134. tr ( γ ν ) = 0 \operatorname{tr}(\gamma^{\nu})=0
  135. tr ( odd num of γ ) = 0 \operatorname{tr}(\mathrm{odd\ num\ of\ }\gamma)=0\,
  136. tr ( γ μ ) = 0. \operatorname{tr}(\gamma^{\mu})=0.\,
  137. γ 5 \gamma^{5}\,
  138. ( γ 5 ) 2 = I 4 , and γ μ γ 5 = - γ 5 γ μ \left(\gamma^{5}\right)^{2}=I_{4},\quad\mathrm{and}\quad\gamma^{\mu}\gamma^{5}% =-\gamma^{5}\gamma^{\mu}\,
  139. γ 5 \gamma^{5}\,
  140. γ \gamma\,
  141. γ 5 \gamma^{5}\,
  142. tr ( γ μ γ ν γ ρ ) \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho})\,
  143. = tr ( γ 5 γ 5 γ μ γ ν γ ρ ) =\operatorname{tr}\left(\gamma^{5}\gamma^{5}\gamma^{\mu}\gamma^{\nu}\gamma^{% \rho}\right)\,
  144. = - tr ( γ 5 γ μ γ ν γ ρ γ 5 ) =-\operatorname{tr}\left(\gamma^{5}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma% ^{5}\right)\,
  145. = - tr ( γ 5 γ 5 γ μ γ ν γ ρ ) =-\operatorname{tr}\left(\gamma^{5}\gamma^{5}\gamma^{\mu}\gamma^{\nu}\gamma^{% \rho}\right)\,
  146. = - tr ( γ μ γ ν γ ρ ) =-\operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\right)\,
  147. tr ( γ μ γ ν γ ρ ) = 0 \operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\right)=0\,
  148. γ 5 \gamma^{5}
  149. γ 5 \gamma^{5}
  150. tr ( γ μ γ ν ) = 4 η μ ν \operatorname{tr}(\gamma^{\mu}\gamma^{\nu})=4\eta^{\mu\nu}
  151. tr ( γ μ γ ν ) \operatorname{tr}(\gamma^{\mu}\gamma^{\nu})\,
  152. = 1 2 ( tr ( γ μ γ ν ) + tr ( γ ν γ μ ) ) =\frac{1}{2}\left(\operatorname{tr}(\gamma^{\mu}\gamma^{\nu})+\operatorname{tr% }(\gamma^{\nu}\gamma^{\mu})\right)\,
  153. = 1 2 tr ( γ μ γ ν + γ ν γ μ ) = 1 2 tr ( { γ μ , γ ν } ) =\frac{1}{2}\operatorname{tr}(\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu% })=\frac{1}{2}\operatorname{tr}\left(\{\gamma^{\mu},\gamma^{\nu}\}\right)\,
  154. = 1 2 2 η μ ν tr ( I 4 ) = 4 η μ ν =\frac{1}{2}2\eta^{\mu\nu}\operatorname{tr}(I_{4})=4\eta^{\mu\nu}\,
  155. tr ( γ μ γ ν γ ρ γ σ ) \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})\,
  156. = tr ( γ μ γ ν ( 2 η ρ σ - γ σ γ ρ ) ) =\operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}(2\eta^{\rho\sigma}-\gamma^{% \sigma}\gamma^{\rho})\right)\,
  157. = 2 η ρ σ tr ( γ μ γ ν ) - tr ( γ μ γ ν γ σ γ ρ ) ( 1 ) =2\eta^{\rho\sigma}\operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}\right)-% \operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho}% \right)\quad\quad(1)\,
  158. γ σ \gamma^{\sigma}\,
  159. tr ( γ μ γ ν γ σ γ ρ ) \operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho}% \right)\,
  160. = tr ( γ μ ( 2 η ν σ - γ σ γ ν ) γ ρ ) =\operatorname{tr}\left(\gamma^{\mu}(2\eta^{\nu\sigma}-\gamma^{\sigma}\gamma^{% \nu})\gamma^{\rho}\right)\,
  161. = 2 η ν σ tr ( γ μ γ ρ ) - tr ( γ μ γ σ γ ν γ ρ ) ( 2 ) =2\eta^{\nu\sigma}\operatorname{tr}\left(\gamma^{\mu}\gamma^{\rho}\right)-% \operatorname{tr}\left(\gamma^{\mu}\gamma^{\sigma}\gamma^{\nu}\gamma^{\rho}% \right)\quad\quad(2)\,
  162. γ σ \gamma^{\sigma}\,
  163. tr ( γ μ γ σ γ ν γ ρ ) \operatorname{tr}\left(\gamma^{\mu}\gamma^{\sigma}\gamma^{\nu}\gamma^{\rho}% \right)\,
  164. = tr ( ( 2 η μ σ - γ σ γ μ ) γ ν γ ρ ) =\operatorname{tr}\left((2\eta^{\mu\sigma}-\gamma^{\sigma}\gamma^{\mu})\gamma^% {\nu}\gamma^{\rho}\right)\,
  165. = 2 η μ σ tr ( γ ν γ ρ ) - tr ( γ σ γ μ γ ν γ ρ ) ( 3 ) =2\eta^{\mu\sigma}\operatorname{tr}\left(\gamma^{\nu}\gamma^{\rho}\right)-% \operatorname{tr}\left(\gamma^{\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}% \right)\quad\quad(3)\,
  166. 2 η ρ σ tr ( γ μ γ ν ) = 2 η ρ σ ( 4 η μ ν ) = 8 η ρ σ η μ ν . 2\eta^{\rho\sigma}\operatorname{tr}\left(\gamma^{\mu}\gamma^{\nu}\right)=2\eta% ^{\rho\sigma}(4\eta^{\mu\nu})=8\eta^{\rho\sigma}\eta^{\mu\nu}.\,
  167. tr ( γ μ γ ν γ ρ γ σ ) = 8 η ρ σ η μ ν - 8 η ν σ η μ ρ + 8 η μ σ η ν ρ \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=8\eta^% {\rho\sigma}\eta^{\mu\nu}-8\eta^{\nu\sigma}\eta^{\mu\rho}+8\eta^{\mu\sigma}% \eta^{\nu\rho}\,
  168. - tr ( γ σ γ μ γ ν γ ρ ) ( 4 ) -\ \operatorname{tr}\left(\gamma^{\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}% \right)\quad\quad\quad\quad\quad\quad(4)\,
  169. tr ( γ σ γ μ γ ν γ ρ ) = tr ( γ μ γ ν γ ρ γ σ ) . \operatorname{tr}\left(\gamma^{\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}% \right)=\operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}% ).\,
  170. 2 tr ( γ μ γ ν γ ρ γ σ ) = 8 η ρ σ η μ ν - 8 η ν σ η μ ρ + 8 η μ σ η ν ρ 2\ \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=8% \eta^{\rho\sigma}\eta^{\mu\nu}-8\eta^{\nu\sigma}\eta^{\mu\rho}+8\eta^{\mu% \sigma}\eta^{\nu\rho}\,
  171. tr ( γ μ γ ν γ ρ γ σ ) = 4 ( η ρ σ η μ ν - η ν σ η μ ρ + η μ σ η ν ρ ) \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma})=4\left% (\eta^{\rho\sigma}\eta^{\mu\nu}-\eta^{\nu\sigma}\eta^{\mu\rho}+\eta^{\mu\sigma% }\eta^{\nu\rho}\right)\,
  172. tr ( γ 5 ) = 0 \operatorname{tr}(\gamma^{5})=0
  173. tr ( γ 5 ) \operatorname{tr}(\gamma^{5})
  174. = tr ( γ 0 γ 0 γ 5 ) =\operatorname{tr}(\gamma^{0}\gamma^{0}\gamma^{5})
  175. γ 0 γ 0 = I 4 \gamma^{0}\gamma^{0}=I_{4}\,
  176. = - tr ( γ 0 γ 5 γ 0 ) =-\operatorname{tr}(\gamma^{0}\gamma^{5}\gamma^{0})
  177. γ 5 \gamma^{5}\,
  178. γ 0 \gamma^{0}\,
  179. = - tr ( γ 0 γ 0 γ 5 ) =-\operatorname{tr}(\gamma^{0}\gamma^{0}\gamma^{5})
  180. = - tr ( γ 5 ) =-\operatorname{tr}(\gamma^{5})\,
  181. γ 0 \gamma^{0}\,
  182. tr ( γ 5 ) \operatorname{tr}(\gamma^{5})
  183. 2 tr ( γ 5 ) = 0 2\operatorname{tr}(\gamma^{5})=0\,
  184. tr ( γ μ γ ν γ 5 ) = 0 \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{5})=0\,
  185. γ α \gamma^{\alpha}\,
  186. α \alpha\,
  187. μ \mu\,
  188. ν \nu\,
  189. tr ( γ μ γ ν γ 5 ) = 0 \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{5})=0\,
  190. ( μ ν ρ σ ) (\mu\nu\rho\sigma)\,
  191. tr ( γ μ γ ν γ ρ γ σ γ 5 ) \operatorname{tr}(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{% 5})\,
  192. ϵ μ ν ρ σ \epsilon^{\mu\nu\rho\sigma}\,
  193. ( ϵ 0123 = η 0 μ η 1 ν η 2 ρ η 3 σ ϵ μ ν ρ σ = η 00 η 11 η 22 η 33 ϵ 0123 = - 1 ) (\epsilon^{0123}=\eta^{0\mu}\eta^{1\nu}\eta^{2\rho}\eta^{3\sigma}\epsilon_{\mu% \nu\rho\sigma}=\eta^{00}\eta^{11}\eta^{22}\eta^{33}\epsilon_{0123}=-1)\,
  194. 4 i 4i\,
  195. ( μ ν ρ σ ) = ( 0123 ) (\mu\nu\rho\sigma)=(0123)\,
  196. γ 5 \gamma^{5}\,
  197. n n
  198. Γ = γ μ 1 γ μ 2 γ μ n . \Gamma=\gamma^{\mu 1}\gamma^{\mu 2}\dots\gamma^{\mu n}.
  199. Γ \Gamma
  200. Γ \Gamma^{\dagger}
  201. = γ μ n γ μ 2 γ μ 1 =\gamma^{\mu n\dagger}\dots\gamma^{\mu 2\dagger}\gamma^{\mu 1\dagger}
  202. = γ 0 γ μ n γ 0 γ 0 γ μ 2 γ 0 γ 0 γ μ 1 γ 0 =\gamma^{0}\gamma^{\mu n}\gamma^{0}\dots\gamma^{0}\gamma^{\mu 2}\gamma^{0}% \gamma^{0}\gamma^{\mu 1}\gamma^{0}
  203. γ 0 \gamma^{0}
  204. = γ 0 γ μ n γ μ 2 γ μ 1 γ 0 =\gamma^{0}\gamma^{\mu n}\dots\gamma^{\mu 2}\gamma^{\mu 1}\gamma^{0}
  205. γ 0 \gamma^{0}
  206. γ 0 \gamma^{0}
  207. γ 0 \gamma^{0}
  208. γ 0 Γ γ 0 \gamma^{0}\Gamma^{\dagger}\gamma^{0}
  209. Γ \Gamma
  210. tr ( γ 0 Γ γ 0 ) \operatorname{tr}(\gamma^{0}\Gamma^{\dagger}\gamma^{0})
  211. = tr ( Γ ) =\operatorname{tr}(\Gamma^{\dagger})
  212. = tr ( Γ * ) =\operatorname{tr}(\Gamma^{*})
  213. = tr ( Γ ) =\operatorname{tr}(\Gamma)
  214. ( γ 0 ) = γ 0 \left(\gamma^{0}\right)^{\dagger}=\gamma^{0}\,
  215. ( γ 0 ) 2 = I 4 \left(\gamma^{0}\right)^{2}=I_{4}\,
  216. ( γ k ) = - γ k \left(\gamma^{k}\right)^{\dagger}=-\gamma^{k}\,
  217. ( γ k ) 2 = - I 4 . \left(\gamma^{k}\right)^{2}=-I_{4}.\,
  218. ( γ μ ) = γ 0 γ μ γ 0 . \left(\gamma^{\mu}\right)^{\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}.\,
  219. γ μ S ( Λ ) γ μ S ( Λ ) - 1 \gamma^{\mu}\to S(\Lambda)\gamma^{\mu}{S(\Lambda)}^{-1}
  220. Λ \Lambda
  221. S ( Λ ) S(\Lambda)
  222. a / := γ μ a μ a\!\!\!/:=\gamma^{\mu}a_{\mu}
  223. a a
  224. a / b / = a b - i a μ σ μ ν b ν a\!\!\!/b\!\!\!/=a\cdot b-ia_{\mu}\sigma^{\mu\nu}b_{\nu}
  225. a / a / = a μ a ν γ μ γ ν = 1 2 a μ a ν ( γ μ γ ν + γ ν γ μ ) = η μ ν a μ a ν = a 2 a\!\!\!/a\!\!\!/=a^{\mu}a^{\nu}\gamma_{\mu}\gamma_{\nu}=\frac{1}{2}a^{\mu}a^{% \nu}(\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu})=\eta_{\mu\nu}a^{\mu}a^% {\nu}=a^{2}
  226. tr ( a / b / ) = 4 ( a b ) \operatorname{tr}(a\!\!\!/b\!\!\!/)=4(a\cdot b)
  227. tr ( a / b / c / d / ) = 4 [ ( a b ) ( c d ) - ( a c ) ( b d ) + ( a d ) ( b c ) ] \operatorname{tr}(a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/)=4\left[(a\cdot b)(c\cdot d% )-(a\cdot c)(b\cdot d)+(a\cdot d)(b\cdot c)\right]
  228. tr ( γ 5 a / b / c / d / ) = - 4 i ϵ μ ν ρ σ a μ b ν c ρ d σ \operatorname{tr}(\gamma_{5}a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/)=-4i\epsilon_{\mu% \nu\rho\sigma}a^{\mu}b^{\nu}c^{\rho}d^{\sigma}
  229. γ μ a / γ μ = - 2 a / \gamma_{\mu}a\!\!\!/\gamma^{\mu}=-2a\!\!\!/
  230. γ μ a / b / γ μ = 4 a b \gamma_{\mu}a\!\!\!/b\!\!\!/\gamma^{\mu}=4a\cdot b\,
  231. γ μ a / b / c / γ μ = - 2 c / b / a / \gamma_{\mu}a\!\!\!/b\!\!\!/c\!\!\!/\gamma^{\mu}=-2c\!\!\!/b\!\!\!/a\!\!\!/\,
  232. ϵ μ ν ρ σ \epsilon_{\mu\nu\rho\sigma}\,
  233. σ μ ν = i 2 [ γ μ , γ ν ] . \sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}].
  234. I 2 I_{2}
  235. γ k = ( 0 σ k - σ k 0 ) \gamma^{k}=\begin{pmatrix}0&\sigma^{k}\\ -\sigma^{k}&0\end{pmatrix}
  236. γ 0 = ( I 2 0 0 - I 2 ) , γ k = ( 0 σ k - σ k 0 ) , γ 5 = ( 0 I 2 I 2 0 ) . \gamma^{0}=\begin{pmatrix}I_{2}&0\\ 0&-I_{2}\end{pmatrix},\quad\gamma^{k}=\begin{pmatrix}0&\sigma^{k}\\ -\sigma^{k}&0\end{pmatrix},\quad\gamma^{5}=\begin{pmatrix}0&I_{2}\\ I_{2}&0\end{pmatrix}.
  237. γ k \gamma^{k}
  238. γ 0 \gamma^{0}
  239. γ 5 \gamma^{5}
  240. γ 0 = ( 0 I 2 I 2 0 ) , γ k = ( 0 σ k - σ k 0 ) , γ 5 = ( - I 2 0 0 I 2 ) , \gamma^{0}=\begin{pmatrix}0&I_{2}\\ I_{2}&0\end{pmatrix},\quad\gamma^{k}=\begin{pmatrix}0&\sigma^{k}\\ -\sigma^{k}&0\end{pmatrix},\quad\gamma^{5}=\begin{pmatrix}-I_{2}&0\\ 0&I_{2}\end{pmatrix},
  241. γ μ = ( 0 σ μ σ ¯ μ 0 ) , σ μ ( 1 , σ i ) , σ ¯ μ ( 1 , - σ i ) . \gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\ \bar{\sigma}^{\mu}&0\end{pmatrix},\quad\sigma^{\mu}\equiv(1,\sigma^{i}),\quad% \bar{\sigma}^{\mu}\equiv(1,-\sigma^{i}).
  242. ψ L = 1 2 ( 1 - γ 5 ) ψ = ( I 2 0 0 0 ) ψ , ψ R = 1 2 ( 1 + γ 5 ) ψ = ( 0 0 0 I 2 ) ψ . \psi_{L}=\frac{1}{2}(1-\gamma^{5})\psi=\begin{pmatrix}I_{2}&0\\ 0&0\end{pmatrix}\psi,\quad\psi_{R}=\frac{1}{2}(1+\gamma^{5})\psi=\begin{% pmatrix}0&0\\ 0&I_{2}\end{pmatrix}\psi.
  243. ψ L / R \psi_{L/R}
  244. ψ = ( ψ L ψ R ) , \psi=\begin{pmatrix}\psi_{L}\\ \psi_{R}\end{pmatrix},
  245. ψ L \psi_{L}
  246. ψ R \psi_{R}
  247. γ 0 = ( 0 - I 2 - I 2 0 ) , γ k = ( 0 σ k - σ k 0 ) , γ 5 = ( I 2 0 0 - I 2 ) . \gamma^{0}=\begin{pmatrix}0&-I_{2}\\ -I_{2}&0\end{pmatrix},\quad\gamma^{k}=\begin{pmatrix}0&\sigma^{k}\\ -\sigma^{k}&0\end{pmatrix},\quad\gamma^{5}=\begin{pmatrix}I_{2}&0\\ 0&-I_{2}\end{pmatrix}.
  248. ψ R = ( I 2 0 0 0 ) ψ , ψ L = ( 0 0 0 I 2 ) ψ . \psi_{R}=\begin{pmatrix}I_{2}&0\\ 0&0\end{pmatrix}\psi,\quad\psi_{L}=\begin{pmatrix}0&0\\ 0&I_{2}\end{pmatrix}\psi.
  249. ψ = ( ψ R ψ L ) , \psi=\begin{pmatrix}\psi_{R}\\ \psi_{L}\end{pmatrix},
  250. ψ L \psi_{L}
  251. ψ R \psi_{R}
  252. γ 0 = ( 0 σ 2 σ 2 0 ) , γ 1 = ( i σ 3 0 0 i σ 3 ) \gamma^{0}=\begin{pmatrix}0&\sigma^{2}\\ \sigma^{2}&0\end{pmatrix},\quad\gamma^{1}=\begin{pmatrix}i\sigma^{3}&0\\ 0&i\sigma^{3}\end{pmatrix}
  253. γ 2 = ( 0 - σ 2 σ 2 0 ) , γ 3 = ( - i σ 1 0 0 - i σ 1 ) , γ 5 = ( σ 2 0 0 - σ 2 ) . \gamma^{2}=\begin{pmatrix}0&-\sigma^{2}\\ \sigma^{2}&0\end{pmatrix},\quad\gamma^{3}=\begin{pmatrix}-i\sigma^{1}&0\\ 0&-i\sigma^{1}\end{pmatrix},\quad\gamma^{5}=\begin{pmatrix}\sigma^{2}&0\\ 0&-\sigma^{2}\end{pmatrix}.
  254. i i
  255. i i
  256. C l 1 , 3 ( ) = C l 1 , 3 ( ) Cl_{1,3}(\mathbb{C})=Cl_{1,3}(\mathbb{R})\otimes\mathbb{C}
  257. γ 1 , 2 , 3 = ( 0 i σ 1 , 2 , 3 - i σ 1 , 2 , 3 0 ) , γ 4 = ( 0 I 2 I 2 0 ) \gamma^{1,2,3}=\begin{pmatrix}0&i\sigma^{1,2,3}\\ -i\sigma^{1,2,3}&0\end{pmatrix},\quad\gamma^{4}=\begin{pmatrix}0&I_{2}\\ I_{2}&0\end{pmatrix}
  258. i i
  259. { γ μ , γ ν } = 2 δ μ ν I 4 \{\gamma^{\mu},\gamma^{\nu}\}=2\delta^{\mu\nu}I_{4}
  260. - i -i
  261. γ 5 = i γ 1 γ 2 γ 3 γ 4 = γ 5 + . \gamma^{5}=i\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}=\gamma^{5+}.
  262. γ 5 = i γ 1 γ 2 γ 3 γ 4 = - i ( I 2 0 0 - I 2 ) . \gamma^{5}=i\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}=-i\begin{pmatrix}I_{2}&0% \\ 0&-I_{2}\end{pmatrix}.
  263. γ 1 , 2 , 3 = ( 0 - i σ 1 , 2 , 3 i σ 1 , 2 , 3 0 ) , γ 4 = ( I 2 0 0 - I 2 ) , γ 5 = ( 0 - I 2 - I 2 0 ) \gamma^{1,2,3}=\begin{pmatrix}0&-i\sigma^{1,2,3}\\ i\sigma^{1,2,3}&0\end{pmatrix},\quad\gamma^{4}=\begin{pmatrix}I_{2}&0\\ 0&-I_{2}\end{pmatrix},\quad\gamma^{5}=\begin{pmatrix}0&-I_{2}\\ -I_{2}&0\end{pmatrix}
  264. γ < s u p > 5 γ<sup>5

Gap_theorem.html

  1. Φ \Phi
  2. g ( x ) x g(x)\geq x
  3. x \,x
  4. Φ \Phi
  5. t t
  6. g t g\circ t
  7. g : ω ω g\,:\,\omega\,\to\,\omega
  8. g ( x ) x g(x)\geq x
  9. x \,x
  10. T ( n ) T(n)
  11. D T I M E ( g ( T ( n ) ) ) = D T I M E ( T ( n ) ) DTIME(g(T(n)))=DTIME(T(n))

Gas_engine.html

  1. Q = P η 1 L H V g a s Q=\frac{P}{\eta}\cdot\frac{1}{LHV_{gas}}
  2. Q Q
  3. P {P}
  4. η {\eta}

Gauge_covariant_derivative.html

  1. t 𝐯 := t 𝐯 + ( 𝐯 ) 𝐯 \nabla_{t}\mathbf{v}:=\partial_{t}\mathbf{v}+(\mathbf{v}\cdot\nabla)\mathbf{v}
  2. 𝐯 \mathbf{v}
  3. D μ := μ - i e A μ D_{\mu}:=\partial_{\mu}-ieA_{\mu}
  4. A μ A_{\mu}
  5. ( - , + , + , + ) (-,+,+,+)
  6. ( + , - , - , - ) (+,-,-,-)
  7. ϕ ( x ) U ( x ) ϕ ( x ) e i α ( x ) ϕ ( x ) , \phi(x)\rightarrow U(x)\phi(x)\equiv e^{i\alpha(x)}\phi(x),
  8. ϕ ( x ) ϕ ( x ) U ( x ) ϕ ( x ) e - i α ( x ) , U = U - 1 . \phi^{\dagger}(x)\rightarrow\phi^{\dagger}(x)U^{\dagger}(x)\equiv\phi^{\dagger% }(x)e^{-i\alpha(x)},\qquad U^{\dagger}=U^{-1}.
  9. α ( x ) \alpha(x)
  10. α ( x ) = α a ( x ) t a \alpha(x)=\alpha^{a}(x)t^{a}
  11. μ \partial_{\mu}
  12. μ ϕ ( x ) U ( x ) μ ϕ ( x ) + ( μ U ) ϕ ( x ) e i α ( x ) μ ϕ ( x ) + i ( μ α ) e i α ( x ) ϕ ( x ) \partial_{\mu}\phi(x)\rightarrow U(x)\partial_{\mu}\phi(x)+(\partial_{\mu}U)% \phi(x)\equiv e^{i\alpha(x)}\partial_{\mu}\phi(x)+i(\partial_{\mu}\alpha)e^{i% \alpha(x)}\phi(x)
  13. ϕ μ ϕ \phi^{\dagger}\partial_{\mu}\phi
  14. D μ D_{\mu}
  15. μ \partial_{\mu}
  16. D μ ϕ ( x ) D μ ϕ ( x ) = U ( x ) D μ ϕ ( x ) , D_{\mu}\phi(x)\rightarrow D^{\prime}_{\mu}\phi^{\prime}(x)=U(x)D_{\mu}\phi(x),
  17. D μ = U ( x ) D μ U ( x ) . D^{\prime}_{\mu}=U(x)D_{\mu}U^{\dagger}(x).
  18. x x
  19. D μ ϕ D μ U ϕ = U D μ ϕ + ( δ D μ U + [ D μ , U ] ) ϕ D_{\mu}\phi\rightarrow D^{\prime}_{\mu}U\phi=UD_{\mu}\phi+(\delta D_{\mu}U+[D_% {\mu},U])\phi
  20. D μ D μ D μ + δ D μ , D_{\mu}\rightarrow D^{\prime}_{\mu}\equiv D_{\mu}+\delta D_{\mu},
  21. A μ A μ = A μ + δ A μ . A_{\mu}\rightarrow A^{\prime}_{\mu}=A_{\mu}+\delta A_{\mu}.
  22. D μ D_{\mu}
  23. ( δ D μ U + [ D μ , U ] ) ϕ = 0. (\delta D_{\mu}U+[D_{\mu},U])\phi=0.
  24. D μ = μ - i g A μ , D_{\mu}=\partial_{\mu}-igA_{\mu},
  25. δ D μ - i g δ A μ \delta D_{\mu}\equiv-ig\delta A_{\mu}
  26. δ A μ = [ U , A μ ] U - i g ( μ U ) U \delta A_{\mu}=[U,A_{\mu}]U^{\dagger}-\frac{i}{g}(\partial_{\mu}U)U^{\dagger}
  27. U ( x ) = 1 + i α ( x ) + 𝒪 ( α 2 ) U(x)=1+i\alpha(x)+\mathcal{O}(\alpha^{2})
  28. δ A μ = 1 g ( μ α - i g [ A μ , α ] ) + 𝒪 ( α 2 ) = 1 g D μ α + 𝒪 ( α 2 ) \delta A_{\mu}=\frac{1}{g}(\partial_{\mu}\alpha-ig[A_{\mu},\alpha])+\mathcal{O% }(\alpha^{2})=\frac{1}{g}D_{\mu}\alpha+\mathcal{O}(\alpha^{2})
  29. D μ D_{\mu}
  30. ϕ ( x ) D μ ϕ ( x ) ϕ ( x ) D μ ϕ ( x ) = ϕ ( x ) D μ ϕ ( x ) \phi^{\dagger}(x)D_{\mu}\phi(x)\rightarrow\phi^{\prime\dagger}(x)D^{\prime}_{% \mu}\phi^{\prime}(x)=\phi^{\dagger}(x)D_{\mu}\phi(x)
  31. ψ e i Λ ψ \psi\mapsto e^{i\Lambda}\psi
  32. A μ A μ + 1 e ( μ Λ ) A_{\mu}\mapsto A_{\mu}+{1\over e}(\partial_{\mu}\Lambda)
  33. D μ D_{\mu}
  34. D μ μ - i e A μ - i ( μ Λ ) D_{\mu}\mapsto\partial_{\mu}-ieA_{\mu}-i(\partial_{\mu}\Lambda)
  35. D μ ψ D_{\mu}\psi
  36. D μ ψ e i Λ D μ ψ D_{\mu}\psi\mapsto e^{i\Lambda}D_{\mu}\psi
  37. ψ ¯ := ψ γ 0 \bar{\psi}:=\psi^{\dagger}\gamma^{0}
  38. ψ ¯ ψ ¯ e - i Λ \bar{\psi}\mapsto\bar{\psi}e^{-i\Lambda}
  39. ψ ¯ D μ ψ ψ ¯ D μ ψ \bar{\psi}D_{\mu}\psi\mapsto\bar{\psi}D_{\mu}\psi
  40. ψ ¯ D μ ψ \bar{\psi}D_{\mu}\psi
  41. μ \partial_{\mu}
  42. ψ ¯ μ ψ ψ ¯ μ ψ + i ψ ¯ ( μ Λ ) ψ \bar{\psi}\partial_{\mu}\psi\mapsto\bar{\psi}\partial_{\mu}\psi+i\bar{\psi}(% \partial_{\mu}\Lambda)\psi
  43. D μ := μ - i g A μ α λ α D_{\mu}:=\partial_{\mu}-ig\,A_{\mu}^{\alpha}\,\lambda_{\alpha}
  44. g g
  45. A A
  46. α = 1 8 \alpha=1\dots 8
  47. ψ \psi
  48. λ α \lambda_{\alpha}
  49. α = 1 8 \alpha=1\dots 8
  50. D μ := μ - i g 1 2 Y B μ - i g 2 2 σ j W μ j - i g 3 2 λ α G μ α D_{\mu}:=\partial_{\mu}-i\frac{g_{1}}{2}\,Y\,B_{\mu}-i\frac{g_{2}}{2}\,\sigma_% {j}\,W_{\mu}^{j}-i\frac{g_{3}}{2}\,\lambda_{\alpha}\,G_{\mu}^{\alpha}

Gauss's_constant.html

  1. G = 1 agm ( 1 , 2 ) = 0.8346268 . G=\frac{1}{\mathrm{agm}(1,\sqrt{2})}=0.8346268\dots.
  2. G = 2 π 0 1 d x 1 - x 4 G=\frac{2}{\pi}\int_{0}^{1}\frac{dx}{\sqrt{1-x^{4}}}
  3. G = 1 2 π B ( 1 4 , 1 2 ) G=\frac{1}{2\pi}B(\tfrac{1}{4},\tfrac{1}{2})
  4. Γ ( 1 4 ) = 2 G 2 π 3 \Gamma(\tfrac{1}{4})=\sqrt{2G\sqrt{2\pi^{3}}}
  5. G = [ Γ ( 1 4 ) ] 2 2 2 π 3 G=\frac{[\Gamma(\tfrac{1}{4})]^{2}}{2\sqrt{2\pi^{3}}}
  6. L 1 = π G L_{1}\;=\;\pi G
  7. L 2 = 1 2 G L_{2}\,\,=\,\,\frac{1}{2G}
  8. G = ϑ 01 2 ( e - π ) G=\vartheta_{01}^{2}(e^{-\pi})
  9. G = 32 4 e - π 3 ( n = - ( - 1 ) n e - 2 n π ( 3 n + 1 ) ) 2 . G=\sqrt[4]{32}e^{-\frac{\pi}{3}}\left(\sum_{n=-\infty}^{\infty}(-1)^{n}e^{-2n% \pi(3n+1)}\right)^{2}.
  10. G = m = 1 tanh 2 ( π m 2 ) . G=\prod_{m=1}^{\infty}\tanh^{2}\left(\frac{\pi m}{2}\right).
  11. 1 G = 0 π / 2 sin ( x ) d x = 0 π / 2 cos ( x ) d x {\frac{1}{G}}=\int_{0}^{\pi/2}\sqrt{\sin(x)}dx=\int_{0}^{\pi/2}\sqrt{\cos(x)}dx
  12. G = 0 d x cosh ( π x ) G=\int_{0}^{\infty}{\frac{dx}{\sqrt{\cosh(\pi x)}}}

Gauss's_lemma_(number_theory).html

  1. a , 2 a , 3 a , , p - 1 2 a a,2a,3a,\dots,\frac{p-1}{2}a
  2. ( a p ) = ( - 1 ) n \left(\frac{a}{p}\right)=(-1)^{n}
  3. ( 7 11 ) = ( - 1 ) 3 = - 1. \left(\frac{7}{11}\right)=(-1)^{3}=-1.
  4. Z = a 2 a 3 a p - 1 2 a Z=a\cdot 2a\cdot 3a\cdot\cdots\cdot\frac{p-1}{2}a
  5. Z = a ( p - 1 ) / 2 ( 1 2 3 p - 1 2 ) Z=a^{(p-1)/2}\left(1\cdot 2\cdot 3\cdot\cdots\cdot\frac{p-1}{2}\right)
  6. | x | = { x if 1 x p - 1 2 , p - x if p + 1 2 x p - 1. |x|=\begin{cases}x&\mbox{if }~{}1\leq x\leq\frac{p-1}{2},\\ p-x&\mbox{if }~{}\frac{p+1}{2}\leq x\leq p-1.\end{cases}
  7. Z = ( - 1 ) n ( | a | | 2 a | | 3 a | | p - 1 2 a | ) . Z=(-1)^{n}\left(|a|\cdot|2a|\cdot|3a|\cdot\cdots\cdots\left|\frac{p-1}{2}a% \right|\right).
  8. Z = ( - 1 ) n ( 1 2 3 p - 1 2 ) . Z=(-1)^{n}\left(1\cdot 2\cdot 3\cdot\cdots\cdot\frac{p-1}{2}\right).
  9. 1 2 3 p - 1 2 1\cdot 2\cdot 3\cdot\cdots\cdot\frac{p-1}{2}
  10. a ( p - 1 ) / 2 = ( - 1 ) n . a^{(p-1)/2}=(-1)^{n}.
  11. ( a p ) = n = 1 ( p - 1 ) / 2 sin ( 2 π a n / p ) sin ( 2 π n / p ) , \left(\frac{a}{p}\right)=\prod_{n=1}^{(p-1)/2}\frac{\sin{(2\pi an/p)}}{\sin{(2% \pi n/p)}},
  12. ( p q ) = sgn i = 1 q - 1 2 k = 1 p - 1 2 ( k p - i q ) . \left(\frac{p}{q}\right)=\operatorname{sgn}\prod_{i=1}^{\frac{q-1}{2}}\prod_{k% =1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right).
  13. ( 2 p ) = ( - 1 ) ( p 2 - 1 ) / 8 = { + 1 if p ± 1 ( mod 8 ) - 1 if p ± 3 ( mod 8 ) \left(\frac{2}{p}\right)=(-1)^{(p^{2}-1)/8}=\begin{cases}+1\,\text{ if }p% \equiv\pm 1\;\;(\mathop{{\rm mod}}8)\\ -1\,\text{ if }p\equiv\pm 3\;\;(\mathop{{\rm mod}}8)\end{cases}
  14. 𝒪 k , \mathcal{O}_{k},
  15. 𝔭 𝒪 k \mathfrak{p}\subset\mathcal{O}_{k}
  16. 𝔭 \mathfrak{p}
  17. 𝔭 \mathfrak{p}
  18. 𝒪 k / 𝔭 : N 𝔭 = | 𝒪 k / 𝔭 | . \mathcal{O}_{k}/\mathfrak{p}\;:\;\;\;\mathrm{N}\mathfrak{p}=|\mathcal{O}_{k}/% \mathfrak{p}|.
  19. ζ n 𝒪 k , \zeta_{n}\in\mathcal{O}_{k},
  20. 𝔭 \mathfrak{p}
  21. n 𝔭 . n\not\in\mathfrak{p}.
  22. ( mod 𝔭 ) . \;\;(\mathop{{\rm mod}}\mathfrak{p}).
  23. ζ n r ζ n s ( mod 𝔭 ) , 0 < r < s n . \zeta_{n}^{r}\equiv\zeta_{n}^{s}\;\;(\mathop{{\rm mod}}\mathfrak{p}),\;\;0<r<s% \leq n.
  24. t = s - r , ζ n t 1 ( mod 𝔭 ) , t=s-r,\;\;\zeta_{n}^{t}\equiv 1\;\;(\mathop{{\rm mod}}\mathfrak{p}),
  25. 0 < t < n . 0<t<n.
  26. x n - 1 = ( x - 1 ) ( x - ζ n ) ( x - ζ n 2 ) ( x - ζ n n - 1 ) , x^{n}-1=(x-1)(x-\zeta_{n})(x-\zeta_{n}^{2})\dots(x-\zeta_{n}^{n-1}),
  27. x n - 1 + x n - 2 + + x + 1 = ( x - ζ n ) ( x - ζ n 2 ) ( x - ζ n n - 1 ) . x^{n-1}+x^{n-2}+\dots+x+1=(x-\zeta_{n})(x-\zeta_{n}^{2})\dots(x-\zeta_{n}^{n-1% }).
  28. ( mod 𝔭 ) , \;\;(\mathop{{\rm mod}}\mathfrak{p}),
  29. n ( 1 - ζ n ) ( 1 - ζ n 2 ) ( 1 - ζ n n - 1 ) ( mod 𝔭 ) . n\equiv(1-\zeta_{n})(1-\zeta_{n}^{2})\dots(1-\zeta_{n}^{n-1})\;\;(\mathop{{\rm mod% }}\mathfrak{p}).
  30. 𝔭 \mathfrak{p}
  31. n 0 ( mod 𝔭 ) , n\not\equiv 0\;\;(\mathop{{\rm mod}}\mathfrak{p}),
  32. 𝒪 k / 𝔭 \mathcal{O}_{k}/\mathfrak{p}
  33. ( 𝒪 k / 𝔭 ) × = 𝒪 k / 𝔭 - { 0 } . (\mathcal{O}_{k}/\mathfrak{p})^{\times}=\mathcal{O}_{k}/\mathfrak{p}-\{0\}.
  34. ( 𝒪 k / 𝔭 ) × (\mathcal{O}_{k}/\mathfrak{p})^{\times}
  35. N 𝔭 = | 𝒪 k / 𝔭 | = | ( 𝒪 k / 𝔭 ) × | + 1 1 ( mod n ) . \mathrm{N}\mathfrak{p}=|\mathcal{O}_{k}/\mathfrak{p}|=|(\mathcal{O}_{k}/% \mathfrak{p})^{\times}|+1\equiv 1\;\;(\mathop{{\rm mod}}n).
  36. 𝒪 k : \mathcal{O}_{k}:
  37. α 𝒪 k , α 𝔭 , \alpha\in\mathcal{O}_{k},\;\;\;\alpha\not\in\mathfrak{p},
  38. α N 𝔭 - 1 1 ( mod 𝔭 ) , \alpha^{\mathrm{N}\mathfrak{p}-1}\equiv 1\;\;(\mathop{{\rm mod}}\mathfrak{p}),
  39. N 𝔭 1 ( mod n ) , \mathrm{N}\mathfrak{p}\equiv 1\;\;(\mathop{{\rm mod}}n),
  40. α N 𝔭 - 1 n ζ n s ( mod 𝔭 ) \alpha^{\frac{\mathrm{N}\mathfrak{p}-1}{n}}\equiv\zeta_{n}^{s}\;\;(\mathop{{% \rm mod}}\mathfrak{p})
  41. 𝒪 k , \mathcal{O}_{k},
  42. ( α 𝔭 ) n = ζ n s α N 𝔭 - 1 n ( mod 𝔭 ) . \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}=\zeta_{n}^{s}\equiv\alpha^{\frac{% \mathrm{N}\mathfrak{p}-1}{n}}\;\;(\mathop{{\rm mod}}\mathfrak{p}).
  43. ( α 𝔭 ) n = 1 if and only if there is an η 𝒪 k such that α η n ( mod 𝔭 ) . \left(\frac{\alpha}{\mathfrak{p}}\right)_{n}=1\mbox{ if and only if there is % an }~{}\eta\in\mathcal{O}_{k}\;\;\mbox{ such that }~{}\;\;\alpha\equiv\eta^{n}% \;\;(\mathop{{\rm mod}}\mathfrak{p}).
  44. μ n = { 1 , ζ n , ζ n 2 , , ζ n n - 1 } \mu_{n}=\{1,\zeta_{n},\zeta_{n}^{2},\dots,\zeta_{n}^{n-1}\}
  45. A = { a 1 , a 2 , , a m } A=\{a_{1},a_{2},\dots,a_{m}\}
  46. ( 𝒪 k / 𝔭 ) × / μ n . (\mathcal{O}_{k}/\mathfrak{p})^{\times}/\mu_{n}.
  47. ( mod ) p . \;\;(\mathop{{\rm mod}}\mathfrak{}){p}.
  48. m n = N 𝔭 - 1 mn=\mathrm{N}\mathfrak{p}-1
  49. A μ = { a i ζ n j : 1 i m , 0 j n - 1 } , A\mu=\{a_{i}\zeta_{n}^{j}\;:\;1\leq i\leq m,\;\;\;0\leq j\leq n-1\},
  50. ( 𝒪 k / 𝔭 ) × . (\mathcal{O}_{k}/\mathfrak{p})^{\times}.
  51. ( 𝒪 k / 𝔭 ) × . (\mathcal{O}_{k}/\mathfrak{p})^{\times}.
  52. a 1 M a_{1}\in M
  53. a 1 , a 1 ζ n , a 1 ζ n 2 , , a 1 ζ n n - 1 a_{1},a_{1}\zeta_{n},a_{1}\zeta_{n}^{2},\dots,a_{1}\zeta_{n}^{n-1}
  54. a 2 , a 2 ζ n , a 2 ζ n 2 , , a 2 ζ n n - 1 a_{2},a_{2}\zeta_{n},a_{2}\zeta_{n}^{2},\dots,a_{2}\zeta_{n}^{n-1}
  55. ( mod ) p . \;\;(\mathop{{\rm mod}}\mathfrak{}){p}.
  56. ζ n 𝒪 k \zeta_{n}\in\mathcal{O}_{k}
  57. 𝔭 𝒪 k \mathfrak{p}\subset\mathcal{O}_{k}
  58. γ 𝒪 k , n γ 𝔭 , \gamma\in\mathcal{O}_{k},\;\;n\gamma\not\in\mathfrak{p},
  59. 𝔭 \mathfrak{p}
  60. ( mod 𝔭 ) . \;\;(\mathop{{\rm mod}}\mathfrak{p}).
  61. γ a i ζ n b ( i ) a π ( i ) ( mod 𝔭 ) , \gamma a_{i}\equiv\zeta_{n}^{b(i)}a_{\pi(i)}\;\;(\mathop{{\rm mod}}\mathfrak{p% }),
  62. ( γ 𝔭 ) n = ζ n b ( 1 ) + b ( 2 ) + + b ( m ) . \left(\frac{\gamma}{\mathfrak{p}}\right)_{n}=\zeta_{n}^{b(1)+b(2)+\dots+b(m)}.
  63. γ a j ζ n s a p ( mod 𝔭 ) . \gamma a_{j}\equiv\zeta_{n}^{s}a_{p}\;\;(\mathop{{\rm mod}}\mathfrak{p}).
  64. ζ n s - r γ a i ζ n s a p γ a j ( mod 𝔭 ) \zeta_{n}^{s-r}\gamma a_{i}\equiv\zeta_{n}^{s}a_{p}\equiv\gamma a_{j}\;\;(% \mathop{{\rm mod}}\mathfrak{p})
  65. 𝔭 \mathfrak{p}
  66. ζ n s - r a i a j ( mod 𝔭 ) , \zeta_{n}^{s-r}a_{i}\equiv a_{j}\;\;(\mathop{{\rm mod}}\mathfrak{p}),
  67. ( γ a 1 ) ( γ a 2 ) ( γ a m ) = γ N 𝔭 - 1 n a 1 a 2 a m ( γ 𝔭 ) n a 1 a 2 a m ( mod 𝔭 ) , \begin{aligned}\displaystyle(\gamma a_{1})(\gamma a_{2})\dots(\gamma a_{m})&% \displaystyle=\gamma^{\frac{\mathrm{N}\mathfrak{p}-1}{n}}a_{1}a_{2}\dots a_{m}% \\ &\displaystyle\equiv\left(\frac{\gamma}{\mathfrak{p}}\right)_{n}a_{1}a_{2}% \dots a_{m}\;\;(\mathop{{\rm mod}}\mathfrak{p}),\end{aligned}
  68. ( γ a 1 ) ( γ a 2 ) ( γ a m ) ζ n b ( 1 ) a π ( 1 ) ζ n b ( 2 ) a π ( 2 ) ζ n b ( m ) a π ( m ) ζ n b ( 1 ) + b ( 2 ) + + b ( m ) a π ( 1 ) a π ( 2 ) a π ( m ) ζ n b ( 1 ) + b ( 2 ) + + b ( m ) a 1 a 2 a m ( mod 𝔭 ) , \begin{aligned}\displaystyle(\gamma a_{1})(\gamma a_{2})\dots(\gamma a_{m})&% \displaystyle\equiv{\zeta_{n}^{b(1)}a_{\pi(1)}}{\zeta_{n}^{b(2)}a_{\pi(2)}}% \dots{\zeta_{n}^{b(m)}a_{\pi(m)}}\\ &\displaystyle\equiv\zeta_{n}^{b(1)+b(2)+\dots+b(m)}a_{\pi(1)}a_{\pi(2)}\dots a% _{\pi(m)}\\ &\displaystyle\equiv\zeta_{n}^{b(1)+b(2)+\dots+b(m)}a_{1}a_{2}\dots a_{m}\;\;(% \mathop{{\rm mod}}\mathfrak{p}),\end{aligned}
  69. ( γ 𝔭 ) n a 1 a 2 a m ζ n b ( 1 ) + b ( 2 ) + + b ( m ) a 1 a 2 a m ( mod 𝔭 ) , \left(\frac{\gamma}{\mathfrak{p}}\right)_{n}a_{1}a_{2}\dots a_{m}\equiv\zeta_{% n}^{b(1)+b(2)+\dots+b(m)}a_{1}a_{2}\dots a_{m}\;\;(\mathop{{\rm mod}}\mathfrak% {p}),
  70. 𝔭 \mathfrak{p}
  71. ( γ 𝔭 ) n ζ n b ( 1 ) + b ( 2 ) + + b ( m ) ( mod 𝔭 ) , \left(\frac{\gamma}{\mathfrak{p}}\right)_{n}\equiv\zeta_{n}^{b(1)+b(2)+\dots+b% (m)}\;\;(\mathop{{\rm mod}}\mathfrak{p}),
  72. 𝔭 \mathfrak{p}
  73. 1 , 2 , 3 , , p - 1 2 . 1,2,3,\dots,\frac{p-1}{2}.
  74. ϕ : G H , \phi:G\to H,

Gaussian_polar_coordinates.html

  1. d s 2 = - f ( r ) 2 d t 2 + d r 2 + g ( r ) 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) , ds^{2}=-f(r)^{2}\,dt^{2}+dr^{2}+g(r)^{2}\,\left(d\theta^{2}+\sin(\theta)^{2}\,% d\phi^{2}\right),
  2. - < t < , r 0 < r < r 1 , 0 < θ < π , - π < ϕ < π -\infty<t<\infty,\,r_{0}<r<r_{1},\,0<\theta<\pi,\,-\pi<\phi<\pi

Gauss–Kuzmin_distribution.html

  1. p ( k ) = - log 2 ( 1 - 1 ( 1 + k ) 2 ) . p(k)=-\log_{2}\left(1-\frac{1}{(1+k)^{2}}\right)~{}.
  2. x = 1 k 1 + 1 k 2 + x=\frac{1}{k_{1}+\frac{1}{k_{2}+\cdots}}
  3. lim n { k n = k } = - log 2 ( 1 - 1 ( k + 1 ) 2 ) . \lim_{n\to\infty}\mathbb{P}\left\{k_{n}=k\right\}=-\log_{2}\left(1-\frac{1}{(k% +1)^{2}}\right)~{}.
  4. x n = 1 k n + 1 + 1 k n + 2 + ; x_{n}=\frac{1}{k_{n+1}+\frac{1}{k_{n+2}+\cdots}}~{};
  5. Δ n ( s ) = { x n s } - log 2 ( 1 + s ) \Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s)
  6. | Δ n ( s ) | C exp ( - α n ) . |\Delta_{n}(s)|\leq C\exp(-\alpha\sqrt{n})~{}.
  7. | Δ n ( s ) | C 0.7 n . |\Delta_{n}(s)|\leq C\,0.7^{n}~{}.
  8. Ψ ( s ) = lim n Δ n ( s ) ( - λ ) n \Psi(s)=\lim_{n\to\infty}\frac{\Delta_{n}(s)}{(-\lambda)^{n}}

Gâteaux_derivative.html

  1. d F ( u ; ) : X Y . dF(u;\cdot):X\rightarrow Y.
  2. d F ( u ; α ψ ) = α d F ( u ; ψ ) . dF(u;\alpha\psi)=\alpha dF(u;\psi).\,
  3. F ( x , y ) = { x 3 x 2 + y 2 if ( x , y ) ( 0 , 0 ) 0 if ( x , y ) = ( 0 , 0 ) . F(x,y)=\begin{cases}\frac{x^{3}}{x^{2}+y^{2}}&\mbox{ if }~{}(x,y)\neq(0,0)\\ 0&\mbox{ if }~{}(x,y)=(0,0).\end{cases}
  4. d F ( 0 , 0 ; a , b ) = { a 3 a 2 + b 2 ( a , b ) ( 0 , 0 ) 0 ( a , b ) = ( 0 , 0 ) . dF(0,0;a,b)=\begin{cases}\frac{a^{3}}{a^{2}+b^{2}}&(a,b)\not=(0,0)\\ 0&(a,b)=(0,0).\end{cases}
  5. D F ( u ) : ψ d F ( u ; ψ ) DF(u):\psi\mapsto dF(u;\psi)
  6. d F : U × X Y dF:U\times X\rightarrow Y\,
  7. u D F ( u ) u\mapsto DF(u)\,
  8. U L ( X , Y ) U\to L(X,Y)\,
  9. D F : U × X Y DF:U\times X\to Y
  10. D 2 F : U × X × X Y D^{2}F:U\times X\times X\to Y
  11. D 2 F ( u ) { h , k } = 1 2 d 2 F ( u ; h + k ) - d 2 F ( u ; h ) - d 2 F ( u ; k ) D^{2}F(u)\{h,k\}=\frac{1}{2}d^{2}F(u;h+k)-d^{2}F(u;h)-d^{2}F(u;k)
  12. F ( u + h ) - F ( u ) = 0 1 d F ( u + t h ; h ) d t F(u+h)-F(u)=\int_{0}^{1}dF(u+th;h)\,dt
  13. d ( G F ) ( u ; x ) = d G ( F ( u ) ; d F ( u ; x ) ) d(G\circ F)(u;x)=dG(F(u);dF(u;x))
  14. F ( u + h ) = F ( u ) + d F ( u ; h ) + 1 2 ! d 2 F ( u ; h ) + + 1 ( k - 1 ) ! d k - 1 F ( u ; h ) + R k F(u+h)=F(u)+dF(u;h)+\frac{1}{2!}d^{2}F(u;h)+\dots+\frac{1}{(k-1)!}d^{k-1}F(u;h% )+R_{k}
  15. R k ( u ; h ) = 1 ( k - 1 ) ! 0 1 ( 1 - t ) k - 1 d k F ( u + t h ; h ) d t R_{k}(u;h)=\frac{1}{(k-1)!}\int_{0}^{1}(1-t)^{k-1}d^{k}F(u+th;h)\,dt
  16. X X
  17. Ω \Omega
  18. E : X E:X\rightarrow\mathbb{R}
  19. E ( u ) = Ω F ( u ( x ) ) d x E(u)=\int_{\Omega}F\left(u(x)\right)dx
  20. d E ( u , ψ ) = f ( u ) , ψ . dE(u,\psi)=\langle f(u),\psi\rangle\,.
  21. E ( u + τ ψ ) - E ( u ) τ = 1 τ ( Ω F ( u + τ ψ ) d x - Ω F ( u ) d x ) \frac{E(u+\tau\psi)-E(u)}{\tau}=\frac{1}{\tau}\left(\int_{\Omega}F(u+\tau\psi)% dx-\int_{\Omega}F(u)dx\right)
  22. = 1 τ ( Ω 0 1 d d s F ( u + s τ ψ ) d s d x ) \quad\quad=\frac{1}{\tau}\left(\int_{\Omega}\int_{0}^{1}\frac{d}{ds}F(u+s\tau% \psi)\,ds\,dx\right)
  23. = Ω 0 1 f ( u + s τ ψ ) ψ d s d x . \quad\quad=\int_{\Omega}\int_{0}^{1}f(u+s\tau\psi)\psi\,ds\,dx.
  24. d E ( u , ψ ) = Ω f ( u ( x ) ) ψ ( x ) d x , dE(u,\psi)=\int_{\Omega}f(u(x))\psi(x)\,dx,

Gegenbauer_polynomials.html

  1. 1 ( 1 - 2 x t + t 2 ) α = n = 0 C n ( α ) ( x ) t n . \frac{1}{(1-2xt+t^{2})^{\alpha}}=\sum_{n=0}^{\infty}C_{n}^{(\alpha)}(x)t^{n}.
  2. C 0 α ( x ) \displaystyle C_{0}^{\alpha}(x)
  3. ( 1 - x 2 ) y ′′ - ( 2 α + 1 ) x y + n ( n + 2 α ) y = 0. (1-x^{2})y^{\prime\prime}-(2\alpha+1)xy^{\prime}+n(n+2\alpha)y=0.\,
  4. C n ( α ) ( z ) = ( 2 α ) n n ! 2 F 1 ( - n , 2 α + n ; α + 1 2 ; 1 - z 2 ) . C_{n}^{(\alpha)}(z)=\frac{(2\alpha)_{n}}{n!}\,_{2}F_{1}\left(-n,2\alpha+n;% \alpha+\frac{1}{2};\frac{1-z}{2}\right).
  5. C n ( α ) ( z ) = k = 0 n / 2 ( - 1 ) k Γ ( n - k + α ) Γ ( α ) k ! ( n - 2 k ) ! ( 2 z ) n - 2 k . C_{n}^{(\alpha)}(z)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^{k}\frac{\Gamma(n-k+% \alpha)}{\Gamma(\alpha)k!(n-2k)!}(2z)^{n-2k}.
  6. C n ( α ) ( x ) = ( 2 α ) n ( α + 1 2 ) n P n ( α - 1 / 2 , α - 1 / 2 ) ( x ) . C_{n}^{(\alpha)}(x)=\frac{(2\alpha)_{n}}{(\alpha+\frac{1}{2})_{n}}P_{n}^{(% \alpha-1/2,\alpha-1/2)}(x).
  7. ( θ ) n (\theta)_{n}
  8. θ \theta
  9. C n ( α ) ( x ) = ( - 2 ) n n ! Γ ( n + α ) Γ ( n + 2 α ) Γ ( α ) Γ ( 2 n + 2 α ) ( 1 - x 2 ) - α + 1 / 2 d n d x n [ ( 1 - x 2 ) n + α - 1 / 2 ] . C_{n}^{(\alpha)}(x)=\frac{(-2)^{n}}{n!}\frac{\Gamma(n+\alpha)\Gamma(n+2\alpha)% }{\Gamma(\alpha)\Gamma(2n+2\alpha)}(1-x^{2})^{-\alpha+1/2}\frac{d^{n}}{dx^{n}}% \left[(1-x^{2})^{n+\alpha-1/2}\right].
  10. w ( z ) = ( 1 - z 2 ) α - 1 2 . w(z)=\left(1-z^{2}\right)^{\alpha-\frac{1}{2}}.
  11. - 1 1 C n ( α ) ( x ) C m ( α ) ( x ) ( 1 - x 2 ) α - 1 2 d x = 0. \int_{-1}^{1}C_{n}^{(\alpha)}(x)C_{m}^{(\alpha)}(x)(1-x^{2})^{\alpha-\frac{1}{% 2}}\,dx=0.
  12. - 1 1 [ C n ( α ) ( x ) ] 2 ( 1 - x 2 ) α - 1 2 d x = π 2 1 - 2 α Γ ( n + 2 α ) n ! ( n + α ) [ Γ ( α ) ] 2 . \int_{-1}^{1}\left[C_{n}^{(\alpha)}(x)\right]^{2}(1-x^{2})^{\alpha-\frac{1}{2}% }\,dx=\frac{\pi 2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)[\Gamma(\alpha)]^{% 2}}.
  13. 1 | 𝐱 - 𝐲 | n - 2 = k = 0 | 𝐱 | k | 𝐲 | k + n - 2 C n , k ( α ) ( 𝐱 𝐲 ) . \frac{1}{|\mathbf{x}-\mathbf{y}|^{n-2}}=\sum_{k=0}^{\infty}\frac{|\mathbf{x}|^% {k}}{|\mathbf{y}|^{k+n-2}}C_{n,k}^{(\alpha)}(\mathbf{x}\cdot\mathbf{y}).
  14. C n , k ( ( n - 2 ) / 2 ) ( 𝐱 𝐲 ) C^{((n-2)/2)}_{n,k}(\mathbf{x}\cdot\mathbf{y})
  15. j = 0 n C j α ( x ) < m t p l > ( 2 α + j - 1 j ) 0 ( x - 1 , α 1 / 4 ) . \sum_{j=0}^{n}\frac{C_{j}^{\alpha}(x)}{<}mtpl>{{2\alpha+j-1\choose j}}\geq 0% \qquad(x\geq-1,\,\alpha\geq 1/4).

GEH_statistic.html

  1. G E H = 2 ( M - C ) 2 M + C GEH=\sqrt{\frac{2(M-C)^{2}}{M+C}}

Geiger–Nuttall_law.html

  1. ln λ = - a 1 Z E + a 2 \ln\lambda=-a_{1}\frac{Z}{\sqrt{E}}+a_{2}

Gelfand_pair.html

  1. dim Hom K ( π , 𝐂 ) dim Hom K ( π ~ , 𝐂 ) 1 \dim\mathrm{Hom}_{K}(\pi,\mathbf{C})\cdot\dim\mathrm{Hom}_{K}(\tilde{\pi},% \mathbf{C})\leq 1
  2. π ~ \tilde{\pi}
  3. ψ i \psi_{i}
  4. ψ i \psi_{i}
  5. ν \nu
  6. F ν F_{\nu}
  7. ν \nu
  8. [ 1 K , χ K G ] 1 [1_{K},\chi\downarrow^{G}_{K}]\leq 1
  9. [ 1 K , χ K G ] = [ 1 K G , χ ] [1_{K},\chi\downarrow^{G}_{K}]=[1\uparrow_{K}^{G},\chi]
  10. 1 K G 1\uparrow_{K}^{G}

Gell-Mann–Nishijima_formula.html

  1. Q = I 3 + 1 2 ( B + S ) . Q=I_{3}+\frac{1}{2}(B+S).
  2. Q = I 3 + 1 2 Y . Q=I_{3}+\frac{1}{2}Y.
  3. Q = I 3 + 1 2 ( B + S + C + B + T ) Q=I_{3}+\frac{1}{2}(B+S+C+B^{\prime}+T)
  4. Q = 2 3 [ ( n u - n u ¯ ) + ( n c - n c ¯ ) + ( n t - n t ¯ ) ] - 1 3 [ ( n d - n d ¯ ) + ( n s - n s ¯ ) + ( n b - n b ¯ ) ] Q=\frac{2}{3}\left[\left(n\text{u}-n_{\bar{\,\text{u}}}\right)+\left(n\text{c}% -n_{\bar{\,\text{c}}}\right)+\left(n\text{t}-n_{\bar{\,\text{t}}}\right)\right% ]-\frac{1}{3}\left[\left(n\text{d}-n_{\bar{\,\text{d}}}\right)+\left(n\text{s}% -n_{\bar{\,\text{s}}}\right)+\left(n\text{b}-n_{\bar{\,\text{b}}}\right)\right]
  5. B = 1 3 [ ( n u - n u ¯ ) + ( n c - n c ¯ ) + ( n t - n t ¯ ) + ( n d - n d ¯ ) + ( n s - n s ¯ ) + ( n b - n b ¯ ) ] B=\frac{1}{3}\left[\left(n\text{u}-n_{\bar{\,\text{u}}}\right)+\left(n\text{c}% -n_{\bar{\,\text{c}}}\right)+\left(n\text{t}-n_{\bar{\,\text{t}}}\right)+\left% (n\text{d}-n_{\bar{\,\text{d}}}\right)+\left(n\text{s}-n_{\bar{\,\text{s}}}% \right)+\left(n\text{b}-n_{\bar{\,\text{b}}}\right)\right]
  6. I 3 = 1 2 [ ( n u - n u ¯ ) - ( n d - n d ¯ ) ] I_{3}=\frac{1}{2}[(n\text{u}-n_{\bar{\,\text{u}}})-(n\text{d}-n_{\bar{\,\text{% d}}})]
  7. S = - ( n s - n s ¯ ) ; S=-\left(n\text{s}-n_{\bar{\,\text{s}}}\right);
  8. C = + ( n c - n c ¯ ) ; C=+\left(n\text{c}-n_{\bar{\,\text{c}}}\right);
  9. B = - ( n b - n b ¯ ) ; B^{\prime}=-\left(n\text{b}-n_{\bar{\,\text{b}}}\right);
  10. T = + ( n t - n t ¯ ) T=+\left(n\text{t}-n_{\bar{\,\text{t}}}\right)

General_selection_model.html

  1. Δ q = p q [ q ( W 2 - W 1 ) + p ( W 1 - W 0 ) ] W ¯ \Delta q=\frac{pq\big[q(W_{2}-W_{1})+p(W_{1}-W_{0})\big]}{\overline{W}}
  2. p p
  3. q q
  4. Δ q \Delta q
  5. W 0 , W 1 , W 2 W_{0},W_{1},W_{2}
  6. W ¯ \overline{W}
  7. p q pq
  8. p = q p=q
  9. Δ Q \Delta Q
  10. W ¯ \overline{W}
  11. Δ Q \Delta Q
  12. W ¯ \overline{W}
  13. [ q ( W 2 - W 1 ) + p ( W 1 - W 0 ) ] \big[q(W_{2}-W_{1})+p(W_{1}-W_{0})\big]

Generalizations_of_the_derivative.html

  1. f : \R \R f:\R\to\R
  2. lim h 0 f ( x + h ) - f ( x ) h \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
  3. L ( z ) = f ( x ) z - f ( x ) x + f ( x ) L(z)=f^{\prime}(x)z-f^{\prime}(x)x+f(x)
  4. ( x , f ( x ) ) (x,f(x))
  5. L ( z ) = f ( x ) z L(z)=f^{\prime}(x)z
  6. lim h 0 f ( x + h ) - f ( x ) - A ( x ) h h = 0. \lim_{\|h\|\to 0}\frac{\|f(x+h)-f(x)-A(x)h\|}{\|h\|}=0.
  7. x x
  8. f ′′ + 2 f - 3 f = 4 x - 1 f^{\prime\prime}+2f^{\prime}-3f=4x-1\,
  9. L ( f ) = 4 x - 1 , L(f)=4x-1,\,
  10. L = d 2 d x 2 + 2 d d x - 3 L=\frac{d^{2}}{dx^{2}}+2\frac{d}{dx}-3
  11. f ( x ) = L - 1 ( 4 x - 1 ) . f(x)=L^{-1}(4x-1).\,
  12. Δ = 2 x 2 + 2 y 2 + 2 z 2 . \Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}% +\frac{\partial^{2}}{\partial z^{2}}.
  13. = 2 x 2 + 2 y 2 + 2 z 2 - 1 c 2 2 t 2 . \square=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}% }+\frac{\partial^{2}}{\partial z^{2}}-\frac{1}{c^{2}}\frac{\partial^{2}}{% \partial t^{2}}.
  14. D q f ( x ) = f ( q x ) - f ( x ) ( q - 1 ) x . D_{q}f(x)=\frac{f(qx)-f(x)}{(q-1)x}.
  15. Δ f ( x ) = f ( x + 1 ) - f ( x ) \Delta f(x)=f(x+1)-f(x)\,
  16. ϵ = ( q - 1 ) x \epsilon=(q-1)x
  17. f ( q x ) - f ( x ) ( q - 1 ) x = f ( x + ϵ ) - f ( x ) ϵ . \frac{f(qx)-f(x)}{(q-1)x}=\frac{f(x+\epsilon)-f(x)}{\epsilon}.
  18. z = q x z=qx
  19. lim z x f ( z ) - f ( x ) z - x = lim q 1 f ( q x ) - f ( x ) q x - x = lim q 1 f ( q x ) - f ( x ) ( q - 1 ) x . \lim_{z\to x}\frac{f(z)-f(x)}{z-x}=\lim_{q\to 1}\frac{f(qx)-f(x)}{qx-x}=\lim_{% q\to 1}\frac{f(qx)-f(x)}{(q-1)x}.
  20. ( a d x d + a d - 1 x d - 1 + + a 1 x + a 0 ) = d a d x d - 1 + ( d - 1 ) a d - 1 x d - 2 + + a 1 . (a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots+a_{1}x+a_{0})^{\prime}=da_{d}x^{d-1}+(d-1)a_% {d-1}x^{d-2}+\cdots+a_{1}.
  21. f f f\mapsto f^{\prime}

Generalized_Appell_polynomials.html

  1. { p n ( z ) } \{p_{n}(z)\}
  2. K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = n = 0 p n ( z ) w n K(z,w)=A(w)\Psi(zg(w))=\sum_{n=0}^{\infty}p_{n}(z)w^{n}
  3. K ( z , w ) K(z,w)
  4. A ( w ) = n = 0 a n w n A(w)=\sum_{n=0}^{\infty}a_{n}w^{n}\quad
  5. a 0 0 a_{0}\neq 0
  6. Ψ ( t ) = n = 0 Ψ n t n \Psi(t)=\sum_{n=0}^{\infty}\Psi_{n}t^{n}\quad
  7. Ψ n 0 \Psi_{n}\neq 0
  8. g ( w ) = n = 1 g n w n g(w)=\sum_{n=1}^{\infty}g_{n}w^{n}\quad
  9. g 1 0. g_{1}\neq 0.
  10. p n ( z ) p_{n}(z)
  11. n n
  12. g ( w ) = w g(w)=w
  13. Ψ ( t ) = e t \Psi(t)=e^{t}
  14. g ( w ) = w g(w)=w
  15. Ψ ( t ) = e t \Psi(t)=e^{t}
  16. p n ( z ) = k = 0 n z k Ψ k h k . p_{n}(z)=\sum_{k=0}^{n}z^{k}\Psi_{k}h_{k}.
  17. h k = P a j 0 g j 1 g j 2 g j k h_{k}=\sum_{P}a_{j_{0}}g_{j_{1}}g_{j_{2}}\cdots g_{j_{k}}
  18. n n
  19. k + 1 k+1
  20. { j } \{j\}
  21. j 0 + j 1 + + j k = n . j_{0}+j_{1}+\cdots+j_{k}=n.\,
  22. p n ( z ) = k = 0 n a n - k z k k ! . p_{n}(z)=\sum_{k=0}^{n}\frac{a_{n-k}z^{k}}{k!}.
  23. K ( z , w ) K(z,w)
  24. A ( w ) Ψ ( z g ( w ) ) A(w)\Psi(zg(w))
  25. g 1 = 1 g_{1}=1
  26. K ( z , w ) w = c ( w ) K ( z , w ) + z b ( w ) w K ( z , w ) z \frac{\partial K(z,w)}{\partial w}=c(w)K(z,w)+\frac{zb(w)}{w}\frac{\partial K(% z,w)}{\partial z}
  27. b ( w ) b(w)
  28. c ( w ) c(w)
  29. b ( w ) = w g ( w ) d d w g ( w ) = 1 + n = 1 b n w n b(w)=\frac{w}{g(w)}\frac{d}{dw}g(w)=1+\sum_{n=1}^{\infty}b_{n}w^{n}
  30. c ( w ) = 1 A ( w ) d d w A ( w ) = n = 0 c n w n . c(w)=\frac{1}{A(w)}\frac{d}{dw}A(w)=\sum_{n=0}^{\infty}c_{n}w^{n}.
  31. K ( z , w ) = n = 0 p n ( z ) w n K(z,w)=\sum_{n=0}^{\infty}p_{n}(z)w^{n}
  32. z n + 1 d d z [ p n ( z ) z n ] = - k = 0 n - 1 c n - k - 1 p k ( z ) - z k = 1 n - 1 b n - k d d z p k ( z ) . z^{n+1}\frac{d}{dz}\left[\frac{p_{n}(z)}{z^{n}}\right]=-\sum_{k=0}^{n-1}c_{n-k% -1}p_{k}(z)-z\sum_{k=1}^{n-1}b_{n-k}\frac{d}{dz}p_{k}(z).
  33. g ( w ) = w g(w)=w
  34. b n = 0 b_{n}=0

Generalized_arithmetic_progression.html

  1. a + m b + n c + a+mb+nc+\ldots
  2. a , b , c a,b,c
  3. m , n m,n
  4. 0
  5. m m
  6. M M
  7. k k
  8. L ( C ; P ) L(C;P)
  9. x x
  10. N n N^{n}
  11. x = c 0 + i = 1 m k i x i , x=c_{0}+\sum_{i=1}^{m}k_{i}x_{i},
  12. c 0 c_{0}
  13. C C
  14. x 1 , , x m x_{1},\ldots,x_{m}
  15. P P
  16. k 1 , , k m k_{1},\ldots,k_{m}
  17. N N
  18. L L
  19. C C
  20. P P
  21. N n N^{n}

Generalized_inverse_Gaussian_distribution.html

  1. b K p + 1 ( a b ) a K p ( a b ) \frac{\sqrt{b}\ K_{p+1}(\sqrt{ab})}{\sqrt{a}\ K_{p}(\sqrt{ab})}
  2. ( p - 1 ) + ( p - 1 ) 2 + a b a \frac{(p-1)+\sqrt{(p-1)^{2}+ab}}{a}
  3. ( b a ) [ K p + 2 ( a b ) K p ( a b ) - ( K p + 1 ( a b ) K p ( a b ) ) 2 ] \left(\frac{b}{a}\right)\left[\frac{K_{p+2}(\sqrt{ab})}{K_{p}(\sqrt{ab})}-% \left(\frac{K_{p+1}(\sqrt{ab})}{K_{p}(\sqrt{ab})}\right)^{2}\right]
  4. ( a a - 2 t ) p 2 K p ( b ( a - 2 t ) ) K p ( a b ) \left(\frac{a}{a-2t}\right)^{\frac{p}{2}}\frac{K_{p}(\sqrt{b(a-2t)})}{K_{p}(% \sqrt{ab})}
  5. ( a a - 2 i t ) p 2 K p ( b ( a - 2 i t ) ) K p ( a b ) \left(\frac{a}{a-2it}\right)^{\frac{p}{2}}\frac{K_{p}(\sqrt{b(a-2it)})}{K_{p}(% \sqrt{ab})}
  6. f ( x ) = ( a / b ) p / 2 2 K p ( a b ) x ( p - 1 ) e - ( a x + b / x ) / 2 , x > 0 , f(x)=\frac{(a/b)^{p/2}}{2K_{p}(\sqrt{ab})}x^{(p-1)}e^{-(ax+b/x)/2},\qquad x>0,
  7. H ( f ( x ) ) = 1 2 log ( b a ) + log ( 2 K p ( a b ) ) - ( p - 1 ) [ d d ν K ν ( a b ) ] ν = p K p ( a b ) + a b 2 K p ( a b ) ( K p + 1 ( a b ) + K p - 1 ( a b ) ) H(f(x))=\frac{1}{2}\log\left(\frac{b}{a}\right)+\log\left(2K_{p}\left(\sqrt{ab% }\right)\right)-(p-1)\frac{\left[\frac{d}{d\nu}K_{\nu}\left(\sqrt{ab}\right)% \right]_{\nu=p}}{K_{p}\left(\sqrt{ab}\right)}+\frac{\sqrt{ab}}{2K_{p}\left(% \sqrt{ab}\right)}\left(K_{p+1}\left(\sqrt{ab}\right)+K_{p-1}\left(\sqrt{ab}% \right)\right)
  8. [ d d ν K ν ( a b ) ] ν = p \left[\frac{d}{d\nu}K_{\nu}\left(\sqrt{ab}\right)\right]_{\nu=p}
  9. ν \nu
  10. ν = p \nu=p
  11. { f ( x ) ( x ( a x - 2 p + 2 ) - b ) + 2 x 2 f ( x ) = 0 , f ( 1 ) = e 1 2 ( - a - b ) ( a b ) p / 2 2 K p ( a b ) } \left\{\begin{array}[]{l}f(x)(x(ax-2p+2)-b)+2x^{2}f^{\prime}(x)=0,\\ f(1)=\frac{e^{\frac{1}{2}(-a-b)}\left(\frac{a}{b}\right)^{p/2}}{2K_{p}\left(% \sqrt{ab}\right)}\end{array}\right\}
  12. f ( x ; μ , λ ) = [ λ 2 π x 3 ] 1 / 2 exp - λ ( x - μ ) 2 2 μ 2 x f(x;\mu,\lambda)=\left[\frac{\lambda}{2\pi x^{3}}\right]^{1/2}\exp{\frac{-% \lambda(x-\mu)^{2}}{2\mu^{2}x}}
  13. a = λ / μ 2 a=\lambda/\mu^{2}
  14. b = λ b=\lambda
  15. p = - 1 / 2 p=-1/2
  16. g ( x ; α , β ) = β α 1 Γ ( α ) x α - 1 e - β x g(x;\alpha,\beta)=\beta^{\alpha}\frac{1}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}
  17. a = 2 β a=2\beta
  18. b = 0 b=0
  19. p = α p=\alpha
  20. z z
  21. P ( z | a , b , p ) = GIG ( z | a , b , p ) P(z|a,b,p)=\,\text{GIG}(z|a,b,p)
  22. T T
  23. X = x 1 , , x T X=x_{1},\ldots,x_{T}
  24. z z
  25. P ( X | z , α , β ) = i = 1 T N ( x i | α + β z , z ) P(X|z,\alpha,\beta)=\prod_{i=1}^{T}N(x_{i}|\alpha+\beta z,z)
  26. N ( x | μ , v ) N(x|\mu,v)
  27. μ \mu
  28. v v
  29. z z
  30. P ( z | X , a , b , p , α , β ) = GIG ( z | p - T 2 , a + T β 2 , b + S ) P(z|X,a,b,p,\alpha,\beta)=\,\text{GIG}(z|p-\tfrac{T}{2},a+T\beta^{2},b+S)
  31. S = i = 1 T ( x i - α ) 2 \textstyle S=\sum_{i=1}^{T}(x_{i}-\alpha)^{2}
  32. P ( z | X , a , b , p , α , β ) P ( z | a , b , p ) P ( X | z , α , β ) P(z|X,a,b,p,\alpha,\beta)\propto P(z|a,b,p)P(X|z,\alpha,\beta)
  33. z z

Generalized_Newtonian_fluid.html

  1. τ = μ eff ( γ ˙ ) γ ˙ \tau=\mu_{\operatorname{eff}}(\dot{\gamma})\dot{\gamma}
  2. τ \tau
  3. γ ˙ \dot{\gamma}
  4. μ eff \mu_{\operatorname{eff}}

Generalized_processor_sharing.html

  1. N N
  2. w i w_{i}
  3. i i
  4. ( s , t ] (s,t]
  5. i i
  6. j j
  7. w j O i ( s , t ) w i O j ( s , t ) w_{j}O_{i}(s,t)\geq w_{i}O_{j}(s,t)
  8. O k ( s , t ) O_{k}(s,t)
  9. k k
  10. ( s , t ] (s,t]
  11. i i
  12. R i = w i j = 1 N w j R R_{i}=\frac{w_{i}}{\sum_{j=1}^{N}w_{j}}R
  13. R R
  14. w 1 = 2 , w 2 = w 3 = 1 w_{1}=2,w_{2}=w_{3}=1
  15. 1 / 4 1/4
  16. ( s , t ] (s,t]

Generalized_selection.html

  1. σ φ ( R ) \sigma_{\varphi}(R)
  2. φ \varphi
  3. and \and
  4. $\or$
  5. ¬ \lnot
  6. R R
  7. φ \varphi
  8. P e r s o n Person
  9. σ A g e 30 and W e i g h t 60 ( P e r s o n ) \sigma_{Age\geq 30\ \and\ Weight\leq 60}(Person)
  10. P e r s o n Person
  11. σ A g e 30 and W e i g h t 60 ( P e r s o n ) \sigma_{Age\geq 30\ \and\ Weight\leq 60}(Person)
  12. σ φ ( R ) = { t : t R , φ ( t ) } \sigma_{\varphi}(R)=\{\ t:t\in R,\ \varphi(t)\ \}
  13. σ φ and ψ ( R ) = σ φ ( R ) σ ψ ( R ) \sigma_{\varphi\and\psi}(R)=\sigma_{\varphi}(R)\cap\sigma_{\psi}(R)
  14. σ φ ψ ( R ) = σ φ ( R ) σ ψ ( R ) \sigma_{\varphi\psi}(R)=\sigma_{\varphi}(R)\cup\sigma_{\psi}(R)
  15. σ ¬ φ ( R ) = R - σ φ ( R ) \sigma_{\lnot\varphi}(R)=R-\sigma_{\varphi}(R)

Generalized_suffix_tree.html

  1. D = S 1 , S 2 , , S d D=S_{1},S_{2},\dots,S_{d}
  2. n n
  3. n n
  4. Θ ( n ) \Theta(n)
  5. z z
  6. P P
  7. m m
  8. O ( m + z ) O(m+z)

Generic_filter.html

  1. 1 \aleph_{1}
  2. 1 \aleph_{1}
  3. F E , F\cap E\neq\varnothing,\,