wpmath0000001_7

E1.html

  1. E 1 ( z ) = 1 e - t z t d t E_{1}(z)=\int_{1}^{\infty}\frac{e^{-tz}}{t}\,{\rm d}t~{}

E_(mathematical_constant).html

  1. e e
  2. n n
  3. e = n = 0 1 n ! = 1 + 1 1 + 1 1 2 + 1 1 2 3 + e=\displaystyle\sum\limits_{n=0}^{\infty}\dfrac{1}{n!}=1+\frac{1}{1}+\frac{1}{% 1\cdot 2}+\frac{1}{1\cdot 2\cdot 3}+\cdots
  4. e e
  5. a a
  6. x = 0 x=0
  7. e e
  8. k k
  9. y = 1 / x y=1/x
  10. x = 1 x=1
  11. x = k x=k
  12. e e
  13. e e
  14. γ γ
  15. e e
  16. e e
  17. e e
  18. π \pi
  19. i i
  20. π \pi
  21. e e
  22. e e
  23. e e
  24. lim n ( 1 + 1 n ) n . \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}.
  25. b b
  26. e e
  27. e e
  28. e e
  29. c c
  30. e e
  31. n n
  32. 100 % / n 100\%/n
  33. n n
  34. n = 52 n=52
  35. n = 365 n=365
  36. n n
  37. e e
  38. R R
  39. t t
  40. R R
  41. R = 5 / 100 = 0.05 R=5/100=0.05
  42. e e
  43. n n
  44. n n
  45. n n
  46. 1 / e 1/e
  47. n = 20 n=20
  48. k k
  49. ( 10 6 k ) ( 10 - 6 ) k ( 1 - 10 - 6 ) 10 6 - k . {\left({{10^{6}}\atop{k}}\right)}\left(10^{-6}\right)^{k}(1-10^{-6})^{10^{6}-k}.
  50. k = 0 k=0
  51. ( 1 - 1 10 6 ) 10 6 . \left(1-\frac{1}{10^{6}}\right)^{10^{6}}.
  52. 1 / e 1/e
  53. 1 e = lim n ( 1 - 1 n ) n . \frac{1}{e}=\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^{n}.
  54. e e
  55. n n
  56. n n
  57. p n = 1 - 1 1 ! + 1 2 ! - 1 3 ! + + ( - 1 ) n n ! = k = 0 n ( - 1 ) k k ! . p_{n}=1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^{n}}{n!}=\sum% _{k=0}^{n}\frac{(-1)^{k}}{k!}.
  58. n n
  59. 1 / e 1/e
  60. n ! / e n!/e
  61. n n
  62. e e
  63. e e
  64. π \pi
  65. n ! 2 π n ( n e ) n . n!\sim\sqrt{2\pi n}\,\left(\frac{n}{e}\right)^{n}.
  66. e = lim n n n ! n e=\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}
  67. ϕ ( x ) = 1 2 π e - 1 2 x 2 . \phi(x)=\frac{1}{\sqrt{2\pi}}\,e^{-\frac{\scriptscriptstyle 1}{% \scriptscriptstyle 2}x^{2}}.
  68. 1 / 2 π \scriptstyle\ 1/\sqrt{2\pi}
  69. 1 / 2 π \scriptstyle\ 1/\sqrt{2\pi}
  70. e e
  71. e e
  72. d d x a x = lim h 0 a x + h - a x h = lim h 0 a x a h - a x h = a x ( lim h 0 a h - 1 h ) . \frac{d}{dx}a^{x}=\lim_{h\to 0}\frac{a^{x+h}-a^{x}}{h}=\lim_{h\to 0}\frac{a^{x% }a^{h}-a^{x}}{h}=a^{x}\left(\lim_{h\to 0}\frac{a^{h}-1}{h}\right).
  73. x x
  74. a a
  75. e e
  76. e e
  77. d d x e x = e x . \frac{d}{dx}e^{x}=e^{x}.
  78. e e
  79. e e
  80. a a
  81. d d x log a x = lim h 0 log a ( x + h ) - log a ( x ) h = 1 x ( lim u 0 1 u log a ( 1 + u ) ) , \frac{d}{dx}\log_{a}x=\lim_{h\to 0}\frac{\log_{a}(x+h)-\log_{a}(x)}{h}=\frac{1% }{x}\left(\lim_{u\to 0}\frac{1}{u}\log_{a}(1+u)\right),
  82. u = h / x u=h/x
  83. a a
  84. e e
  85. d d x log e x = 1 x . \frac{d}{dx}\log_{e}x=\frac{1}{x}.
  86. l n ln
  87. a = e a=e
  88. a a
  89. a a
  90. 1 / x 1/x
  91. a a
  92. a a
  93. e e
  94. e e
  95. e e
  96. d d t e t = e t . \frac{d}{dt}e^{t}=e^{t}.
  97. e e
  98. d d t log e t = 1 t . \frac{d}{dt}\log_{e}t=\frac{1}{t}.
  99. e e
  100. e = lim n ( 1 + 1 n ) n e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
  101. e = lim x 0 ( 1 + x ) 1 x e=\lim_{x\to 0}\left(1+x\right)^{\frac{1}{x}}
  102. e e
  103. e = n = 0 1 n ! = 1 0 ! + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + , e=\sum_{n=0}^{\infty}\frac{1}{n!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac% {1}{3!}+\frac{1}{4!}+\cdots\,,
  104. n ! n!
  105. n n
  106. e e
  107. 1 e 1 t d t = 1. \int_{1}^{e}\frac{1}{t}\,dt=1.
  108. d d x e x = e x \frac{d}{dx}e^{x}=e^{x}
  109. e x d x = e x + C . \int e^{x}\,dx=e^{x}+C.
  110. e e
  111. ( 1 + 1 x ) x < e < ( 1 + 1 x ) x + 1 \left(1+\frac{1}{x}\right)^{x}<e<\left(1+\frac{1}{x}\right)^{x+1}
  112. e x x + 1 e^{x}\geq x+1
  113. x = 0 x=0
  114. e e
  115. x x \sqrt[x]{x}
  116. x = e x=e
  117. f ( x ) = x x . f(x)=\sqrt[x]{x}.
  118. x = e x=e
  119. e y y + 1 e^{y}\geq y+1
  120. y = ( x - e ) / e y=(x-e)/e
  121. e x / e x e^{x/e}\geq x
  122. e 1 / e x 1 / x e^{1/e}\geq x^{1/x}
  123. x = 1 / e x=1/e
  124. f ( x ) = x x f(x)=x^{x}\,
  125. x x
  126. f ( x ) = x x n \!\ f(x)=x^{x^{n}}
  127. x x
  128. x = 1 / e x=1/e
  129. n > 0 n>0
  130. x x x x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}
  131. x {{}^{\infty}}x
  132. e e
  133. e e
  134. e e
  135. e e
  136. e e
  137. e x = 1 + x 1 ! + x 2 2 ! + x 3 3 ! + = n = 0 x n n ! e^{x}=1+{x\over 1!}+{x^{2}\over 2!}+{x^{3}\over 3!}+\cdots=\sum_{n=0}^{\infty}% \frac{x^{n}}{n!}
  138. x x
  139. x x
  140. e i x = cos x + i sin x , e^{ix}=\cos x+i\sin x,\,\!
  141. x x
  142. x = π x=π
  143. e i π + 1 = 0 e^{i\pi}+1=0\,\!
  144. ln ( - 1 ) = i π . \ln(-1)=i\pi.\,\!
  145. ( cos x + i sin x ) n = ( e i x ) n = e i n x = cos ( n x ) + i sin ( n x ) , (\cos x+i\sin x)^{n}=\left(e^{ix}\right)^{n}=e^{inx}=\cos(nx)+i\sin(nx),
  146. cos x + i sin x \cos x+i\sin x\,
  147. c i s ( x ) cis(x)
  148. y ( x ) = C e x y(x)=Ce^{x}\,
  149. y = y . y^{\prime}=y.\,
  150. e e
  151. lim n ( 1 + 1 n ) n , \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n},
  152. e = n = 0 1 n ! e=\sum_{n=0}^{\infty}\frac{1}{n!}
  153. x = 1 x=1
  154. e = [ 2 ; 1 , 𝟐 , 1 , 1 , 𝟒 , 1 , 1 , 𝟔 , 1 , 1 , , 𝟐 𝐧 , 1 , 1 , ] = [ 1 ; 𝟎 , 1 , 1 , 𝟐 , 1 , 1 , 𝟒 , 1 , 1 , , 𝟐 𝐧 , 1 , 1 , ] , e=[2;1,\mathbf{2},1,1,\mathbf{4},1,1,\mathbf{6},1,1,...,\mathbf{2n},1,1,...]=[% 1;\mathbf{0},1,1,\mathbf{2},1,1,\mathbf{4},1,1,...,\mathbf{2n},1,1,...],
  155. e = 2 + 1 1 + 1 𝟐 + 1 1 + 1 1 + 1 𝟒 + 1 1 + 1 1 + = 1 + 1 𝟎 + 1 1 + 1 1 + 1 𝟐 + 1 1 + 1 1 + 1 𝟒 + 1 1 + 1 1 + . e=2+\cfrac{1}{1+\cfrac{1}{\mathbf{2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\mathbf% {4}+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}=1+\cfrac{1}{\mathbf{0}+\cfrac{1}{1+% \cfrac{1}{1+\cfrac{1}{\mathbf{2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\mathbf{4}+% \cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}}}.
  156. e e
  157. e = [ 1 ; 0.5 , 12 , 5 , 28 , 9 , 44 , 13 , , 4 ( 4 n - 1 ) , ( 4 n + 1 ) , ] , e=[1;0.5,12,5,28,9,44,13,\ldots,4(4n-1),(4n+1),\ldots],
  158. e = 1 + 2 1 + 1 6 + 1 10 + 1 14 + 1 18 + 1 22 + 1 26 + . e=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{2% 2+\cfrac{1}{26+\ddots\,}}}}}}}.
  159. e e
  160. e e
  161. e e
  162. V V
  163. n n
  164. n n
  165. V = min { n X 1 + X 2 + + X n > 1 } . V=\min{\left\{n\mid X_{1}+X_{2}+\cdots+X_{n}>1\right\}}.
  166. V V
  167. e e
  168. E ( V ) = e E(V)=e
  169. e e
  170. e e
  171. e e
  172. e e
  173. e e
  174. e e
  175. e e
  176. π \pi
  177. e e
  178. e e
  179. e e
  180. e e
  181. e e
  182. e e
  183. e e
  184. π \pi
  185. d c c = d y d s r d x \frac{dc}{c}=\frac{dyds}{rdx}
  186. c = e d y d s r d x c=e^{\int\frac{dyds}{rdx}}
  187. d c c = d y d s r d x \frac{dc}{c}=\frac{dyds}{rdx}
  188. c = e d y d s r d x c=e^{\int\frac{dyds}{rdx}}
  189. 1 e n 1 t d t = n . \int_{1}^{e^{n}}\frac{1}{t}\,dt=n.
  190. e e

Earley_parser.html

  1. O ( n 3 ) {O}(n^{3})
  2. O ( n 2 ) {O}(n^{2})

Earned_value_management.html

  1. EV = Start Current PV ( Completed ) \begin{aligned}\displaystyle\mathrm{EV}&\displaystyle=\sum_{\mathrm{Start}}^{% \mathrm{Current}}\mathrm{PV(Completed)}\end{aligned}
  2. S V = E V - P V \displaystyle SV=EV-PV
  3. S P I = E V P V \displaystyle SPI={EV\over PV}
  4. C V = E V - A C \displaystyle CV=EV-AC
  5. C P I = E V A C \displaystyle CPI={EV\over AC}
  6. E A C = A C + ( B A C - E V ) C P I = B A C C P I \begin{aligned}\displaystyle EAC=AC+{(BAC-EV)\over CPI}={BAC\over CPI}\end{aligned}
  7. E T C = E A C - A C \displaystyle ETC=EAC-AC
  8. T C P I B A C = B A C - E V B A C - A C TCPI_{BAC}={BAC-EV\over BAC-AC}
  9. T C P I E A C = B A C - E V E A C - A C TCPI_{EAC}={BAC-EV\over EAC-AC}
  10. I E A C = A C + ( B A C - E V ) C P I IEAC=\sum AC+{\left(BAC-\sum EV\right)\over CPI}

Earth.html

  1. × 10 7 \times 10^{7}
  2. × 10 7 \times 10^{7}
  3. × 10 1 8 \times 10^{1}8
  4. × 10 2 2 \times 10^{2}2
  5. ( 1 3 332 , 946 ) 1 3 = 0.01 \left(\frac{1}{3\cdot 332,946}\right)^{\frac{1}{3}}=0.01

Eclipse.html

  1. L = r R o R s - R o L\ =\ \frac{r\cdot R_{o}}{R_{s}-R_{o}}
  2. × 10 6 \times 10^{6}
  3. × 10 5 \times 10^{5}

Eclipse_cycle.html

  1. 1 / 13 {1}/{13}
  2. 181 / 2 18{1}/{2}
  3. 21 / 3 2{1}/{3}
  4. 1 / 20 {1}/{20}
  5. EY = SM × DM SM-DM \mbox{EY}~{}=\frac{\mbox{SM}~{}\times\mbox{DM}~{}}{\mbox{SM-DM}~{}}

Ecliptic_coordinate_system.html

  1. λ λ
  2. β β
  3. Δ Δ
  4. l l
  5. b b
  6. r r
  7. x x
  8. y y
  9. z z
  10. l l
  11. λ \lambda
  12. b b
  13. β \beta
  14. r r
  15. Δ \mathit{\Delta}
  16. x = r cos b cos l x=r\cos b\cos l
  17. y = r cos b sin l y=r\cos b\sin l
  18. z = r sin b z=r\sin b
  19. [ x e q u a t o r i a l y e q u a t o r i a l z e q u a t o r i a l ] = [ 1 0 0 0 cos ϵ - sin ϵ 0 sin ϵ cos ϵ ] [ x e c l i p t i c y e c l i p t i c z e c l i p t i c ] \begin{bmatrix}x_{equatorial}\\ y_{equatorial}\\ z_{equatorial}\\ \end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&\cos\epsilon&-\sin\epsilon\\ 0&\sin\epsilon&\cos\epsilon\\ \end{bmatrix}\!\cdot\!\begin{bmatrix}x_{ecliptic}\\ y_{ecliptic}\\ z_{ecliptic}\\ \end{bmatrix}
  20. [ x e c l i p t i c y e c l i p t i c z e c l i p t i c ] = [ 1 0 0 0 cos ϵ sin ϵ 0 - sin ϵ cos ϵ ] [ x e q u a t o r i a l y e q u a t o r i a l z e q u a t o r i a l ] \begin{bmatrix}x_{ecliptic}\\ y_{ecliptic}\\ z_{ecliptic}\\ \end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&\cos\epsilon&\sin\epsilon\\ 0&-\sin\epsilon&\cos\epsilon\\ \end{bmatrix}\!\cdot\!\begin{bmatrix}x_{equatorial}\\ y_{equatorial}\\ z_{equatorial}\\ \end{bmatrix}
  21. ϵ \epsilon
  22. x x
  23. y y
  24. z z

Ecology.html

  1. d N d T = b N - d N = ( b - d ) N = r N , \frac{\operatorname{d}N}{\operatorname{d}T}=bN-dN=(b-d)N=rN,
  2. d N d T = a N ( 1 - N K ) , \frac{dN}{dT}=aN\left(1-\frac{N}{K}\right),

Econometrics.html

  1. Δ Unemployment \Delta\ \,\text{Unemployment}
  2. β 0 \beta_{0}
  3. β 1 \beta_{1}
  4. ε \varepsilon
  5. Δ Unemployment = β 0 + β 1 Growth + ε . \Delta\ \text{Unemployment}=\beta_{0}+\beta_{1}\,\text{Growth}+\varepsilon.
  6. β 0 \beta_{0}
  7. β 1 \beta_{1}
  8. β 1 \beta_{1}
  9. β 0 \beta_{0}
  10. β 1 \beta_{1}
  11. ln ( wage ) = β 0 + β 1 ( years of education ) + ε . \ln(\,\text{wage})=\beta_{0}+\beta_{1}(\,\text{years of education})+\varepsilon.
  12. β 1 \beta_{1}
  13. ε \varepsilon
  14. β 0 and β 1 \beta_{0}\mbox{ and }~{}\beta_{1}
  15. ε \varepsilon
  16. ε \varepsilon
  17. ϵ \epsilon
  18. β 1 \beta_{1}

Economic_surplus.html

  1. C S = 1 2 Q 𝑚𝑘𝑡 ( P 𝑚𝑎𝑥 - P 𝑚𝑘𝑡 ) CS=\frac{1}{2}Q_{\mathit{mkt}}\left({P_{\mathit{max}}-P_{\mathit{mkt}}}\right)
  2. C S = P 𝑚𝑘𝑡 P 𝑚𝑎𝑥 D ( P ) d P , CS=\int^{P_{\mathit{max}}}_{P_{\mathit{mkt}}}D(P)\,dP,
  3. D ( P 𝑚𝑎𝑥 ) = 0. D(P_{\mathit{max}})=0.
  4. P = P m k t P=P_{mkt}
  5. Δ C S = 1 2 ( Q 1 + Q 0 ) ( P 1 - P 0 ) \Delta CS=\frac{1}{2}\left({Q_{1}+Q_{0}}\right)\left({P_{1}-P_{0}}\right)

Economy_of_Egypt.html

  1. \approx
  2. \approx
  3. \approx

Ectopic_pregnancy.html

  1. h C G r a t i o = h C G a t 48 h h C G a t 0 h hCG~{}ratio=\frac{hCG~{}at~{}48h}{hCG~{}at~{}0h}

Einsteinium.html

  1. U 92 238 + 6 ( n , γ ) - 2 β - 94 244 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow[-2\ \beta^{-}]{+\ 6\ (n,\gamma)}\ ^{244% }_{\ 94}Pu}
  2. U 92 238 + 15 n 6 β - 98 253 Cf β - 99 253 Es \mathrm{{}^{238}_{\ 92}U\ \xrightarrow[6\beta^{-}]{+\ 15n}\ ^{253}_{\ 98}Cf\ % \xrightarrow{\beta^{-}}\ ^{253}_{\ 99}Es}
  3. Cf 98 252 ( n , γ ) 98 253 Cf β - 17.81 d 99 253 Es ( n , γ ) 99 254 Es β - 100 254 Fm \mathrm{{}^{252}_{\ 98}Cf\ \xrightarrow{(n,\gamma)}\ ^{253}_{\ 98}Cf\ % \xrightarrow[17.81\ d]{\beta^{-}}\ ^{253}_{\ 99}Es\ \xrightarrow{(n,\gamma)}\ % ^{254}_{\ 99}Es\ \xrightarrow[]{\beta^{-}}\ ^{254}_{100}Fm}
  4. 3 ¯ \overline{3}
  5. Es 99 253 20 d 𝛼 97 249 Bk β - 314 d 98 249 Cf \mathrm{{}^{253}_{\ 99}Es\ \xrightarrow[20\ d]{\alpha}\ ^{249}_{\ 97}Bk\ % \xrightarrow[314\ d]{\beta^{-}}\ ^{249}_{\ 98}Cf}
  6. Cf 98 249 + 1 2 D 99 248 Es + 3 0 1 n ( 99 248 Es 27 min ϵ 98 248 Cf ) \mathrm{{}^{249}_{\ 98}Cf\ +\ ^{2}_{1}D\ \longrightarrow\ ^{248}_{\ 99}Es\ +\ % 3\ ^{1}_{0}n\quad(^{248}_{\ 99}Es\ \xrightarrow[27\ min]{\epsilon}\ ^{248}_{\ % 98}Cf)}
  7. Bk 97 249 + α 99 249 , 250 , 251 , 252 Es \mathrm{{}^{249}_{\ 97}Bk\ \xrightarrow{+\alpha}\ ^{249,\ 250,\ 251,\ 252}_{\ % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 99}Es}
  8. Cf 98 252 ( n , γ ) 98 253 Cf β - 17.81 d 99 253 Es \mathrm{{}^{252}_{\ 98}Cf\ \xrightarrow{(n,\gamma)}\ ^{253}_{\ 98}Cf\ % \xrightarrow[17.81\ d]{\beta^{-}}\ ^{253}_{\ 99}Es}
  9. × 10 1 5 \times 10^{1}5
  10. × 10 - 14 \times 10^{-}14
  11. × 10 - 7 \times 10^{-}7
  12. 3 ¯ \overline{3}
  13. 3 ¯ \overline{3}
  14. 3 ¯ \overline{3}
  15. Es 99 254 + 20 48 Ca 119 302 Uue * n o a t o m s \,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}\to\ \ no\ atoms

Elastic_collision.html

  1. m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 . \,\!m_{1}\vec{u}_{1}+m_{2}\vec{u}_{2}=m_{1}\vec{v}_{1}+m_{2}\vec{v}_{2}.
  2. m 1 u 1 2 2 + m 2 u 2 2 2 = m 1 v 1 2 2 + m 2 v 2 2 2 . \frac{m_{1}u_{1}^{2}}{2}+\frac{m_{2}u_{2}^{2}}{2}=\frac{m_{1}v_{1}^{2}}{2}+% \frac{m_{2}v_{2}^{2}}{2}.
  3. v 1 = u 1 ( m 1 - m 2 ) + 2 m 2 u 2 m 1 + m 2 v_{1}=\frac{u_{1}(m_{1}-m_{2})+2m_{2}u_{2}}{m_{1}+m_{2}}
  4. v 2 = u 2 ( m 2 - m 1 ) + 2 m 1 u 1 m 1 + m 2 v_{2}=\frac{u_{2}(m_{2}-m_{1})+2m_{1}u_{1}}{m_{1}+m_{2}}
  5. v 1 = u 1 \ v_{1}=u_{1}
  6. v 2 = u 2 \ v_{2}=u_{2}
  7. v 1 - v 2 = u 2 - u 1 \ v_{1}-v_{2}=u_{2}-u_{1}
  8. m 1 ( v 1 2 - u 1 2 ) = m 2 ( u 2 2 - v 2 2 ) \ m_{1}\left(v_{1}^{2}-u_{1}^{2}\right)=m_{2}\left(u_{2}^{2}-v_{2}^{2}\right)
  9. m 1 ( v 1 - u 1 ) ( v 1 + u 1 ) = m 2 ( u 2 - v 2 ) ( u 2 + v 2 ) \Rightarrow m_{1}\left(v_{1}-u_{1}\right)\left(v_{1}+u_{1}\right)=m_{2}\left(u% _{2}-v_{2}\right)\left(u_{2}+v_{2}\right)
  10. m 1 ( v 1 - u 1 ) = m 2 ( u 2 - v 2 ) \ m_{1}(v_{1}-u_{1})=m_{2}(u_{2}-v_{2})
  11. v 1 + u 1 = u 2 + v 2 \ v_{1}+u_{1}=u_{2}+v_{2}
  12. v 1 - v 2 = u 2 - u 1 \Rightarrow v_{1}-v_{2}=u_{2}-u_{1}
  13. t \ t
  14. t \ t^{\prime}
  15. x ¯ ( t ) = m 1 x 1 ( t ) + m 2 x 2 ( t ) m 1 + m 2 \bar{x}(t)=\frac{m_{1}x_{1}(t)+m_{2}x_{2}(t)}{m_{1}+m_{2}}
  16. x ¯ ( t ) = m 1 x 1 ( t ) + m 2 x 2 ( t ) m 1 + m 2 \bar{x}(t^{\prime})=\frac{m_{1}x_{1}(t^{\prime})+m_{2}x_{2}(t^{\prime})}{m_{1}% +m_{2}}
  17. v x ¯ = m 1 u 1 + m 2 u 2 m 1 + m 2 \ v_{\bar{x}}=\frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}
  18. v x ¯ = m 1 v 1 + m 2 v 2 m 1 + m 2 \ v_{\bar{x}}^{\prime}=\frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}
  19. v x ¯ \ v_{\bar{x}}
  20. v x ¯ \ v_{\bar{x}}^{\prime}
  21. v x ¯ = v x ¯ \ v_{\bar{x}}=\ v_{\bar{x}}^{\prime}
  22. v 1 \ v_{1}
  23. v 2 \ v_{2}
  24. u 1 \ u_{1}
  25. v 1 \ v_{1}
  26. p = m v 1 - v 2 c 2 p=\frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  27. p 1 = - p 2 p_{1}=-p_{2}
  28. p 1 2 = p 2 2 p_{1}^{2}=p_{2}^{2}
  29. m 1 2 c 4 + p 1 2 c 2 + m 2 2 c 4 + p 2 2 c 2 = E \sqrt{m_{1}^{2}c^{4}+p_{1}^{2}c^{2}}+\sqrt{m_{2}^{2}c^{4}+p_{2}^{2}c^{2}}=E
  30. p 1 = ± E 4 - 2 E 2 m 1 2 c 4 - 2 E 2 m 2 2 c 4 + m 1 4 c 8 - 2 m 1 2 m 2 2 c 8 + m 2 4 c 8 c E p_{1}=\pm\frac{\sqrt{E^{4}-2E^{2}m_{1}^{2}c^{4}-2E^{2}m_{2}^{2}c^{4}+m_{1}^{4}% c^{8}-2m_{1}^{2}m_{2}^{2}c^{8}+m_{2}^{4}c^{8}}}{cE}
  31. u 1 = - v 1 u_{1}=-v_{1}
  32. m 1 m_{1}
  33. m 2 m_{2}
  34. u 1 u_{1}
  35. u 2 u_{2}
  36. v 1 v_{1}
  37. v 2 v_{2}
  38. p 1 p_{1}
  39. p 2 p_{2}
  40. c c
  41. E E
  42. m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 = 0 m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}={0}\,\!
  43. m 1 u 1 2 + m 2 u 2 2 = m 1 v 1 2 + m 2 v 2 2 m_{1}u_{1}^{2}+m_{2}u_{2}^{2}=m_{1}v_{1}^{2}+m_{2}v_{2}^{2}\,\!
  44. ( m 2 u 2 ) 2 2 m 1 + ( m 2 u 2 ) 2 2 m 2 = ( m 2 v 2 ) 2 2 m 1 + ( m 2 v 2 ) 2 2 m 2 \frac{(m_{2}u_{2})^{2}}{2m_{1}}+\frac{(m_{2}u_{2})^{2}}{2m_{2}}=\frac{(m_{2}v_% {2})^{2}}{2m_{1}}+\frac{(m_{2}v_{2})^{2}}{2m_{2}}\,\!
  45. ( m 1 + m 2 ) ( m 2 u 2 ) 2 = ( m 1 + m 2 ) ( m 2 v 2 ) 2 (m_{1}+m_{2})(m_{2}u_{2})^{2}=(m_{1}+m_{2})(m_{2}v_{2})^{2}\,\!
  46. u 2 = - v 2 u_{2}=-v_{2}\,\!
  47. ( m 1 u 1 ) 2 2 m 1 + ( m 1 u 1 ) 2 2 m 2 = ( m 1 v 1 ) 2 2 m 1 + ( m 1 v 1 ) 2 2 m 2 \frac{(m_{1}u_{1})^{2}}{2m_{1}}+\frac{(m_{1}u_{1})^{2}}{2m_{2}}=\frac{(m_{1}v_% {1})^{2}}{2m_{1}}+\frac{(m_{1}v_{1})^{2}}{2m_{2}}\,\!
  48. ( m 1 + m 2 ) ( m 1 u 1 ) 2 = ( m 1 + m 2 ) ( m 1 v 1 ) 2 (m_{1}+m_{2})(m_{1}u_{1})^{2}=(m_{1}+m_{2})(m_{1}v_{1})^{2}\,\!
  49. u 1 = - v 1 u_{1}=-v_{1}\,\!
  50. u 1 = - v 1 u_{1}=-v_{1}
  51. m 1 u 1 1 - u 1 2 / c 2 + m 2 u 2 1 - u 2 2 / c 2 = m 1 v 1 1 - v 1 2 / c 2 + m 2 v 2 1 - v 2 2 / c 2 = p T \frac{m_{1}\;u_{1}}{\sqrt{1-u_{1}^{2}/c^{2}}}+\frac{m_{2}\;u_{2}}{\sqrt{1-u_{2% }^{2}/c^{2}}}=\frac{m_{1}\;v_{1}}{\sqrt{1-v_{1}^{2}/c^{2}}}+\frac{m_{2}\;v_{2}% }{\sqrt{1-v_{2}^{2}/c^{2}}}=p_{T}
  52. m 1 c 2 1 - u 1 2 / c 2 + m 2 c 2 1 - u 2 2 / c 2 = m 1 c 2 1 - v 1 2 / c 2 + m 2 c 2 1 - v 2 2 / c 2 = E \frac{m_{1}c^{2}}{\sqrt{1-u_{1}^{2}/c^{2}}}+\frac{m_{2}c^{2}}{\sqrt{1-u_{2}^{2% }/c^{2}}}=\frac{m_{1}c^{2}}{\sqrt{1-v_{1}^{2}/c^{2}}}+\frac{m_{2}c^{2}}{\sqrt{% 1-v_{2}^{2}/c^{2}}}=E
  53. p T p_{T}
  54. E E
  55. v c v_{c}
  56. v c v_{c}
  57. v c = p T c 2 E v_{c}=\frac{p_{T}c^{2}}{E}
  58. u 1 u_{1}^{\prime}
  59. u 2 u_{2}^{\prime}
  60. u 1 = u 1 - v c 1 - u 1 v c c 2 u_{1}^{\prime}=\frac{u_{1}-v_{c}}{1-\frac{u_{1}v_{c}}{c^{2}}}
  61. u 2 = u 2 - v c 1 - u 2 v c c 2 u_{2}^{\prime}=\frac{u_{2}-v_{c}}{1-\frac{u_{2}v_{c}}{c^{2}}}
  62. v 1 = - u 1 v_{1}^{\prime}=-u_{1}^{\prime}
  63. v 2 = - u 2 v_{2}^{\prime}=-u_{2}^{\prime}
  64. v 1 = v 1 + v c 1 + v 1 v c c 2 v_{1}=\frac{v_{1}^{\prime}+v_{c}}{1+\frac{v_{1}^{\prime}v_{c}}{c^{2}}}
  65. v 2 = v 2 + v c 1 + v 2 v c c 2 v_{2}=\frac{v_{2}^{\prime}+v_{c}}{1+\frac{v_{2}^{\prime}v_{c}}{c^{2}}}
  66. u 1 c u_{1}\ll c
  67. u 2 c u_{2}\ll c
  68. p T p_{T}
  69. m 1 u 1 + m 2 u 2 m_{1}u_{1}+m_{2}u_{2}
  70. v c v_{c}
  71. m 1 u 1 + m 2 u 2 m 1 + m 2 \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}
  72. u 1 u_{1}^{\prime}
  73. u 1 - v c u_{1}-v_{c}
  74. m 1 u 1 + m 2 u 1 - m 1 u 1 - m 2 u 2 m 1 + m 2 = m 2 ( u 1 - u 2 ) m 1 + m 2 \frac{m_{1}u_{1}+m_{2}u_{1}-m_{1}u_{1}-m_{2}u_{2}}{m_{1}+m_{2}}=\frac{m_{2}(u_% {1}-u_{2})}{m_{1}+m_{2}}
  75. u 2 u_{2}^{\prime}
  76. m 1 ( u 2 - u 1 ) m 1 + m 2 \frac{m_{1}(u_{2}-u_{1})}{m_{1}+m_{2}}
  77. v 1 v_{1}^{\prime}
  78. m 2 ( u 2 - u 1 ) m 1 + m 2 \frac{m_{2}(u_{2}-u_{1})}{m_{1}+m_{2}}
  79. v 2 v_{2}^{\prime}
  80. m 1 ( u 1 - u 2 ) m 1 + m 2 \frac{m_{1}(u_{1}-u_{2})}{m_{1}+m_{2}}
  81. v 1 v_{1}
  82. v 1 + v c v_{1}^{\prime}+v_{c}
  83. m 2 u 2 - m 2 u 1 + m 1 u 1 + m 2 u 2 m 1 + m 2 = u 1 ( m 1 - m 2 ) + 2 m 2 u 2 m 1 + m 2 \frac{m_{2}u_{2}-m_{2}u_{1}+m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}}=\frac{u_{1}(m_% {1}-m_{2})+2m_{2}u_{2}}{m_{1}+m_{2}}
  84. v 2 v_{2}
  85. u 2 ( m 2 - m 1 ) + 2 m 1 u 1 m 1 + m 2 \frac{u_{2}(m_{2}-m_{1})+2m_{1}u_{1}}{m_{1}+m_{2}}
  86. s s
  87. v / c = tanh ( s ) = e s - e - s e s + e - s v/c=\mbox{tanh}~{}(s)={\frac{e^{s}-e^{-s}}{e^{s}+e^{-s}}}
  88. e s = c + v c - v e^{s}=\sqrt{\frac{c+v}{c-v}}
  89. E = m c 2 1 - v 2 c 2 = m c 2 cosh ( s ) E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=mc^{2}\mbox{cosh}~{}(s)
  90. p = m v 1 - v 2 c 2 = m c sinh ( s ) p=\frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=mc\mbox{ sinh}~{}(s)
  91. m 1 {m_{1}}
  92. m 2 {m_{2}}
  93. v 1 {v_{1}}
  94. v 2 {v_{2}}
  95. u 1 {u_{1}}
  96. u 2 {u_{2}}
  97. s 1 {s_{1}}
  98. s 2 {s_{2}}
  99. s 3 {s_{3}}
  100. s 4 {s_{4}}
  101. c {c}
  102. m 1 cosh ( s 1 ) + m 2 cosh ( s 2 ) = m 1 cosh ( s 3 ) + m 2 cosh ( s 4 ) m_{1}\mbox{cosh}~{}(s_{1})+m_{2}\mbox{cosh}~{}(s_{2})=m_{1}\mbox{cosh}~{}(s_{3% })+m_{2}\mbox{cosh}~{}(s_{4})
  103. m 1 sinh ( s 1 ) + m 2 sinh ( s 2 ) = m 1 sinh ( s 3 ) + m 2 sinh ( s 4 ) m_{1}\mbox{sinh}~{}(s_{1})+m_{2}\mbox{sinh}~{}(s_{2})=m_{1}\mbox{sinh}~{}(s_{3% })+m_{2}\mbox{sinh}~{}(s_{4})
  104. m 1 e s 1 + m 2 e s 2 = m 1 e s 3 + m 2 e s 4 m_{1}e^{s_{1}}+m_{2}e^{s_{2}}=m_{1}e^{s_{3}}+m_{2}e^{s_{4}}
  105. cosh ( s ) 2 - sinh ( s ) 2 = 1 \mbox{cosh}~{}^{2}(s)-\mbox{sinh}~{}^{2}(s)=1
  106. 2 m 1 m 2 ( cosh ( s 1 ) sinh ( s 2 ) - cosh ( s 2 ) sinh ( s 1 ) ) = 2 m 1 m 2 ( cosh ( s 3 ) sinh ( s 4 ) - cosh ( s 4 ) sinh ( s 3 ) ) 2m_{1}m_{2}(\mbox{cosh}~{}(s_{1})\mbox{sinh}~{}(s_{2})-\mbox{cosh}~{}(s_{2})% \mbox{sinh}~{}(s_{1}))=2m_{1}m_{2}(\mbox{cosh}~{}(s_{3})\mbox{sinh}~{}(s_{4})-% \mbox{cosh}~{}(s_{4})\mbox{sinh}~{}(s_{3}))
  107. cosh ( s 1 - s 2 ) = cosh ( s 3 - s 4 ) \mbox{cosh}~{}(s_{1}-s_{2})=\mbox{cosh}~{}(s_{3}-s_{4})
  108. cosh ( s ) \mbox{cosh}~{}(s)
  109. s 1 - s 2 = s 3 - s 4 s_{1}-s_{2}=s_{3}-s_{4}
  110. s 1 - s 2 = - s 3 + s 4 s_{1}-s_{2}=-s_{3}+s_{4}
  111. s 2 s_{2}
  112. e s 1 e^{s_{1}}
  113. e s 2 e^{s_{2}}
  114. e s 1 = e s 4 m 1 e s 3 + m 2 e s 4 m 1 e s 4 + m 2 e s 3 e^{s_{1}}=e^{s_{4}}{\frac{m_{1}e^{s_{3}}+m_{2}e^{s_{4}}}{m_{1}e^{s_{4}}+m_{2}e% ^{s_{3}}}}
  115. e s 2 = e s 3 m 1 e s 3 + m 2 e s 4 m 1 e s 4 + m 2 e s 3 e^{s_{2}}=e^{s_{3}}{\frac{m_{1}e^{s_{3}}+m_{2}e^{s_{4}}}{m_{1}e^{s_{4}}+m_{2}e% ^{s_{3}}}}
  116. v 1 / c = tanh ( s 1 ) = e s 1 - e - s 1 e s 1 + e - s 1 v_{1}/c=\mbox{tanh}~{}(s_{1})={\frac{e^{s_{1}}-e^{-s_{1}}}{e^{s_{1}}+e^{-s_{1}% }}}
  117. v 2 / c = tanh ( s 2 ) = e s 2 - e - s 2 e s 2 + e - s 2 v_{2}/c=\mbox{tanh}~{}(s_{2})={\frac{e^{s_{2}}-e^{-s_{2}}}{e^{s_{2}}+e^{-s_{2}% }}}
  118. e s 3 = c + u 1 c - u 1 e^{s_{3}}=\sqrt{\frac{c+u_{1}}{c-u_{1}}}
  119. e s 4 = c + u 2 c - u 2 e^{s_{4}}=\sqrt{\frac{c+u_{2}}{c-u_{2}}}
  120. Z = ( 1 - u 1 2 / c 2 ) ( 1 - u 2 2 / c 2 ) Z=\sqrt{(1-u_{1}^{2}/c^{2})(1-u_{2}^{2}/c^{2})}
  121. v 1 = 2 m 1 m 2 c 2 u 2 Z + 2 m 2 2 c 2 u 2 - ( m 1 2 + m 2 2 ) u 1 u 2 2 + ( m 1 2 - m 2 2 ) c 2 u 1 2 m 1 m 2 c 2 Z - 2 m 2 2 u 1 u 2 - ( m 1 2 - m 2 2 ) u 2 2 + ( m 1 2 + m 2 2 ) c 2 v_{1}=\frac{2m_{1}m_{2}c^{2}u_{2}Z+2m_{2}^{2}c^{2}u_{2}-(m_{1}^{2}+m_{2}^{2})u% _{1}u_{2}^{2}+(m_{1}^{2}-m_{2}^{2})c^{2}u_{1}}{2m_{1}m_{2}c^{2}Z-2m_{2}^{2}u_{% 1}u_{2}-(m_{1}^{2}-m_{2}^{2})u_{2}^{2}+(m_{1}^{2}+m_{2}^{2})c^{2}}
  122. v 2 = 2 m 1 m 2 c 2 u 1 Z + 2 m 1 2 c 2 u 1 - ( m 1 2 + m 2 2 ) u 1 2 u 2 + ( m 2 2 - m 1 2 ) c 2 u 2 2 m 1 m 2 c 2 Z - 2 m 1 2 u 1 u 2 - ( m 2 2 - m 1 2 ) u 1 2 + ( m 1 2 + m 2 2 ) c 2 v_{2}=\frac{2m_{1}m_{2}c^{2}u_{1}Z+2m_{1}^{2}c^{2}u_{1}-(m_{1}^{2}+m_{2}^{2})u% _{1}^{2}u_{2}+(m_{2}^{2}-m_{1}^{2})c^{2}u_{2}}{2m_{1}m_{2}c^{2}Z-2m_{1}^{2}u_{% 1}u_{2}-(m_{2}^{2}-m_{1}^{2})u_{1}^{2}+(m_{1}^{2}+m_{2}^{2})c^{2}}
  123. ϑ 1 \vartheta_{1}
  124. ϑ 2 \vartheta_{2}
  125. θ \theta
  126. tan ϑ 1 = m 2 sin θ m 1 + m 2 cos θ , ϑ 2 = π - θ 2 . \tan\vartheta_{1}=\frac{m_{2}\sin\theta}{m_{1}+m_{2}\cos\theta},\qquad% \vartheta_{2}=\frac{{\pi}-{\theta}}{2}.
  127. v 1 = v 1 m 1 2 + m 2 2 + 2 m 1 m 2 cos θ m 1 + m 2 , v 2 = v 1 2 m 1 m 1 + m 2 sin θ 2 . v^{\prime}_{1}=v_{1}\frac{\sqrt{m_{1}^{2}+m_{2}^{2}+2m_{1}m_{2}\cos\theta}}{m_% {1}+m_{2}},\qquad v^{\prime}_{2}=v_{1}\frac{2m_{1}}{m_{1}+m_{2}}\sin\frac{% \theta}{2}.
  128. v 1 x = v 1 cos ( θ 1 - φ ) ( m 1 - m 2 ) + 2 m 2 v 2 cos ( θ 2 - φ ) m 1 + m 2 cos ( φ ) + v 1 sin ( θ 1 - φ ) cos ( φ + π 2 ) v 1 y = v 1 cos ( θ 1 - φ ) ( m 1 - m 2 ) + 2 m 2 v 2 cos ( θ 2 - φ ) m 1 + m 2 sin ( φ ) + v 1 sin ( θ 1 - φ ) sin ( φ + π 2 ) \begin{aligned}\displaystyle v^{\prime}_{1x}&\displaystyle=\frac{v_{1}\cos(% \theta_{1}-\varphi)(m_{1}-m_{2})+2m_{2}v_{2}\cos(\theta_{2}-\varphi)}{m_{1}+m_% {2}}\cos(\varphi)\\ &\displaystyle\quad+v_{1}\sin(\theta_{1}-\varphi)\cos(\varphi+\frac{\pi}{2})\\ \displaystyle v^{\prime}_{1y}&\displaystyle=\frac{v_{1}\cos(\theta_{1}-\varphi% )(m_{1}-m_{2})+2m_{2}v_{2}\cos(\theta_{2}-\varphi)}{m_{1}+m_{2}}\sin(\varphi)% \\ &\displaystyle\quad+v_{1}\sin(\theta_{1}-\varphi)\sin(\varphi+\frac{\pi}{2})% \end{aligned}
  129. v 1 x = v 1 cos θ 1 , v 1 y = v 1 sin θ 1 v_{1x}=v_{1}\cos\theta_{1},\;v_{1y}=v_{1}\sin\theta_{1}
  130. 𝐯 1 = 𝐯 1 - 2 m 2 m 1 + m 2 𝐯 1 - 𝐯 2 , 𝐱 1 - 𝐱 2 𝐱 1 - 𝐱 2 2 ( 𝐱 1 - 𝐱 2 ) , 𝐯 2 = 𝐯 2 - 2 m 1 m 1 + m 2 𝐯 2 - 𝐯 1 , 𝐱 2 - 𝐱 1 𝐱 2 - 𝐱 1 2 ( 𝐱 2 - 𝐱 1 ) \begin{aligned}\displaystyle\mathbf{v}^{\prime}_{1}&\displaystyle=\mathbf{v}_{% 1}-\frac{2m_{2}}{m_{1}+m_{2}}\ \frac{\langle\mathbf{v}_{1}-\mathbf{v}_{2},\,% \mathbf{x}_{1}-\mathbf{x}_{2}\rangle}{\|\mathbf{x}_{1}-\mathbf{x}_{2}\|^{2}}\ % (\mathbf{x}_{1}-\mathbf{x}_{2}),\\ \displaystyle\mathbf{v}^{\prime}_{2}&\displaystyle=\mathbf{v}_{2}-\frac{2m_{1}% }{m_{1}+m_{2}}\ \frac{\langle\mathbf{v}_{2}-\mathbf{v}_{1},\,\mathbf{x}_{2}-% \mathbf{x}_{1}\rangle}{\|\mathbf{x}_{2}-\mathbf{x}_{1}\|^{2}}\ (\mathbf{x}_{2}% -\mathbf{x}_{1})\end{aligned}

Electric_charge.html

  1. 1 / 3 {1}/{3}
  2. 2 / 3 {2}/{3}
  3. I = d Q d t . I=\frac{dQ}{dt}.
  4. t i t_{i}
  5. t f t_{f}
  6. Q = t i t f I d t Q=\int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}I\,\mathrm{d}t

Electric_current.html

  1. I I
  2. I I
  3. C C
  4. I I
  5. I I
  6. I I
  7. I I
  8. I = V R I=\frac{V}{R}
  9. Q I 2 R Q\propto I^{2}R
  10. I = Q t , I={Q\over t}\,,
  11. I = d Q d t . I=\frac{\mathrm{d}Q}{\mathrm{d}t}\,.
  12. I = J d A I=\int\vec{J}\cdot d\vec{A}
  13. I I
  14. J \vec{J}
  15. d A d\vec{A}
  16. J \vec{J}
  17. E \vec{E}
  18. σ \sigma
  19. J = σ E \vec{J}=\sigma\vec{E}\,
  20. σ \sigma
  21. ρ \rho
  22. J = E ρ \vec{J}=\frac{\vec{E}}{\rho}
  23. E = ρ J \vec{E}=\rho\vec{J}
  24. D D
  25. α q \alpha_{q}
  26. J = σ E + D q n , J=\sigma E+Dq\nabla n,
  27. q q
  28. n n
  29. n n
  30. p p
  31. I = V R , I={V\over R}\,,
  32. I I
  33. V V
  34. R R
  35. I = n A v Q , I=nAvQ\,,
  36. I I
  37. n n
  38. A A
  39. v v
  40. Q Q

Electric_field.html

  1. 𝐄 \mathbf{E}
  2. 𝐅 \mathbf{F}
  3. q q
  4. 𝐅 = q . 𝐄 \mathbf{F}=q.\mathbf{E}
  5. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  6. × 𝐄 = 0 \nabla\times\mathbf{E}=0
  7. s y m b o l E ( s y m b o l r ) = 1 4 π ε 0 d s y m b o l r ρ ( s y m b o l r ) s y m b o l r - s y m b o l r | s y m b o l r - s y m b o l r | 3 symbol{E}(symbol{r})={1\over 4\pi\varepsilon_{0}}\int dsymbol{r^{\prime}}\rho(% symbol{r^{\prime}}){symbol{r}-symbol{r^{\prime}}\over|symbol{r}-symbol{r^{% \prime}}|^{3}}
  8. ρ ( 𝐫 ) \mathbf{\rho}(\mathbf{r})
  9. 𝐫 \mathbf{r}
  10. ε 0 \varepsilon_{0}
  11. q q
  12. 𝐫 𝟎 \mathbf{r_{0}}
  13. ρ ( 𝐫 ) = q δ ( 𝐫 - 𝐫 𝟎 ) \rho(\mathbf{r})=q\delta(\mathbf{r-r_{0}})
  14. 𝐄 1 \mathbf{E}_{1}
  15. 𝐄 2 \mathbf{E}_{2}
  16. ρ 1 \rho_{1}
  17. ρ 2 \rho_{2}
  18. ρ 1 + ρ 2 \rho_{1}+\rho_{2}
  19. 𝐄 1 + 𝐄 2 \mathbf{E}_{1}+\mathbf{E}_{2}
  20. q 1 , q 2 , , q n q_{1},q_{2},...,q_{n}
  21. 𝐫 1 , 𝐫 2 , 𝐫 n \mathbf{r}_{1},\mathbf{r}_{2},...\mathbf{r}_{n}
  22. 𝐄 ( 𝐫 ) = i = 1 N 𝐄 i ( 𝐫 ) = 1 4 π ε 0 i = 1 N q i 𝐫 - 𝐫 i | 𝐫 - 𝐫 i | 3 \mathbf{E}(\mathbf{r})=\sum_{i=1}^{N}\mathbf{E}_{i}(\mathbf{r})=\frac{1}{4\pi% \varepsilon_{0}}\sum_{i=1}^{N}q_{i}\frac{\mathbf{r}-\mathbf{r}_{i}}{|\mathbf{r% }-\mathbf{r}_{i}|^{3}}
  23. Φ \Phi
  24. 𝐄 = - Φ \mathbf{E}=-\nabla\Phi
  25. 𝐅 = q ( Q 4 π ε 0 𝐫 ^ | 𝐫 | 2 ) = q 𝐄 \mathbf{F}=q\left(\frac{Q}{4\pi\varepsilon_{0}}\frac{\mathbf{\hat{r}}}{|% \mathbf{r}|^{2}}\right)=q\mathbf{E}
  26. 𝐅 = m ( - G M 𝐫 ^ | 𝐫 | 2 ) = m 𝐠 \mathbf{F}=m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^{2}}\right)=m\mathbf% {g}
  27. 𝐫 ^ = 𝐫 | 𝐫 | \mathbf{\hat{r}}=\mathbf{\frac{r}{|r|}}
  28. E = - Δ ϕ d E=-\frac{\Delta\phi}{d}
  29. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  30. Φ \Phi
  31. 𝐄 = - Φ - 𝐀 t \mathbf{E}=-\nabla\Phi-\frac{\partial\mathbf{A}}{\partial t}
  32. × 𝐄 = - ( × 𝐀 ) t = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial(\nabla\times\mathbf{A})}{\partial t}=-% \frac{\partial\mathbf{B}}{\partial t}
  33. u E M = ε 2 | 𝐄 | 2 + 1 2 μ | 𝐁 | 2 u_{EM}=\frac{\varepsilon}{2}|\mathbf{E}|^{2}+\frac{1}{2\mu}|\mathbf{B}|^{2}
  34. μ \mu
  35. u E S = 1 2 ε | 𝐄 | 2 , u_{ES}=\frac{1}{2}\varepsilon|\mathbf{E}|^{2}\,,
  36. U E S = 1 2 ε V | 𝐄 | 2 d V , U_{ES}=\frac{1}{2}\varepsilon\int_{V}|\mathbf{E}|^{2}\,\mathrm{d}V\,,
  37. 𝐃 = ε 0 𝐄 + 𝐏 \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}\!
  38. 𝐃 ( 𝐫 ) = ε 𝐄 ( 𝐫 ) \mathbf{D(r)}=\varepsilon\mathbf{E(r)}
  39. D i = ε i j E j D_{i}=\varepsilon_{ij}E_{j}

Electric_potential.html

  1. Φ Φ
  2. V V
  3. V V
  4. 𝐄 = ( - V 𝐄 ) = - 2 V 𝐄 = ρ / ε 0 , \mathbf{\nabla}\cdot\mathbf{E}=\mathbf{\nabla}\cdot\left(-\mathbf{\nabla}V_{% \mathbf{E}}\right)=-\nabla^{2}V_{\mathbf{E}}=\rho/\varepsilon_{0},\,
  5. U 𝐄 = q V . U_{\mathbf{E}}=q\,V.\,
  6. V 𝐄 = 1 4 π ε 0 Q r , V_{\mathbf{E}}=\frac{1}{4\pi\varepsilon_{0}}\frac{Q}{r},\,
  7. C 𝐄 d s y m b o l \textstyle\int_{C}\mathbf{E}\cdot\mathrm{d}symbol{\ell}
  8. × 𝐄 𝟎 \mathbf{\nabla}\times\mathbf{E}\neq\mathbf{0}
  9. 𝐁 = × 𝐀 , \mathbf{B}=\mathbf{\nabla}\times\mathbf{A},\,
  10. 𝐅 = 𝐄 + 𝐀 t \mathbf{F}=\mathbf{E}+\frac{\partial\mathbf{A}}{\partial t}
  11. 𝐄 = - V - 𝐀 t , \mathbf{E}=-\mathbf{\nabla}V-\frac{\partial\mathbf{A}}{\partial t},\,
  12. - a b 𝐄 d s y m b o l V ( b ) - V ( a ) , -\int_{a}^{b}\mathbf{E}\cdot\mathrm{d}symbol{\ell}\neq V_{(b)}-V_{(a)},\,

Electrical_impedance.html

  1. Z Z
  2. | Z | θ |Z|∠θ
  3. ω ω
  4. Ω Ω
  5. Z \scriptstyle Z
  6. Z = | Z | e j arg ( Z ) \ Z=|Z|e^{j\arg(Z)}
  7. | Z | \scriptstyle|Z|
  8. arg ( Z ) \scriptstyle\arg(Z)
  9. θ \scriptstyle\theta
  10. j \scriptstyle j
  11. i \scriptstyle i
  12. Z = R + j X \ Z=R+jX
  13. R \scriptstyle R
  14. X \scriptstyle X
  15. V = I Z = I | Z | e j arg ( Z ) \ V=IZ=I|Z|e^{j\arg(Z)}
  16. | Z | \scriptstyle|Z|
  17. Z \scriptstyle Z
  18. I \scriptstyle I
  19. θ = arg ( Z ) \scriptstyle\theta\;=\;\arg(Z)
  20. θ 2 π T \scriptstyle\frac{\theta}{2\pi}T
  21. V \scriptstyle V
  22. I \scriptstyle I
  23. V = | V | e j ( ω t + ϕ V ) I = | I | e j ( ω t + ϕ I ) \begin{aligned}\displaystyle V&\displaystyle=|V|e^{j(\omega t+\phi_{V})}\\ \displaystyle I&\displaystyle=|I|e^{j(\omega t+\phi_{I})}\end{aligned}
  24. Z = V I \ Z=\frac{V}{I}
  25. | V | e j ( ω t + ϕ V ) = | I | e j ( ω t + ϕ I ) | Z | e j θ = | I | | Z | e j ( ω t + ϕ I + θ ) \begin{aligned}\displaystyle|V|e^{j(\omega t+\phi_{V})}&\displaystyle=|I|e^{j(% \omega t+\phi_{I})}|Z|e^{j\theta}\\ &\displaystyle=|I||Z|e^{j(\omega t+\phi_{I}+\theta)}\end{aligned}
  26. t t
  27. | V | = | I | | Z | ϕ V = ϕ I + θ \begin{aligned}\displaystyle|V|&\displaystyle=|I||Z|\\ \displaystyle\phi_{V}&\displaystyle=\phi_{I}+\theta\end{aligned}
  28. cos ( ω t + ϕ ) = 1 2 [ e j ( ω t + ϕ ) + e - j ( ω t + ϕ ) ] \ \cos(\omega t+\phi)=\frac{1}{2}\Big[e^{j(\omega t+\phi)}+e^{-j(\omega t+\phi% )}\Big]
  29. cos ( ω t + ϕ ) = { e j ( ω t + ϕ ) } \ \cos(\omega t+\phi)=\Re\Big\{e^{j(\omega t+\phi)}\Big\}
  30. e j ω t \scriptstyle e^{j\omega t}
  31. π / 2 \pi/2
  32. π / 2 \pi/2
  33. Z R = R \ Z_{R}=R
  34. Z L = j ω L \ Z_{L}=j\omega L
  35. Z C = 1 j ω C \ Z_{C}=\frac{1}{j\omega C}
  36. j \displaystyle j
  37. Z L \displaystyle Z_{L}
  38. v R ( t ) = i R ( t ) R v_{\,\text{R}}\left(t\right)={i_{\,\text{R}}\left(t\right)}R
  39. v R ( t ) = V p sin ( ω t ) v_{\,\text{R}}(t)=V_{p}\sin(\omega t)
  40. v R ( t ) i R ( t ) = V p sin ( ω t ) I p sin ( ω t ) = R \frac{v_{\,\text{R}}\left(t\right)}{i_{\,\text{R}}\left(t\right)}=\frac{V_{p}% \sin(\omega t)}{I_{p}\sin\left(\omega t\right)}=R
  41. R \scriptstyle R
  42. Z resistor = R Z_{\,\text{resistor}}=R
  43. i C ( t ) = C d v C ( t ) d t i_{\,\text{C}}(t)=C\frac{\operatorname{d}v_{\,\text{C}}(t)}{\operatorname{d}t}
  44. v C ( t ) = V p sin ( ω t ) v_{\,\text{C}}(t)=V_{p}\sin(\omega t)\,
  45. d v C ( t ) d t = ω V p cos ( ω t ) \frac{\operatorname{d}v_{\,\text{C}}(t)}{\operatorname{d}t}=\omega V_{p}\cos% \left(\omega t\right)
  46. v C ( t ) i C ( t ) = V p sin ( ω t ) ω V p C cos ( ω t ) = sin ( ω t ) ω C sin ( ω t + π 2 ) \frac{v_{\,\text{C}}\left(t\right)}{i_{\,\text{C}}\left(t\right)}=\frac{V_{p}% \sin(\omega t)}{\omega V_{p}C\cos\left(\omega t\right)}=\frac{\sin(\omega t)}{% \omega C\sin\left(\omega t+\frac{\pi}{2}\right)}
  47. 1 ω C \scriptstyle\frac{1}{\omega C}
  48. Z capacitor = 1 ω C e - j π 2 \ Z_{\,\text{capacitor}}=\frac{1}{\omega C}e^{-j\frac{\pi}{2}}
  49. Z capacitor = - j 1 ω C = 1 j ω C \ Z_{\,\text{capacitor}}=-j\frac{1}{\omega C}=\frac{1}{j\omega C}
  50. v L ( t ) = L d i L ( t ) d t v_{\,\text{L}}(t)=L\frac{\operatorname{d}i_{\,\text{L}}(t)}{\operatorname{d}t}
  51. i L ( t ) = I p sin ( ω t ) i_{\,\text{L}}(t)=I_{p}\sin(\omega t)
  52. d i L ( t ) d t = ω I p cos ( ω t ) \frac{\operatorname{d}i_{\,\text{L}}(t)}{\operatorname{d}t}=\omega I_{p}\cos% \left(\omega t\right)
  53. v L ( t ) i L ( t ) = ω I p L cos ( ω t ) I p sin ( ω t ) = ω L sin ( ω t + π 2 ) sin ( ω t ) \frac{v_{\,\text{L}}\left(t\right)}{i_{\,\text{L}}\left(t\right)}=\frac{\omega I% _{p}L\cos(\omega t)}{I_{p}\sin\left(\omega t\right)}=\frac{\omega L\sin\left(% \omega t+\frac{\pi}{2}\right)}{\sin(\omega t)}
  54. ω L \scriptstyle\omega L
  55. Z inductor = ω L e j π 2 \ Z_{\,\text{inductor}}=\omega Le^{j\frac{\pi}{2}}
  56. Z inductor = j ω L \ Z_{\,\text{inductor}}=j\omega L
  57. R R\,
  58. s L sL\,
  59. 1 s C \frac{1}{sC}\,
  60. | Z | = Z Z * = R 2 + X 2 |Z|=\sqrt{ZZ^{*}}=\sqrt{R^{2}+X^{2}}
  61. θ = arctan ( X R ) \theta=\arctan{\left(\frac{X}{R}\right)}
  62. R \scriptstyle R
  63. R = | Z | cos θ \ R=|Z|\cos{\theta}\quad
  64. X \scriptstyle X
  65. θ \scriptstyle\theta
  66. X = | Z | sin θ \ X=|Z|\sin{\theta}\quad
  67. X C = ( ω C ) - 1 = ( 2 π f C ) - 1 X_{C}=(\omega C)^{-1}=(2\pi fC)^{-1}\quad
  68. X L \scriptstyle{X_{L}}
  69. f \scriptstyle{f}
  70. L \scriptstyle{L}
  71. X L = ω L = 2 π f L X_{L}=\omega L=2\pi fL\quad
  72. \scriptstyle{\mathcal{E}}
  73. B \scriptstyle{B}
  74. = - d Φ B d t \mathcal{E}=-{{d\Phi_{B}}\over dt}\quad
  75. N N
  76. = - N d Φ B d t \mathcal{E}=-N{d\Phi_{B}\over dt}\quad
  77. X = X L - X C {X=X_{L}-X_{C}}
  78. Z = R + j X \ Z=R+jX
  79. Z eq = Z 1 + Z 2 + + Z n \ Z_{\,\text{eq}}=Z_{1}+Z_{2}+\cdots+Z_{n}\quad
  80. Z eq = R + j X = ( R 1 + R 2 + + R n ) + j ( X 1 + X 2 + + X n ) \ Z_{\,\text{eq}}=R+jX=(R_{1}+R_{2}+\cdots+R_{n})+j(X_{1}+X_{2}+\cdots+X_{n})\quad
  81. 1 Z eq = 1 Z 1 + 1 Z 2 + + 1 Z n \frac{1}{Z_{\,\text{eq}}}=\frac{1}{Z_{1}}+\frac{1}{Z_{2}}+\cdots+\frac{1}{Z_{n}}
  82. 1 Z eq = 1 Z 1 + 1 Z 2 = Z 1 + Z 2 Z 1 Z 2 \frac{1}{Z_{\,\text{eq}}}=\frac{1}{Z_{1}}+\frac{1}{Z_{2}}=\frac{Z_{1}+Z_{2}}{Z% _{1}Z_{2}}
  83. Z eq = Z 1 Z 2 Z 1 + Z 2 \ Z_{\,\text{eq}}=\frac{Z_{1}Z_{2}}{Z_{1}+Z_{2}}
  84. Z eq \scriptstyle Z_{\,\text{eq}}
  85. R eq \scriptstyle R_{\,\text{eq}}
  86. X eq \scriptstyle X_{\,\text{eq}}
  87. Z eq = R eq + j X eq R eq = ( X 1 R 2 + X 2 R 1 ) ( X 1 + X 2 ) + ( R 1 R 2 - X 1 X 2 ) ( R 1 + R 2 ) ( R 1 + R 2 ) 2 + ( X 1 + X 2 ) 2 X eq = ( X 1 R 2 + X 2 R 1 ) ( R 1 + R 2 ) - ( R 1 R 2 - X 1 X 2 ) ( X 1 + X 2 ) ( R 1 + R 2 ) 2 + ( X 1 + X 2 ) 2 \begin{aligned}\displaystyle Z_{\,\text{eq}}&\displaystyle=R_{\,\text{eq}}+jX_% {\,\text{eq}}\\ \displaystyle R_{\,\text{eq}}&\displaystyle=\frac{(X_{1}R_{2}+X_{2}R_{1})(X_{1% }+X_{2})+(R_{1}R_{2}-X_{1}X_{2})(R_{1}+R_{2})}{(R_{1}+R_{2})^{2}+(X_{1}+X_{2})% ^{2}}\\ \displaystyle X_{\,\text{eq}}&\displaystyle=\frac{(X_{1}R_{2}+X_{2}R_{1})(R_{1% }+R_{2})-(R_{1}R_{2}-X_{1}X_{2})(X_{1}+X_{2})}{(R_{1}+R_{2})^{2}+(X_{1}+X_{2})% ^{2}}\end{aligned}

Electrical_length.html

  1. ϵ r \epsilon\text{r}
  2. κ = v p c = 1 ϵ r \kappa=\frac{v_{p}}{c}=\frac{1}{\sqrt{\epsilon\text{r}}}
  3. ϵ eff \epsilon\text{eff}
  4. κ = 1 ϵ eff \kappa=\frac{1}{\sqrt{\epsilon\text{eff}}}
  5. ϵ eff \epsilon\text{eff}
  6. ϵ eff = ( 1 - F ) + F ϵ r \epsilon\text{eff}=(1-F)+F\epsilon\text{r}
  7. ϵ eff \epsilon\text{eff}
  8. ϵ eff \epsilon\text{eff}

Electrical_resistance_and_conductance.html

  1. R = V I , G = I V = 1 R R={V\over I},\qquad G={I\over V}=\frac{1}{R}
  2. d V d I \frac{dV}{dI}\,\!
  3. V I V\propto I
  4. R R
  5. G G
  6. R = ρ A , R=\rho\frac{\ell}{A},
  7. G = σ A . G=\sigma\frac{A}{\ell}.
  8. \ell
  9. ρ = 1 / σ \rho=1/\sigma
  10. R static = V I R_{\mathrm{static}}=\frac{V}{I}\,
  11. R diff = d V d I R_{\mathrm{diff}}=\frac{dV}{dI}\,
  12. V ( t ) = Re ( V 0 e j ω t ) , I ( t ) = Re ( I 0 e j ω t ) , Z = V 0 I 0 , Y = I 0 V 0 V(t)=\,\text{Re}(V_{0}e^{j\omega t}),\quad I(t)=\,\text{Re}(I_{0}e^{j\omega t}% ),\quad Z=\frac{V_{0}}{I_{0}},\quad Y=\frac{I_{0}}{V_{0}}
  13. ω \omega
  14. j = - 1 j=\sqrt{-1}
  15. Z = R + j X , Y = G + j B Z=R+jX,\quad Y=G+jB
  16. Z = 1 / Y Z=1/Y
  17. R = 1 / G R=1/G
  18. P = I 2 R P=I^{2}R
  19. R ( T ) = R 0 [ 1 + α ( T - T 0 ) ] R(T)=R_{0}[1+\alpha(T-T_{0})]
  20. α \alpha
  21. T 0 T_{0}
  22. R 0 R_{0}
  23. T 0 T_{0}
  24. α \alpha
  25. α \alpha
  26. α \alpha
  27. α 15 \alpha_{15}
  28. α \alpha

Electrical_resistivity_and_conductivity.html

  1. η = ρ / μ 0 1 / σ μ 0 \eta=\rho/\mu_{0}\equiv 1/\sigma\mu_{0}
  2. μ 0 \mu_{0}
  3. m A 2 Ω N - 1 mA^{2}\Omega N^{-1}
  4. ρ = R A , \rho=R\frac{A}{\ell},\,\!
  5. \ell
  6. R = ρ A . R=\rho\frac{\ell}{A}.\,\!
  7. R = ρ / A R=\rho\ell/A
  8. A = 1 m 2 A=1\,\text{m}^{2}
  9. = 1 m \ell=1\,\text{m}
  10. σ = 1 ρ . \sigma=\frac{1}{\rho}.\,\!
  11. ρ = E J , \rho=\frac{E}{J},\,\!
  12. σ = 1 ρ = J E . \sigma=\frac{1}{\rho}=\frac{J}{E}.\,\!
  13. n e e e Φ / k B T e . n_{e}\propto e^{e\Phi/k_{B}T_{e}}.
  14. E = - ( k B T e / e ) ( n e / n e ) . \vec{E}=-(k_{B}T_{e}/e)(\nabla n_{e}/n_{e}).
  15. ρ ( T ) = ρ 0 [ 1 + α ( T - T 0 ) ] \rho(T)=\rho_{0}[1+\alpha(T-T_{0})]
  16. α \alpha
  17. T 0 T_{0}
  18. ρ 0 \rho_{0}
  19. T 0 T_{0}
  20. α \alpha
  21. α \alpha
  22. α \alpha
  23. α 15 \alpha_{15}
  24. ρ ( T ) = ρ ( 0 ) + A ( T Θ R ) n 0 Θ R T x n ( e x - 1 ) ( 1 - e - x ) d x \rho(T)=\rho(0)+A\left(\frac{T}{\Theta_{R}}\right)^{n}\int_{0}^{\frac{\Theta_{% R}}{T}}\frac{x^{n}}{(e^{x}-1)(1-e^{-x})}dx
  25. ρ ( 0 ) \rho(0)
  26. Θ R \Theta_{R}
  27. ρ = ρ 0 e - a T \rho=\rho_{0}e^{-aT}\,
  28. 1 / T = A + B ln ( ρ ) + C ( ln ( ρ ) ) 3 1/T=A+B\ln(\rho)+C(\ln(\rho))^{3}\,
  29. ρ = A exp ( T - 1 / n ) \rho=A\exp(T^{-1/n})
  30. J = σ E E = ρ J . J=\sigma E\,\,\rightleftharpoons\,\,E=\rho J.\,\!
  31. 𝐉 = σ 𝐄 𝐄 = ρ 𝐉 \mathbf{J}=\sigma\mathbf{E}\,\,\rightleftharpoons\,\,\mathbf{E}=\rho\mathbf{J}\,\!
  32. J i = σ i j E j E i = ρ i j J j . J_{i}=\sigma_{ij}E_{j}\,\,\rightleftharpoons\,\,E_{i}=\rho_{ij}J_{j}.\,\!
  33. R = ρ / A R=\rho\ell/A
  34. J = σ E E = ρ J J=\sigma E\,\,\rightleftharpoons\,\,E=\rho J\,\!
  35. 𝐉 ( 𝐫 ) = σ ( 𝐫 ) 𝐄 ( 𝐫 ) 𝐄 ( 𝐫 ) = ρ ( 𝐫 ) 𝐉 ( 𝐫 ) , \mathbf{J}(\mathbf{r})=\sigma(\mathbf{r})\mathbf{E}(\mathbf{r})\,\,% \rightleftharpoons\,\,\mathbf{E}(\mathbf{r})=\rho(\mathbf{r})\mathbf{J}(% \mathbf{r}),\,\!

Electricity.html

  1. P = work done per unit time = Q V t = I V P=\,\text{work done per unit time}=\frac{QV}{t}=IV\,

Electro-optic_modulator.html

  1. π \pi
  2. V π V_{\pi}
  3. ω \omega
  4. A e i ω t . Ae^{i\omega t}.
  5. Ω \Omega
  6. β \beta
  7. A e i ω t + i β sin ( Ω t ) . Ae^{i\omega t+i\beta\sin(\Omega t)}.
  8. β \beta
  9. A e i ω t ( 1 + i β sin ( Ω t ) ) , Ae^{i\omega t}\left(1+i\beta\sin(\Omega t)\right),
  10. A e i ω t ( 1 + β 2 ( e i Ω t - e - i Ω t ) ) = A ( e i ω t + β 2 e i ( ω + Ω ) t - β 2 e i ( ω - Ω ) t ) . Ae^{i\omega t}\left(1+\frac{\beta}{2}(e^{i\Omega t}-e^{-i\Omega t})\right)=A% \left(e^{i\omega t}+\frac{\beta}{2}e^{i(\omega+\Omega)t}-\frac{\beta}{2}e^{i(% \omega-\Omega)t}\right).
  11. ω + Ω \omega+\Omega
  12. ω - Ω \omega-\Omega
  13. A e i ω t + i β sin ( Ω t ) = A e i ω t ( J 0 ( β ) + k = 1 J k ( β ) e i k Ω t + k = 1 ( - 1 ) k J k ( β ) e - i k Ω t ) , Ae^{i\omega t+i\beta\sin(\Omega t)}=Ae^{i\omega t}\left(J_{0}(\beta)+\sum_{k=1% }^{\infty}J_{k}(\beta)e^{ik\Omega t}+\sum_{k=1}^{\infty}(-1)^{k}J_{k}(\beta)e^% {-ik\Omega t}\right),
  14. ( 1 + β sin ( Ω t ) ) A e i ω t = A e i ω t + A β 2 i ( e i ( ω + Ω ) t - e i ( ω - Ω ) t ) . \left(1+\beta\sin(\Omega t)\right)Ae^{i\omega t}=Ae^{i\omega t}+\frac{A\beta}{% 2i}\left(e^{i(\omega+\Omega)t}-e^{i(\omega-\Omega)t}\right).

Electrochemistry.html

  1. W m a x = W e l e c t r i c a l = - n F E c e l l W_{max}=W_{electrical}=-nFE_{cell}
  2. Δ G = - n F E c e l l \Delta G=-nFE_{cell}
  3. Δ G = - R T ln K = - n F E c e l l \Delta G^{\circ}=-RT\ln K=-nFE^{\circ}_{cell}
  4. E c e l l o = R T n F ln K E^{o}_{cell}=\frac{RT}{nF}\ln K
  5. E c e l l o = 0.0591 V n log K E^{o}_{cell}=\frac{0.0591\,\mathrm{V}}{n}\log K
  6. Δ G = Δ G + R T ln Q \Delta G=\Delta G^{\circ}+RT\ln Q
  7. n F Δ E = n F Δ E - R T ln Q nF\Delta E=nF\Delta E^{\circ}-RT\ln Q
  8. Δ E = Δ E - R T n F ln Q \Delta E=\Delta E^{\circ}-\frac{RT}{nF}\ln Q
  9. Δ E = Δ E - 0.05916 V n log Q \Delta E=\Delta E^{\circ}-\frac{0.05916\,\mathrm{V}}{n}\log Q
  10. E = E - 0.05916 V 2 log [ Cu 2 + ] d i l u t e d [ Cu 2 + ] c o n c e n t r a t e d E=E^{\circ}-\frac{0.05916\,\mathrm{V}}{2}\log\frac{[\mathrm{Cu^{2+}}]_{diluted% }}{[\mathrm{Cu^{2+}}]_{concentrated}}
  11. E = 0 - 0.05916 V 2 log 0.05 2.0 = 0.0474 V E=0-\frac{0.05916\,\mathrm{V}}{2}\log\frac{0.05}{2.0}=0.0474\,\mathrm{V}
  12. E = 0 - 0.0257 V 2 ln 0.05 2.0 = 0.0474 V E=0-\frac{0.0257\,\mathrm{V}}{2}\ln\frac{0.05}{2.0}=0.0474\,\mathrm{V}
  13. m = 1 96485 ( C mol - 1 ) Q M n m=\frac{1}{96485\mathrm{(C\cdot mol^{-1})}}\cdot\frac{QM}{n}

Electrolysis.html

  1. m = k q m=k\cdot q

Electromagnetic_field.html

  1. E = h ν E=\,h\,\nu
  2. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  3. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  4. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  5. × 𝐁 = μ 0 𝐉 + μ 0 ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}
  6. ρ \rho
  7. ϵ 0 \epsilon_{0}
  8. μ 0 \mu_{0}
  9. 𝐄 1 = 𝐄 2 \mathbf{E}_{1}=\mathbf{E}_{2}
  10. 𝐇 1 = 𝐇 2 \mathbf{H}_{1}=\mathbf{H}_{2}
  11. 𝐃 1 = 𝐃 2 \mathbf{D}_{1}=\mathbf{D}_{2}
  12. 𝐁 1 = 𝐁 2 \mathbf{B}_{1}=\mathbf{B}_{2}
  13. ( ε ) (\varepsilon)
  14. < m t p l > tan θ 1 tan θ 2 = ε r 2 ε r 1 \frac{<}{m}tpl>{{\tan\theta_{1}}}{{\tan\theta_{2}}}=\frac{{\varepsilon_{r2}}}{% {\varepsilon_{r1}}}
  15. ( μ ) (\mu)
  16. < m t p l > tan θ 1 tan θ 2 = μ r 2 μ r 1 \frac{<}{m}tpl>{{\tan\theta_{1}}}{{\tan\theta_{2}}}=\frac{{\mu_{r2}}}{{\mu_{r1% }}}
  17. ρ \rho
  18. ( 2 - 1 c 2 2 t 2 ) 𝐄 = 0 \left(\nabla^{2}-{1\over{c}^{2}}{\partial^{2}\over\partial t^{2}}\right)% \mathbf{E}\ \ =\ \ 0
  19. ( 2 - 1 c 2 2 t 2 ) 𝐁 = 0 \left(\nabla^{2}-{1\over{c}^{2}}{\partial^{2}\over\partial t^{2}}\right)% \mathbf{B}\ \ =\ \ 0

Electromagnetic_induction.html

  1. = - d Φ B d t \mathcal{E}=-{{d\Phi_{\mathrm{B}}}\over dt}
  2. \mathcal{E}
  3. = - N d Φ B d t \mathcal{E}=-N{{d\Phi_{\mathrm{B}}}\over dt}
  4. Φ B = Σ ( t ) 𝐁 ( 𝐫 , t ) d 𝐀 , \Phi_{\mathrm{B}}=\iint\limits_{\Sigma(t)}\mathbf{B}(\mathbf{r},t)\cdot d% \mathbf{A}\ ,
  5. \mathcal{E}
  6. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)
  7. = 1 q wire 𝐅 d s y m b o l = wire ( 𝐄 + 𝐯 × 𝐁 ) d s y m b o l \mathcal{E}=\frac{1}{q}\oint_{\mathrm{wire}}\mathbf{F}\cdot dsymbol{\ell}=% \oint_{\mathrm{wire}}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\cdot dsymbol% {\ell}
  8. × \nabla\times
  9. = - B v , \mathcal{E}=-B\ell v,

Electromagnetic_radiation.html

  1. v = f λ \displaystyle v=f\lambda
  2. E = h f = h c λ E=hf=\frac{hc}{\lambda}\,\!
  3. λ \lambda
  4. p = E c = h f c = h λ . p={E\over c}={hf\over c}={h\over\lambda}.
  5. 𝐄 = 0 ( 1 ) \nabla\cdot\mathbf{E}=0\qquad\qquad\qquad\ \ (1)
  6. × 𝐄 = - 𝐁 t ( 2 ) \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\qquad\qquad\ (2)
  7. 𝐁 = 0 ( 3 ) \nabla\cdot\mathbf{B}=0\qquad\qquad\qquad\ \ (3)
  8. × 𝐁 = μ 0 ϵ 0 𝐄 t ( 4 ) \nabla\times\mathbf{B}=\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t% }\qquad\quad\ (4)
  9. \nabla
  10. 𝐄 = 𝐁 = 𝟎 , \mathbf{E}=\mathbf{B}=\mathbf{0},
  11. × ( × 𝐀 ) = ( 𝐀 ) - 2 𝐀 \nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf% {A}\right)-\nabla^{2}\mathbf{A}
  12. × ( × 𝐄 ) = × ( - 𝐁 t ) ( 5 ) \nabla\times\left(\nabla\times\mathbf{E}\right)=\nabla\times\left(-\frac{% \partial\mathbf{B}}{\partial t}\right)\qquad\qquad\qquad\quad\ \ \ (5)\,
  13. × ( × 𝐄 ) = ( 𝐄 ) - 2 𝐄 = - 2 𝐄 ( 6 ) \nabla\times\left(\nabla\times\mathbf{E}\right)=\nabla\left(\nabla\cdot\mathbf% {E}\right)-\nabla^{2}\mathbf{E}=-\nabla^{2}\mathbf{E}\qquad\ \ (6)\,
  14. × ( - 𝐁 t ) = - t ( × 𝐁 ) = - μ 0 ϵ 0 2 𝐄 t 2 ( 7 ) \nabla\times\left(-\frac{\partial\mathbf{B}}{\partial t}\right)=-\frac{% \partial}{\partial t}\left(\nabla\times\mathbf{B}\right)=-\mu_{0}\epsilon_{0}% \frac{\partial^{2}\mathbf{E}}{\partial t^{2}}\quad\ \ \ \ (7)
  15. 2 𝐄 = μ 0 ϵ 0 2 𝐄 t 2 \nabla^{2}\mathbf{E}=\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}
  16. 2 𝐁 = μ 0 ϵ 0 2 𝐁 t 2 . \nabla^{2}\mathbf{B}=\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{B}}{\partial t% ^{2}}.
  17. 2 f = 1 c 0 2 2 f t 2 \nabla^{2}f=\frac{1}{{c_{0}}^{2}}\frac{\partial^{2}f}{\partial t^{2}}\,
  18. f = 0 \Box f=0
  19. \Box
  20. = 2 - 1 c 0 2 2 t 2 = 2 x 2 + 2 y 2 + 2 z 2 - 1 c 0 2 2 t 2 \Box=\nabla^{2}-\frac{1}{{c_{0}}^{2}}\frac{\partial^{2}}{\partial t^{2}}=\frac% {\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{% \partial^{2}}{\partial z^{2}}-\frac{1}{{c_{0}}^{2}}\frac{\partial^{2}}{% \partial t^{2}}
  21. c 0 = 1 μ 0 ϵ 0 c_{0}=\frac{1}{\sqrt{\mu_{0}\epsilon_{0}}}
  22. ϵ 0 \epsilon_{0}
  23. μ 0 \mu_{0}
  24. 𝐄 = 𝐄 0 f ( 𝐤 ^ 𝐱 - c 0 t ) \mathbf{E}=\mathbf{E}_{0}f\left(\hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)
  25. 𝐄 0 \mathbf{E}_{0}
  26. f f
  27. 𝐤 ^ \hat{\mathbf{k}}
  28. 𝐱 {\mathbf{x}}
  29. f ( 𝐤 ^ 𝐱 - c 0 t ) f\left(\hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)
  30. 2 f ( 𝐤 ^ 𝐱 - c 0 t ) = 1 c 0 2 2 t 2 f ( 𝐤 ^ 𝐱 - c 0 t ) , \nabla^{2}f\left(\hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)=\frac{1}{{c_{0}% }^{2}}\frac{\partial^{2}}{\partial t^{2}}f\left(\hat{\mathbf{k}}\cdot\mathbf{x% }-c_{0}t\right),
  31. 𝐤 ^ \hat{\mathbf{k}}
  32. 𝐄 = 𝐤 ^ 𝐄 0 f ( 𝐤 ^ 𝐱 - c 0 t ) = 0 \nabla\cdot\mathbf{E}=\hat{\mathbf{k}}\cdot\mathbf{E}_{0}f^{\prime}\left(\hat{% \mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)=0
  33. 𝐄 𝐤 ^ = 0 \mathbf{E}\cdot\hat{\mathbf{k}}=0
  34. × 𝐄 = 𝐤 ^ × 𝐄 0 f ( 𝐤 ^ 𝐱 - c 0 t ) = - 𝐁 t \nabla\times\mathbf{E}=\hat{\mathbf{k}}\times\mathbf{E}_{0}f^{\prime}\left(% \hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)=-\frac{\partial\mathbf{B}}{% \partial t}
  35. 𝐁 = 1 c 0 𝐤 ^ × 𝐄 \mathbf{B}=\frac{1}{c_{0}}\hat{\mathbf{k}}\times\mathbf{E}
  36. 𝐄 , 𝐁 \mathbf{E},\mathbf{B}
  37. E 0 = c 0 B 0 E_{0}=c_{0}B_{0}
  38. 𝐄 × 𝐁 \mathbf{E}\times\mathbf{B}

Electromagnetic_spectrum.html

  1. f = c λ , or f = E h , or E = h c λ , f=\frac{c}{\lambda},\quad\,\text{or}\quad f=\frac{E}{h},\quad\,\text{or}\quad E% =\frac{hc}{\lambda},

Electromagnetism.html

  1. Φ \Phi

Electromotive_force.html

  1. \mathcal{E}
  2. \mathcal{E}
  3. = - A B s y m b o l E c s d s y m b o l , \mathcal{E}=-\int_{A}^{B}symbol{E_{cs}\cdot}dsymbol{\ell}\ ,
  4. = C s y m b o l E d s y m b o l , \mathcal{E}=\oint_{C}symbol{E\cdot}dsymbol{\ell}\ ,
  5. = C s y m b o l [ E + v × B ] d s y m b o l \mathcal{E}=\oint_{C}symbol{\left[E+v\times B\right]\cdot}dsymbol{\ell}
  6. + 1 q C 𝐞𝐟𝐟𝐞𝐜𝐭𝐢𝐯𝐞 𝐜𝐡𝐞𝐦𝐢𝐜𝐚𝐥 𝐟𝐨𝐫𝐜𝐞𝐬 d s y m b o l +\frac{1}{q}\oint_{C}\mathrm{\mathbf{effective\ chemical\ forces\ \cdot}}\ % dsymbol{\ell}
  7. + 1 q C 𝐞𝐟𝐟𝐞𝐜𝐭𝐢𝐯𝐞 𝐭𝐡𝐞𝐫𝐦𝐚𝐥 𝐟𝐨𝐫𝐜𝐞𝐬 d s y m b o l , +\frac{1}{q}\oint_{C}\mathrm{\mathbf{effective\ thermal\ forces\ \cdot}}\ % dsymbol{\ell}\ ,
  8. d G = - S d T + V d P + d Z , dG=-SdT+VdP+\mathcal{E}dZ\ ,
  9. ( T ) Z = - ( S Z ) T \left(\frac{\partial\mathcal{E}}{\partial T}\right)_{Z}=-\left(\frac{\partial S% }{\partial Z}\right)_{T}
  10. Δ Z = - n 0 F 0 , \Delta Z=-n_{0}F_{0}\ ,
  11. Δ H = - n 0 F 0 ( - T d d T ) , \Delta H=-n_{0}F_{0}\left(\mathcal{E}-T\frac{d\mathcal{E}}{dT}\right)\ ,
  12. Zn ( s ) Zn ( aq ) 2 + + 2 e - \mathrm{Zn_{(s)}\rightarrow Zn^{2+}_{(aq)}+2e^{-}\ }
  13. Zn 2 + \mathrm{Zn}^{2+}
  14. SO 4 2 - \mathrm{SO}_{4}^{2-}
  15. Cu ( aq ) 2 + + 2 e - Cu ( s ) \mathrm{Cu^{2+}_{(aq)}+2e^{-}\rightarrow Cu_{(s)}\ }
  16. I = I L - I 0 ( e q V / ( m k T ) - 1 ) , I=I_{L}-I_{0}\left(e^{qV/(mkT)}-1\right)\ ,
  17. V oc = m k T q ln ( I L I 0 + 1 ) , V\text{oc}=m\ \frac{kT}{q}\ \ln\left(\frac{I\text{L}}{I_{0}}+1\right)\ ,

Electron.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. [ u r a d i c a l , u 3 ] / 2 {[u^{\prime}radical^{\prime},u^{\prime}3^{\prime}]}/{2}
  5. S = s ( s + 1 ) h 2 π = 3 2 \begin{aligned}\displaystyle S&\displaystyle=\sqrt{s(s+1)}\cdot\frac{h}{2\pi}% \\ &\displaystyle=\frac{\sqrt{3}}{2}\hbar\\ \end{aligned}
  6. 1 / 2 {1}/{2}
  7. ħ / 2 {ħ}/{2}
  8. Δ λ = h m e c ( 1 - cos θ ) , \textstyle\Delta\lambda=\frac{h}{m_{\mathrm{e}}c}(1-\cos\theta),
  9. 1 / 137 {1}/{137}
  10. γ = 1 / 1 - v 2 / c 2 \scriptstyle\gamma=1/\sqrt{1-{v^{2}}/{c^{2}}}
  11. K e = ( γ - 1 ) m e c 2 , \displaystyle K_{\mathrm{e}}=(\gamma-1)m_{\mathrm{e}}c^{2},
  12. E p = e 2 8 π ε 0 r , E_{\mathrm{p}}=\frac{e^{2}}{8\pi\varepsilon_{0}r},
  13. E p = m 0 c 2 , \textstyle E_{\mathrm{p}}=m_{0}c^{2},

Electronegativity.html

  1. χ A - χ B = ( eV ) - 1 / 2 E d ( AB ) - [ E d ( AA ) + E d ( BB ) ] / 2 \chi_{\rm A}-\chi_{\rm B}=({\rm eV})^{-1/2}\sqrt{E_{\rm d}({\rm AB})-[E_{\rm d% }({\rm AA})+E_{\rm d}({\rm BB})]/2}
  2. E d ( AB ) = [ E d ( AA ) + E d ( BB ) ] / 2 + ( χ A - χ B ) 2 e V E_{\rm d}({\rm AB})=[E_{\rm d}({\rm AA})+E_{\rm d}({\rm BB})]/2+(\chi_{\rm A}-% \chi_{\rm B})^{2}eV
  3. E d ( AB ) = E d ( AA ) E d ( BB ) + 1.3 ( χ A - χ B ) 2 e V E_{\rm d}({\rm AB})=\sqrt{E_{\rm d}({\rm AA})E_{\rm d}({\rm BB})}+1.3(\chi_{% \rm A}-\chi_{\rm B})^{2}eV
  4. χ = ( E i + E ea ) / 2 \chi=(E_{\rm i}+E_{\rm ea})/2\,
  5. χ = 0.187 ( E i + E ea ) + 0.17 \chi=0.187(E_{\rm i}+E_{\rm ea})+0.17\,
  6. χ = ( 1.97 × 10 - 3 ) ( E i + E ea ) + 0.19. \chi=(1.97\times 10^{-3})(E_{\rm i}+E_{\rm ea})+0.19.
  7. μ ( Mulliken ) = - χ ( Mulliken ) = - ( E i + E ea ) / 2 \mu(\rm Mulliken)=-\chi(\rm Mulliken)=-(E_{\rm i}+E_{\rm ea})/2\,
  8. χ = 3590 Z eff r cov 2 + 0.744 \chi=3590{{Z_{\rm eff}}\over{r^{2}_{\rm cov}}}+0.744
  9. χ = n s ε s + n p ε p n s + n p \chi={n_{\rm s}\varepsilon_{\rm s}+n_{\rm p}\varepsilon_{\rm p}\over n_{\rm s}% +n_{\rm p}}

Electronvolt.html

  1. 1 eV / c 2 = ( 1.60217646 × 10 - 19 C ) 1 V ( 2.99792458 × 10 8 m / s ) 2 = 1.783 × 10 - 36 kg . 1\;\,\text{eV}/c^{2}=\frac{(1.60217646\times 10^{-19}\;\,\text{C})\cdot 1\;\,% \text{V}}{(2.99792458\times 10^{8}\;\,\text{m}/\,\text{s})^{2}}=1.783\times 10% ^{-36}\;\,\text{kg}.
  2. p = 1 GeV / c = ( 1 × 10 9 ) ( 1.60217646 × 10 - 19 1 C ) V ( 2.99792458 × 10 8 m / s ) = 5.344286 × 10 - 19 kg m / s . p=1\;\,\text{GeV}/c=\frac{(1\times 10^{9})\cdot(1.60217646\times 10^{-19}\;1\;% \,\text{C})\cdot\,\text{V}}{(2.99792458\times 10^{8}\;\,\text{m}/\,\text{s})}=% 5.344286\times 10^{-19}\;\,\text{kg}\cdot\,\text{m}/\,\text{s}.
  3. = h 2 π = 1.054 571 726 ( 47 ) × 10 - 34 J s = 6.582 119 28 ( 15 ) × 10 - 16 eV s . \hbar={{h}\over{2\pi}}=1.054\ 571\ 726(47)\times 10^{-34}\ \mbox{J s}~{}=6.582% \ 119\ 28(15)\times 10^{-16}\ \mbox{eV s}~{}.
  4. 1 k B = 1.602 176 53 ( 14 ) × 10 - 19 J/eV 1.380 6505 ( 24 ) × 10 - 23 J/K = 11 604.505 ( 20 ) K/eV . {1\over k_{\,\text{B}}}={1.602\,176\,53(14)\times 10^{-19}\,\text{ J/eV}\over 1% .380\,6505(24)\times 10^{-23}\,\text{ J/K}}=11\,604.505(20)\,\text{ K/eV}.
  5. 290 K 11604 K / e V \frac{290K}{11604K/eV}
  6. E = h ν = h c λ E=h\nu=\frac{hc}{\lambda}
  7. = ( 4.13566 7516 × 10 - 15 eV s ) ( 299 792 458 m/s ) λ =\frac{(4.13566\,7516\times 10^{-15}\,\mbox{eV}~{}\,\mbox{s}~{})(299\,792\,458% \,\mbox{m/s}~{})}{\lambda}
  8. E (eV) = 4.13566 7516 feVs ν (PHz) E\mbox{(eV)}~{}=4.13566\,7516\,\mbox{feVs}~{}\cdot\nu\ \mbox{(PHz)}~{}
  9. = 1 239.84193 eV nm λ (nm) . =\frac{1\,239.84193\,\mbox{eV}~{}\,\mbox{nm}~{}}{\lambda\ \mbox{(nm)}~{}}.

Electroplating.html

  1. M z + + R e d s o l u t i o n catalytic surface M s o l i d + O x y s o l u t i o n M^{z+}+Red_{solution}\stackrel{\,\text{catalytic surface}}{\Longrightarrow}M_{% solid}+Oxy_{solution}

Electroweak_interaction.html

  1. ( γ Z 0 ) = ( cos θ W sin θ W - sin θ W cos θ W ) ( B 0 W 0 ) \begin{pmatrix}\gamma\\ Z^{0}\end{pmatrix}=\begin{pmatrix}\cos\theta_{W}&\sin\theta_{W}\\ -\sin\theta_{W}&\cos\theta_{W}\end{pmatrix}\begin{pmatrix}B^{0}\\ W^{0}\end{pmatrix}
  2. M Z = M W cos θ W M_{Z}=\frac{M_{W}}{\cos\theta_{W}}
  3. E W = g + f + h + y . \mathcal{L}_{EW}=\mathcal{L}_{g}+\mathcal{L}_{f}+\mathcal{L}_{h}+\mathcal{L}_{% y}.
  4. g \mathcal{L}_{g}
  5. g = - 1 4 W a μ ν W μ ν a - 1 4 B μ ν B μ ν \mathcal{L}_{g}=-\frac{1}{4}W_{a}^{\mu\nu}W_{\mu\nu}^{a}-\frac{1}{4}B^{\mu\nu}% B_{\mu\nu}
  6. W a μ ν W^{a\mu\nu}
  7. a = 1 , 2 , 3 a=1,2,3
  8. B μ ν B^{\mu\nu}
  9. f \mathcal{L}_{f}
  10. f = Q ¯ i i D / Q i + u ¯ i i D / u i + d ¯ i i D / d i + L ¯ i i D / L i + e ¯ i i D / e i \mathcal{L}_{f}=\overline{Q}_{i}iD\!\!\!\!/\;Q_{i}+\overline{u}_{i}iD\!\!\!\!/% \;u_{i}+\overline{d}_{i}iD\!\!\!\!/\;d_{i}+\overline{L}_{i}iD\!\!\!\!/\;L_{i}+% \overline{e}_{i}iD\!\!\!\!/\;e_{i}
  11. i i
  12. Q Q
  13. u u
  14. d d
  15. L L
  16. e e
  17. h = | D μ h | 2 - λ ( | h | 2 - v 2 2 ) 2 \mathcal{L}_{h}=|D_{\mu}h|^{2}-\lambda\left(|h|^{2}-\frac{v^{2}}{2}\right)^{2}
  18. y = - y u i j ϵ a b h b Q ¯ i a u j c - y d i j h Q ¯ i d j c - y e i j h L ¯ i e j c + h . c . \mathcal{L}_{y}=-y_{u\,ij}\epsilon^{ab}\,h_{b}^{\dagger}\,\overline{Q}_{ia}u_{% j}^{c}-y_{d\,ij}\,h\,\overline{Q}_{i}d^{c}_{j}-y_{e\,ij}\,h\,\overline{L}_{i}e% ^{c}_{j}+h.c.
  19. E W = K + N + C + H + H V + W W V + W W V V + Y \mathcal{L}_{EW}=\mathcal{L}_{K}+\mathcal{L}_{N}+\mathcal{L}_{C}+\mathcal{L}_{% H}+\mathcal{L}_{HV}+\mathcal{L}_{WWV}+\mathcal{L}_{WWVV}+\mathcal{L}_{Y}
  20. K \mathcal{L}_{K}
  21. K = f f ¯ ( i / - m f ) f - 1 4 A μ ν A μ ν - 1 2 W μ ν + W - μ ν + m W 2 W μ + W - μ - 1 4 Z μ ν Z μ ν + 1 2 m Z 2 Z μ Z μ + 1 2 ( μ H ) ( μ H ) - 1 2 m H 2 H 2 \begin{aligned}\displaystyle\mathcal{L}_{K}=\sum_{f}\overline{f}(i\partial\!\!% \!/\!\;-m_{f})f-\frac{1}{4}A_{\mu\nu}A^{\mu\nu}-\frac{1}{2}W^{+}_{\mu\nu}W^{-% \mu\nu}+m_{W}^{2}W^{+}_{\mu}W^{-\mu}\\ \displaystyle\qquad-\frac{1}{4}Z_{\mu\nu}Z^{\mu\nu}+\frac{1}{2}m_{Z}^{2}Z_{\mu% }Z^{\mu}+\frac{1}{2}(\partial^{\mu}H)(\partial_{\mu}H)-\frac{1}{2}m_{H}^{2}H^{% 2}\end{aligned}
  22. A μ ν A_{\mu\nu}
  23. Z μ ν Z_{\mu\nu}
  24. W μ ν - W^{-}_{\mu\nu}
  25. W μ ν + ( W μ ν - ) W^{+}_{\mu\nu}\equiv(W^{-}_{\mu\nu})^{\dagger}
  26. X μ ν = μ X ν - ν X μ + g f a b c X μ b X ν c X_{\mu\nu}=\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu}+gf^{abc}X^{b}_{\mu}X^{c% }_{\nu}
  27. N \mathcal{L}_{N}
  28. C \mathcal{L}_{C}
  29. N = e J μ e m A μ + g cos θ W ( J μ 3 - sin 2 θ W J μ e m ) Z μ \mathcal{L}_{N}=eJ_{\mu}^{em}A^{\mu}+\frac{g}{\cos\theta_{W}}(J_{\mu}^{3}-\sin% ^{2}\theta_{W}J_{\mu}^{em})Z^{\mu}
  30. J μ e m J_{\mu}^{em}
  31. J μ 3 J_{\mu}^{3}
  32. J μ e m = f q f f ¯ γ μ f J_{\mu}^{em}=\sum_{f}q_{f}\overline{f}\gamma_{\mu}f
  33. J μ 3 = f I f 3 f ¯ γ μ 1 - γ 5 2 f J_{\mu}^{3}=\sum_{f}I^{3}_{f}\overline{f}\gamma_{\mu}\frac{1-\gamma^{5}}{2}f
  34. q f q_{f}
  35. I f 3 I_{f}^{3}
  36. C = - g 2 [ u ¯ i γ μ 1 - γ 5 2 M i j C K M d j + ν ¯ i γ μ 1 - γ 5 2 e i ] W μ + + h . c . \mathcal{L}_{C}=-\frac{g}{\sqrt{2}}\left[\overline{u}_{i}\gamma^{\mu}\frac{1-% \gamma^{5}}{2}M^{CKM}_{ij}d_{j}+\overline{\nu}_{i}\gamma^{\mu}\frac{1-\gamma^{% 5}}{2}e_{i}\right]W_{\mu}^{+}+h.c.
  37. H \mathcal{L}_{H}
  38. H = - g m H 2 4 m W H 3 - g 2 m H 2 32 m W 2 H 4 \mathcal{L}_{H}=-\frac{gm_{H}^{2}}{4m_{W}}H^{3}-\frac{g^{2}m_{H}^{2}}{32m_{W}^% {2}}H^{4}
  39. H V \mathcal{L}_{HV}
  40. H V = ( g m W H + g 2 4 H 2 ) ( W μ + W - μ + 1 2 cos 2 θ W Z μ Z μ ) \mathcal{L}_{HV}=\left(gm_{W}H+\frac{g^{2}}{4}H^{2}\right)\left(W_{\mu}^{+}W^{% -\mu}+\frac{1}{2\cos^{2}\theta_{W}}Z_{\mu}Z^{\mu}\right)
  41. W W V \mathcal{L}_{WWV}
  42. W W V = - i g [ ( W μ ν + W - μ - W + μ W μ ν - ) ( A ν sin θ W - Z ν cos θ W ) + W ν - W μ + ( A μ ν sin θ W - Z μ ν cos θ W ) ] \mathcal{L}_{WWV}=-ig[(W_{\mu\nu}^{+}W^{-\mu}-W^{+\mu}W_{\mu\nu}^{-})(A^{\nu}% \sin\theta_{W}-Z^{\nu}\cos\theta_{W})+W_{\nu}^{-}W_{\mu}^{+}(A^{\mu\nu}\sin% \theta_{W}-Z^{\mu\nu}\cos\theta_{W})]
  43. W W V V \mathcal{L}_{WWVV}
  44. W W V V = - g 2 4 { [ 2 W μ + W - μ + ( A μ sin θ W - Z μ cos θ W ) 2 ] 2 - [ W μ + W ν - + W ν + W μ - + ( A μ sin θ W - Z μ cos θ W ) ( A ν sin θ W - Z ν cos θ W ) ] 2 } \begin{aligned}\displaystyle\mathcal{L}_{WWVV}=-\frac{g^{2}}{4}\Big\{&% \displaystyle[2W_{\mu}^{+}W^{-\mu}+(A_{\mu}\sin\theta_{W}-Z_{\mu}\cos\theta_{W% })^{2}]^{2}\\ &\displaystyle-[W_{\mu}^{+}W_{\nu}^{-}+W_{\nu}^{+}W_{\mu}^{-}+(A_{\mu}\sin% \theta_{W}-Z_{\mu}\cos\theta_{W})(A_{\nu}\sin\theta_{W}-Z_{\nu}\cos\theta_{W})% ]^{2}\Big\}\end{aligned}
  45. Y \mathcal{L}_{Y}
  46. Y = - f g m f 2 m W f ¯ f H \mathcal{L}_{Y}=-\sum_{f}\frac{gm_{f}}{2m_{W}}\overline{f}fH
  47. 1 - γ 5 2 \frac{1-\gamma^{5}}{2}

Elementary_algebra.html

  1. 3 x 2 - 2 x y + c 3x^{2}-2xy+c
  2. x , y x,y
  3. a , b , c a,b,c
  4. x , y x,y
  5. z z
  6. 3 × x 2 3\times x^{2}
  7. 3 x 2 3x^{2}
  8. 2 × x × y 2\times x\times y
  9. 2 x y 2xy
  10. x 2 x^{2}
  11. x x
  12. 1 x 2 1x^{2}
  13. x 2 x^{2}
  14. 3 x 1 3x^{1}
  15. 3 x 3x
  16. x 0 x^{0}
  17. 1 1
  18. 0 0 0^{0}
  19. x 2 x^{2}
  20. x 2 x^{2}
  21. x 2 x^{2}
  22. 3 x 3x
  23. T = Y + 20 T=Y+20
  24. 60 × 5 = 300 60\times 5=300
  25. s = 60 × m s=60\times m
  26. π = c / d \pi=c/d
  27. ( a + b ) = ( b + a ) (a+b)=(b+a)
  28. x + x + x x+x+x
  29. 3 x 3x
  30. x × x × x x\times x\times x
  31. x 3 x^{3}
  32. 2 x 2 + 3 a b - x 2 + a b 2x^{2}+3ab-x^{2}+ab
  33. x 2 + 4 a b x^{2}+4ab
  34. x 2 x^{2}
  35. a b ab
  36. x ( 2 x + 3 ) x(2x+3)
  37. ( x × 2 x ) + ( x × 3 ) (x\times 2x)+(x\times 3)
  38. 2 x 2 + 3 x 2x^{2}+3x
  39. 6 x 5 + 3 x 2 6x^{5}+3x^{2}
  40. 3 x 2 3x^{2}
  41. 3 x 2 ( 2 x 3 + 1 ) 3x^{2}(2x^{3}+1)
  42. = =
  43. c 2 = a 2 + b 2 c^{2}=a^{2}+b^{2}
  44. c 2 c^{2}
  45. a a
  46. b b
  47. a + b = b + a a+b=b+a
  48. x 2 - 1 = 8 x^{2}-1=8
  49. x = 3 x=3
  50. x = - 3 x=-3
  51. a > b a>b
  52. > >
  53. a < b a<b
  54. < <
  55. b = b b=b
  56. a = b a=b
  57. b = a b=a
  58. a = b a=b
  59. b = c b=c
  60. a = c a=c
  61. a = b a=b
  62. c = d c=d
  63. a + c = b + d a+c=b+d
  64. a c = b d ac=bd
  65. a = b a=b
  66. a + c = b + c a+c=b+c
  67. f f
  68. a = b a=b
  69. f ( a ) = f ( b ) f(a)=f(b)
  70. < <
  71. Align g t ; &gt;
  72. a < b a<b
  73. b < c b<c
  74. a < c a<c
  75. a < b a<b
  76. c < d c<d
  77. a + c < b + d a+c<b+d
  78. a < b a<b
  79. c > 0 c>0
  80. a c < b c ac<bc
  81. a < b a<b
  82. c < 0 c<0
  83. b c < a c bc<ac
  84. < <
  85. Align g t ; &gt;
  86. a < b a<b
  87. b > a b>a
  88. a 2 := a * a a^{2}:=a*a
  89. a a
  90. 3 2 3^{2}
  91. 2 x + 4 = 12 2x+4=12
  92. x x
  93. 2 x + 4 = 12 2x+4=12
  94. 2 x + 4 - 4 = 12 - 4 2x+4-4=12-4
  95. 2 x = 8 2x=8
  96. 2 x 2 = 8 2 \frac{2x}{2}=\frac{8}{2}
  97. x = 4 x=4
  98. a x + b = c ax+b=c\,
  99. b b
  100. a a
  101. x = c - b a x=\frac{c-b}{a}
  102. y = x + 22 y=x+22
  103. y y
  104. x x
  105. y + 10 = 2 × ( x + 10 ) y+10=2\times(x+10)
  106. y = 2 × ( x + 10 ) - 10 y=2\times(x+10)-10
  107. y = 2 x + 20 - 10 y=2x+20-10
  108. y = 2 x + 10 y=2x+10
  109. y = 2 x + 10 y=2x+10
  110. y = x + 22 y=x+22
  111. y y
  112. ( y - y ) = ( 2 x - x ) + 10 - 22 (y-y)=(2x-x)+10-22
  113. 0 = x - 12 0=x-12
  114. 12 = x 12=x
  115. x = 12 x=12
  116. x 2 x^{2}
  117. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  118. a a
  119. a x 2 ax^{2}
  120. a 0 a\neq 0
  121. a a
  122. x 2 + p x + q = 0 x^{2}+px+q=0\,
  123. p = b / a p=b/a
  124. q = c / a q=c/a
  125. x = - b ± b 2 - 4 a c 2 a , x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},
  126. x = - b + b 2 - 4 a c 2 a and x = - b - b 2 - 4 a c 2 a x=\frac{-b+\sqrt{b^{2}-4ac}}{2a}\quad\,\text{and}\quad x=\frac{-b-\sqrt{b^{2}-% 4ac}}{2a}
  127. x 2 + 3 x - 10 = 0 , x^{2}+3x-10=0,\,
  128. ( x + 5 ) ( x - 2 ) = 0. (x+5)(x-2)=0.\,
  129. x = 2 x=2
  130. x = - 5 x=-5
  131. x 2 + 1 = 0 x^{2}+1=0\,
  132. ( x + 1 ) 2 = 0. (x+1)^{2}=0.\,
  133. [ x - ( - 1 ) ] [ x - ( - 1 ) ] = 0. [x-(-1)][x-(-1)]=0.
  134. x 2 + x + 1 = 0 x^{2}+x+1=0
  135. x = - 1 + - 3 2 and x = - 1 - - 3 2 . x=\frac{-1+\sqrt{-3}}{2}\quad\quad\,\text{and}\quad\quad x=\frac{-1-\sqrt{-3}}% {2}.
  136. - 3 \sqrt{-3}
  137. a x = b a^{x}=b
  138. a > 0 a>0
  139. X = log a b = ln b ln a X=\log_{a}b=\frac{\ln b}{\ln a}
  140. b > 0 b>0
  141. 3 2 x - 1 + 1 = 10 3\cdot 2^{x-1}+1=10
  142. 2 x - 1 = 3 2^{x-1}=3\,
  143. x - 1 = log 2 3 x-1=\log_{2}3\,
  144. x = log 2 3 + 1. x=\log_{2}3+1.\,
  145. l o g a ( x ) = b log_{a}(x)=b
  146. a > 0 a>0
  147. X = a b . X=a^{b}.\,
  148. 4 log 5 ( x - 3 ) - 2 = 6 4\log_{5}(x-3)-2=6\,
  149. log 5 ( x - 3 ) = 2 \log_{5}(x-3)=2\,
  150. x - 3 = 5 2 = 25 x-3=5^{2}=25\,
  151. x = 28. x=28.\,
  152. x \sqrt{x}
  153. x 3 \sqrt[3]{x}
  154. x n \sqrt[n]{x}
  155. x n \sqrt[n]{x}
  156. x 1 n x^{\frac{1}{n}}
  157. x 3 2 \sqrt[2]{x^{3}}
  158. x x
  159. x 3 2 x^{\frac{3}{2}}
  160. x m n = a \sqrt[n]{x^{m}}=a
  161. x m n = a x^{\frac{m}{n}}=a
  162. m m
  163. n n
  164. m m
  165. m m
  166. a 0 a\geq 0
  167. m m
  168. n n
  169. a < 0 a<0
  170. m m
  171. n n
  172. a < 0 a<0
  173. x = a n m x=\sqrt[m]{a^{n}}
  174. x = ( a m ) n x=\left(\sqrt[m]{a}\right)^{n}
  175. x = ± a n m x=\pm\sqrt[m]{a^{n}}
  176. x = ± ( a m ) n x=\pm\left(\sqrt[m]{a}\right)^{n}
  177. x = ± a n m x=\pm\sqrt[m]{a^{n}}
  178. ( x + 5 ) 2 / 3 = 4 , (x+5)^{2/3}=4,\,
  179. x + 5 \displaystyle x+5
  180. x - y = - 1 x-y=-1
  181. 3 x + y = 9 3x+y=9
  182. { 4 x + 2 y = 14 2 x - y = 1. \begin{cases}4x+2y&=14\\ 2x-y&=1.\end{cases}\,
  183. 4 x + 2 y = 14 4x+2y=14\,
  184. 4 x - 2 y = 2. 4x-2y=2.\,
  185. 8 x = 16 8x=16\,
  186. x = 2. x=2.\,
  187. x = 2 x=2
  188. y = 3 y=3
  189. x x
  190. { x = 2 y = 3. \begin{cases}x=2\\ y=3.\end{cases}\,
  191. y y
  192. x x
  193. { 4 x + 2 y = 14 2 x - y = 1. \begin{cases}4x+2y&=14\\ 2x-y&=1.\end{cases}\,
  194. y y
  195. 2 x - y = 1 2x-y=1\,
  196. 2 x 2x
  197. 2 x - 2 x - y \displaystyle 2x-2x-y
  198. y = 2 x - 1. y=2x-1.\,
  199. y y
  200. 4 x + 2 ( 2 x - 1 ) \displaystyle 4x+2(2x-1)
  201. 8 x - 2 + 2 \displaystyle 8x-2+2
  202. x = 2 x=2\,
  203. { x = 2 y = 3. \begin{cases}x=2\\ y=3.\end{cases}\,
  204. y y
  205. x x
  206. { x + y = 1 0 x + 0 y = 2 . \begin{cases}\begin{aligned}\displaystyle x+y&\displaystyle=1\\ \displaystyle 0x+0y&\displaystyle=2\,.\end{aligned}\end{cases}
  207. { 4 x + 2 y = 12 - 2 x - y = - 4 . \begin{cases}\begin{aligned}\displaystyle 4x+2y&\displaystyle=12\\ \displaystyle-2x-y&\displaystyle=-4\,.\end{aligned}\end{cases}
  208. 0 x + 0 y = 4 , 0x+0y=4\,,
  209. x x
  210. y y
  211. { 4 x + 2 y = 12 - 2 x - y = - 6 \begin{cases}\begin{aligned}\displaystyle 4x+2y&\displaystyle=12\\ \displaystyle-2x-y&\displaystyle=-6\end{aligned}\end{cases}\,
  212. y y
  213. y = - 2 x + 6 y=-2x+6\,
  214. 4 x + 2 ( - 2 x + 6 ) = 12 \displaystyle 4x+2(-2x+6)=12
  215. x x
  216. x x
  217. x x
  218. y = - 2 x + 6 y=-2x+6
  219. { x + 2 y = 10 y - z = 2. \begin{cases}\begin{aligned}\displaystyle x+2y&\displaystyle=10\\ \displaystyle y-z&\displaystyle=2.\end{aligned}\end{cases}

Elementary_function.html

  1. e tan ( x ) 1 + x 2 sin ( 1 + ln 2 x ) \dfrac{e^{\tan(x)}}{1+x^{2}}\sin\left(\sqrt{1+\ln^{2}x}\,\right)
  2. - i ln ( x + i 1 - x 2 ) -i\ln(x+i\sqrt{1-x^{2}})
  3. arccos ( x ) \arccos(x)
  4. arccos ( x ) \arccos(x)
  5. erf ( x ) = 2 π 0 x e - t 2 d t , \mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\,dt,
  6. ( u + v ) = u + v \partial(u+v)=\partial u+\partial v
  7. ( u v ) = u v + u v . \partial(u\cdot v)=\partial u\cdot v+u\cdot\partial v\,.

ElGamal_encryption.html

  1. G G
  2. G G
  3. G G\,
  4. q q\,
  5. g g
  6. x x
  7. { 1 , , q - 1 } \{1,\ldots,q-1\}
  8. h := g x h:=g^{x}
  9. h h\,
  10. G , q , g G,q,g\,
  11. x x
  12. m m
  13. ( G , q , g , h ) (G,q,g,h)
  14. y y
  15. { 1 , , q - 1 } \{1,\ldots,q-1\}
  16. c 1 := g y c_{1}:=g^{y}
  17. s := h y s:=h^{y}
  18. m m
  19. m m^{\prime}
  20. G G
  21. c 2 := m s c_{2}:=m^{\prime}\cdot s
  22. ( c 1 , c 2 ) = ( g y , m h y ) = ( g y , m ( g x ) y ) (c_{1},c_{2})=(g^{y},m^{\prime}\cdot h^{y})=(g^{y},m^{\prime}\cdot(g^{x})^{y})
  23. h y h^{y}
  24. m m^{\prime}
  25. y y
  26. y y
  27. ( c 1 , c 2 ) (c_{1},c_{2})
  28. x x
  29. s := c 1 x s:=c_{1}{}^{x}
  30. m := c 2 s - 1 m^{\prime}:=c_{2}\cdot s^{-1}
  31. m m
  32. s - 1 s^{-1}
  33. s s
  34. G G
  35. G G
  36. c 2 s - 1 = m h y ( g x y ) - 1 = m g x y g - x y = m . c_{2}\cdot s^{-1}=m^{\prime}\cdot h^{y}\cdot(g^{xy})^{-1}=m^{\prime}\cdot g^{% xy}\cdot g^{-xy}=m^{\prime}.
  37. G G
  38. G G
  39. G G
  40. ( c 1 , c 2 ) (c_{1},c_{2})
  41. m m
  42. ( c 1 , 2 c 2 ) (c_{1},2c_{2})
  43. 2 m 2m
  44. G G
  45. s s\,
  46. ( c 1 , c 2 ) (c_{1},c_{2})\,
  47. x x\,
  48. s = c 1 < m t p l > q - x = g ( q - x ) y s^{\prime}=c_{1}^{<}mtpl>{{q-x}}=g^{(q-x)y}
  49. s s^{\prime}\,
  50. s s\,
  51. s s = g x y g ( q - x ) y = ( g q ) y = e y = e , s\cdot s^{\prime}=g^{xy}\cdot g^{(q-x)y}=(g^{q})^{y}=e^{y}=e,
  52. e e\,
  53. G G\,
  54. m = c 2 s m^{\prime}=c_{2}\cdot s^{\prime}
  55. m m\,
  56. c 2 s = m s s = m e = m . c_{2}\cdot s^{\prime}=m^{\prime}\cdot s\cdot s^{\prime}=m^{\prime}\cdot e=m^{% \prime}.

Elias_delta_coding.html

  1. x x
  2. log 2 ( x ) + 2 log 2 ( log 2 ( x ) + 1 ) + 1 \lfloor\log_{2}(x)\rfloor+2\lfloor\log_{2}(\lfloor\log_{2}(x)\rfloor+1)\rfloor+1
  3. γ \gamma^{\prime}
  4. γ \gamma

Elias_gamma_coding.html

  1. x x
  2. 2 log 2 ( x ) + 1 2\lfloor\log_{2}(x)\rfloor+1
  3. { x 2 x + 1 when x 0 x - 2 x when x < 0 \begin{cases}x\mapsto 2x+1&\mathrm{when~{}}x\geq 0\\ x\mapsto-2x&\mathrm{when~{}}x<0\\ \end{cases}

Ellipse.html

  1. ( x a ) 2 + ( y b ) 2 = 1 \displaystyle{\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}}=1
  2. x = a cos θ . {x}={a}\cos\theta.
  3. y = b sin θ . {y}={b}\sin\theta.
  4. x 2 = a 2 cos 2 θ . {x}^{2}={a}^{2}\cos^{2}\theta.
  5. y 2 = b 2 sin 2 θ . {y}^{2}={b}^{2}\sin^{2}\theta.
  6. x 2 a 2 = cos 2 θ . \frac{{x}^{2}}{{a}^{2}}=\cos^{2}\theta.
  7. y 2 b 2 = sin 2 θ . \frac{{y}^{2}}{{b}^{2}}=\sin^{2}\theta.
  8. x 2 a 2 + y 2 b 2 = cos 2 θ + sin 2 θ . \frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=\cos^{2}\theta+\sin^{2}\theta.
  9. x 2 a 2 + y 2 b 2 = 1. \frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1.
  10. f = a 2 - b 2 . f=\sqrt{a^{2}-b^{2}}.
  11. P F 1 + P F 2 = ( x + f ) 2 + y 2 + ( x - f ) 2 + y 2 = 2 a PF_{1}+PF_{2}=\sqrt{(x+f)^{2}+y^{2}}+\sqrt{(x-f)^{2}+y^{2}}=2a
  12. ε \varepsilon
  13. e = ε = a 2 - b 2 a 2 = 1 - ( b a ) 2 = f / a e=\varepsilon=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{1-\left(\frac{b}{a}\right% )^{2}}=f/a
  14. g = 1 - b a = 1 - 1 - e 2 , g=1-\frac{b}{a}=1-\sqrt{1-e^{2}},
  15. e = g ( 2 - g ) . e=\sqrt{g(2-g)}.
  16. A ellipse A\text{ellipse}
  17. A ellipse = π a b A\text{ellipse}=\pi ab
  18. 1 / 2 {1}/{2}
  19. A x 2 + B x y + C y 2 = 1 Ax^{2}+Bxy+Cy^{2}=1
  20. 2 π 4 A C - B 2 \frac{2\pi}{\sqrt{4AC-B^{2}}}
  21. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  22. y = ± a 2 b 2 - b 2 x 2 a 2 y=\pm\sqrt{\frac{a^{2}b^{2}-b^{2}x^{2}}{a^{2}}}
  23. A ellipse A\text{ellipse}
  24. A ellipse = - a a 2 b 1 - x 2 / a 2 d x = b a - a a 2 a 2 - x 2 d x . \begin{aligned}\displaystyle A\text{ellipse}&\displaystyle=\int_{-a}^{a}2b% \sqrt{1-x^{2}/a^{2}}\,dx\\ &\displaystyle=\frac{b}{a}\int_{-a}^{a}2\sqrt{a^{2}-x^{2}}\,dx.\end{aligned}
  25. a a
  26. π a 2 \pi a^{2}
  27. A ellipse = b a A circle = π a b . A\text{ellipse}=\frac{b}{a}A\text{circle}=\pi ab.
  28. T ( r , θ ) = ( r a cos θ , r b sin θ ) . {T}(r,\theta)=(ra\cos\theta,rb\sin\theta).
  29. θ \theta
  30. 0 r 1 0\leqslant r\leqslant 1
  31. 0 θ 2 π 0\leqslant\theta\leqslant 2\pi
  32. d r d θ dr\,d\theta
  33. d A ellipse = det ( T r T θ ) d r d θ = det ( a cos θ - r a sin θ b sin θ r b cos θ ) d r d θ = a b r d r d θ . \begin{aligned}\displaystyle dA\text{ellipse}&\displaystyle=\det\begin{pmatrix% }\frac{\partial{T}}{\partial r}&\frac{\partial{T}}{\partial\theta}\\ \end{pmatrix}\,dr\,d\theta\\ &\displaystyle=\det\begin{pmatrix}a\cos\theta&-ra\sin\theta\\ b\sin\theta&rb\cos\theta\end{pmatrix}\,dr\,d\theta\\ &\displaystyle=abr\,dr\,d\theta.\end{aligned}
  34. A ellipse = ellipse d A ellipse = ellipse a b r d r d θ = a b 0 2 π 0 1 r d r d θ = a b π . A\text{ellipse}=\iint\text{ellipse}dA\text{ellipse}=\iint\text{ellipse}abr\,dr% \,d\theta=ab\int_{0}^{2\pi}\int_{0}^{1}r\,dr\,d\theta=ab\pi.
  35. C C
  36. C = 4 a E ( e ) C=4aE(e)
  37. E E
  38. C = 2 π a [ 1 - ( 1 2 ) 2 e 2 - ( 1 3 2 4 ) 2 e 4 3 - ( 1 3 5 2 4 6 ) 2 e 6 5 - ] C=2\pi a\left[{1-\left({1\over 2}\right)^{2}e^{2}-\left({1\cdot 3\over 2\cdot 4% }\right)^{2}{e^{4}\over 3}-\left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)% ^{2}{e^{6}\over 5}-\cdots}\right]
  39. C = 2 π a [ 1 - n = 1 ( ( 2 n - 1 ) ! ! 2 n n ! ) 2 e 2 n 2 n - 1 ] , C=2\pi a\left[1-\sum_{n=1}^{\infty}\left(\frac{(2n-1)!!}{2^{n}n!}\right)^{2}% \frac{e^{2n}}{2n-1}\right],
  40. n ! ! n!!
  41. h = ( a - b ) 2 / ( a + b ) 2 h=(a-b)^{2}/(a+b)^{2}
  42. C = π ( a + b ) [ 1 + n = 1 ( ( 2 n - 1 ) ! ! 2 n n ! ) 2 h n ( 2 n - 1 ) 2 ] . C=\pi(a+b)\left[1+\sum_{n=1}^{\infty}\left(\frac{(2n-1)!!}{2^{n}n!}\right)^{2}% \frac{h^{n}}{(2n-1)^{2}}\right].
  43. C π [ 3 ( a + b ) - ( 3 a + b ) ( a + 3 b ) ] = π [ 3 ( a + b ) - 10 a b + 3 ( a 2 + b 2 ) ] C\approx\pi\left[3(a+b)-\sqrt{(3a+b)(a+3b)}\right]=\pi\left[3(a+b)-\sqrt{10ab+% 3(a^{2}+b^{2})}\right]
  44. C π ( a + b ) ( 1 + 3 h 10 + 4 - 3 h ) . C\approx\pi\left(a+b\right)\left(1+\frac{3h}{10+\sqrt{4-3h}}\right).
  45. h 3 h^{3}
  46. h 5 h^{5}
  47. x 2 a 2 + y 2 b 2 = 1 \tfrac{x^{2}}{a^{2}}+\tfrac{y^{2}}{b^{2}}=1
  48. C 2 π a , C\leq 2\pi a,
  49. π ( a + b ) C 4 ( a + b ) , \pi(a+b)\leq C\leq 4(a+b),
  50. 4 a 2 + b 2 C 2 π a 2 + b 2 , 4\sqrt{a^{2}+b^{2}}\leq C\leq\sqrt{2}\pi\sqrt{a^{2}+b^{2}},
  51. C 4 ( a - b ) + 2 π . C\leq 4(a-b)+2\pi.
  52. 2 π a 2\pi a
  53. 4 a 2 + b 2 4\sqrt{a^{2}+b^{2}}
  54. 2 b < s u p > 2 2\frac{b}{<sup>2}
  55. 1 a 2 b 2 ( x 2 a 4 + y 2 b 4 ) - 3 2 \frac{1}{a^{2}b^{2}}\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}\right)^{-% \frac{3}{2}}
  56. F 1 P F 2 \angle F_{1}PF_{2}
  57. ( X , Y ) (X,Y)
  58. A X 2 + B X Y + C Y 2 + D X + E Y + F = 0 ~{}AX^{2}+BXY+CY^{2}+DX+EY+F=0
  59. B 2 - 4 A C < 0. B^{2}-4AC<0.
  60. | A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F | \begin{vmatrix}A&B/2&D/2\\ B/2&C&E/2\\ D/2&E/2&F\end{vmatrix}
  61. Δ = ( A C - B 2 4 ) F + B E D 4 - C D 2 4 - A E 2 4 \Delta=\left(AC-\frac{B^{2}}{4}\right)F+\frac{BED}{4}-\frac{CD^{2}}{4}-\frac{% AE^{2}}{4}
  62. a a
  63. b b
  64. ( x c , y c ) (x_{c},y_{c})
  65. Θ \Theta
  66. A \displaystyle A
  67. x c a n 2 a 2 + y c a n 2 b 2 = 1 \frac{x_{can}^{2}}{a^{2}}+\frac{y_{can}^{2}}{b^{2}}=1
  68. x c a n = ( x - x c ) cos Θ + ( y - y c ) sin Θ x_{can}=(x-x_{c})\cos\Theta+(y-y_{c})\sin\Theta
  69. y c a n = - ( x - x c ) sin Θ + ( y - y c ) cos Θ y_{can}=-(x-x_{c})\sin\Theta+(y-y_{c})\cos\Theta
  70. a > b a>b
  71. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  72. ( x , y ) (x,y)
  73. ( X c , Y c ) (X_{c},Y_{c})
  74. x x
  75. ( X a , Y a ) (X_{a},Y_{a})
  76. y y
  77. ( - Y a , X a ) (-Y_{a},X_{a})
  78. x = X a ( X - X c ) + Y a ( Y - Y c ) x=X_{a}(X-X_{c})+Y_{a}(Y-Y_{c})
  79. y = - Y a ( X - X c ) + X a ( Y - Y c ) y=-Y_{a}(X-X_{c})+X_{a}(Y-Y_{c})
  80. ( 0 , 0 ) (0,0)
  81. ( - e a , 0 ) (-ea,0)
  82. ( + e a , 0 ) (+ea,0)
  83. ( X c , Y c ) (X_{c},Y_{c})
  84. ( x - X c ) 2 a 2 + ( y - Y c ) 2 b 2 = 1 \frac{(x-X_{c})^{2}}{a^{2}}+\frac{(y-Y_{c})^{2}}{b^{2}}=1
  85. \reals 2 \reals^{2}
  86. X 2 + Y 2 = 1 X^{2}+Y^{2}=1\,
  87. Y = ± b 1 - ( X / a ) 2 = ± ( a 2 - X 2 ) ( 1 - e 2 ) Y=\pm b\sqrt{1-(X/a)^{2}}=\pm\sqrt{(a^{2}-X^{2})(1-e^{2})}
  88. ( X , Y ) (X,Y)
  89. a + e X a+eX
  90. a - e X a-eX
  91. ( X ( t ) , Y ( t ) ) (X(t),Y(t))
  92. X ( t ) = X c + a cos t cos φ - b sin t sin φ X(t)=X_{c}+a\,\cos t\,\cos\varphi-b\,\sin t\,\sin\varphi
  93. Y ( t ) = Y c + a cos t sin φ + b sin t cos φ Y(t)=Y_{c}+a\,\cos t\,\sin\varphi+b\,\sin t\,\cos\varphi
  94. ( X c , Y c ) (X_{c},Y_{c})
  95. φ \varphi
  96. X X
  97. X ( t ) = a cos t X(t)=a\,\cos t
  98. Y ( t ) = b sin t Y(t)=b\,\sin t
  99. ( X ( t ) , Y ( t ) ) (X(t),Y(t))
  100. ϕ \phi
  101. θ \theta
  102. - cot ϕ = a b tan t = tan θ ( 1 - g ) 2 = tan θ 1 - e 2 , -\cot\phi=\frac{a}{b}\tan t=\frac{\tan\theta}{(1-g)^{2}}=\frac{\tan\theta}{1-e% ^{2}},
  103. - tan t = b a cot ϕ = ( 1 - e 2 ) cot ϕ = ( 1 - g ) cot ϕ = - tan θ ( 1 - e 2 ) = - a b tan θ . -\tan t=\frac{b}{a}\cot\phi=\sqrt{(1-e^{2})}\cot\phi=(1-g)\cot\phi=\frac{-\tan% \theta}{\sqrt{(1-e^{2})}}=-\frac{a}{b}\tan\theta.
  104. θ \theta
  105. r ( θ ) = a b ( b cos θ ) 2 + ( a sin θ ) 2 r(\theta)=\frac{ab}{\sqrt{(b\cos\theta)^{2}+(a\sin\theta)^{2}}}
  106. θ = 0 \theta=0
  107. r ( θ ) = a ( 1 - e 2 ) 1 ± e cos θ r(\theta)=\frac{a(1-e^{2})}{1\pm e\cos\theta}
  108. θ = 0 \theta=0
  109. ϕ \phi
  110. r = a ( 1 - e 2 ) 1 - e cos ( θ - ϕ ) . r=\frac{a(1-e^{2})}{1-e\cos(\theta-\phi)}.
  111. θ \theta
  112. a ( 1 - e 2 ) a(1-e^{2})
  113. l l
  114. r ( θ ) = P ( θ ) + Q ( θ ) R ( θ ) r(\theta)=\frac{P(\theta)+Q(\theta)}{R(\theta)}
  115. P ( θ ) = r 0 [ ( b 2 - a 2 ) cos ( θ + θ 0 - 2 φ ) + ( a 2 + b 2 ) cos ( θ - θ 0 ) ] P(\theta)=r_{0}\left[\left(b^{2}-a^{2}\right)\cos\left(\theta+\theta_{0}-2% \varphi\right)+\left(a^{2}+b^{2}\right)\cos\left(\theta-\theta_{0}\right)\right]
  116. Q ( θ ) = 2 a b R ( θ ) - 2 r 0 2 sin 2 ( θ - θ 0 ) Q(\theta)=\sqrt{2}ab\sqrt{R(\theta)-2r_{0}^{2}\sin^{2}\left(\theta-\theta_{0}% \right)}
  117. R ( θ ) = ( b 2 - a 2 ) cos ( 2 θ - 2 φ ) + a 2 + b 2 R(\theta)=\left(b^{2}-a^{2}\right)\cos(2\theta-2\varphi)+a^{2}+b^{2}
  118. α \alpha
  119. α = sin - 1 ( e ) = cos - 1 ( b a ) = 2 tan - 1 ( a - b a + b ) ; \alpha=\sin^{-1}(e)=\cos^{-1}\left(\frac{b}{a}\right)=2\tan^{-1}\left(\sqrt{% \frac{a-b}{a+b}}\right);\,\!
  120. e e
  121. e \displaystyle e
  122. r a r_{a}
  123. r p r_{p}
  124. a a
  125. r a r_{a}
  126. r p r_{p}
  127. a a
  128. b b
  129. l l
  130. a = r a + r p 2 b = r a r p l = 2 1 r a + 1 r p = 2 r a r p r a + r p \begin{aligned}\displaystyle a&\displaystyle=\frac{r_{a}+r_{p}}{2}\\ \displaystyle b&\displaystyle=\sqrt{r_{a}\cdot r_{p}}\\ \displaystyle l&\displaystyle=\frac{2}{\frac{1}{r_{a}}+\frac{1}{r_{p}}}=\frac{% 2r_{a}r_{p}}{r_{a}+r_{p}}\end{aligned}
  131. θ \theta

Ellipsis.html

  1. 1 , 2 , 3 , , 100 . 1,2,3,\ldots,100\,.
  2. 1 + 2 + 3 + + 100 1+2+3+\cdots+100\,
  3. 1 + 2 + 3 + + 100 = n = 1 100 n 1+2+3+\cdots+100\ =\sum_{n=1}^{100}n
  4. 1 × 2 × 3 × × 100 = n = 1 100 n = 100 ! 1\times 2\times 3\times\cdots\times 100\ =\prod_{n=1}^{100}n=100!
  5. π = 3.14159265 \pi=3.14159265\ldots
  6. 1 + 4 + 9 + + n 2 + + 400 . 1+4+9+\cdots+n^{2}+\cdots+400\,.
  7. { ± π 2 , ± 3 π 2 , ± 5 π 2 , } . \left\{\pm\frac{\pi}{2},\pm\frac{3\pi}{2},\pm\frac{5\pi}{2},\ldots\right\}\,.
  8. I n = [ 1 0 0 0 1 0 0 0 1 ] . I_{n}=\begin{bmatrix}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{bmatrix}.
  9. \ldots\,\!
  10. \cdots\,\!
  11. \ddots\,\!
  12. \vdots\,\!

Elliptic_curve.html

  1. y 2 = x 3 + a x + b y^{2}=x^{3}+ax+b
  2. y 2 = x 3 + a x + b y^{2}=x^{3}+ax+b
  3. Δ = - 16 ( 4 a 3 + 27 b 2 ) \Delta=-16(4a^{3}+27b^{2})
  4. s = y P - y Q x P - x Q s=\frac{y_{P}-y_{Q}}{x_{P}-x_{Q}}
  5. x R = s 2 - x P - x Q y R = y P + s ( x R - x P ) \begin{aligned}\displaystyle x_{R}&\displaystyle=s^{2}-x_{P}-x_{Q}\\ \displaystyle y_{R}&\displaystyle=y_{P}+s(x_{R}-x_{P})\end{aligned}
  6. s = 3 x P 2 - p 2 y P x R = s 2 - 2 x P y R = y P + s ( x R - x P ) \begin{aligned}\displaystyle s&\displaystyle=\frac{3{x_{P}}^{2}-p}{2y_{P}}\\ \displaystyle x_{R}&\displaystyle=s^{2}-2x_{P}\\ \displaystyle y_{R}&\displaystyle=y_{P}+s(x_{R}-x_{P})\end{aligned}
  7. ( z ) 2 = 4 ( z ) 3 - g 2 ( z ) - g 3 \wp^{\prime}(z)^{2}=4\wp(z)^{3}-g_{2}\wp(z)-g_{3}
  8. ( z ) \wp(z)
  9. ( z ) \wp^{\prime}(z)
  10. z [ 1 : ( z ) : ( z ) ] z\mapsto[1:\wp(z):\wp^{\prime}(z)]
  11. y 2 = x ( x - 1 ) ( x - λ ) y^{2}=x(x-1)(x-\lambda)
  12. g 2 = 4 1 3 3 ( λ 2 - λ + 1 ) g_{2}=\frac{4^{\frac{1}{3}}}{3}(\lambda^{2}-\lambda+1)
  13. g 3 = 1 27 ( λ + 1 ) ( 2 λ 2 - 5 λ + 2 ) g_{3}=\frac{1}{27}(\lambda+1)(2\lambda^{2}-5\lambda+2)
  14. Δ = g 2 3 - 27 g 3 2 = λ 2 ( λ - 1 ) 2 \Delta=g_{2}^{3}-27g_{3}^{2}=\lambda^{2}(\lambda-1)^{2}
  15. a n ω 1 + b n ω 2 \frac{a}{n}\omega_{1}+\frac{b}{n}\omega_{2}
  16. h ( P ) = l o g m a x ( p , q ) h(P)=logmax(p,q)
  17. p / q {p}/{q}
  18. 1 / 4 {1}/{4}
  19. 1 / 4 {1}/{4}
  20. y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}
  21. Z ( E ( 𝐅 p ) ) = exp ( card [ E ( 𝐅 p n ) ] T n n ) Z(E(\mathbf{F}_{p}))=\exp\left(\sum\mathrm{card}\left[E({\mathbf{F}}_{p^{n}})% \right]\frac{T^{n}}{n}\right)
  22. Z ( E ( 𝐅 p ) ) = 1 - a p T + p T 2 ( 1 - T ) ( 1 - p T ) Z(E(\mathbf{F}_{p}))=\frac{1-a_{p}T+pT^{2}}{(1-T)(1-pT)}
  23. L ( E ( 𝐐 ) , s ) = p ( 1 - a p p - s + ε ( p ) p 1 - 2 s ) - 1 L(E(\mathbf{Q}),s)=\prod_{p}\left(1-a_{p}p^{-s}+\varepsilon(p)p^{1-2s}\right)^% {-1}
  24. 2 x 2 + y 2 + 8 z 2 = n 2x^{2}+y^{2}+8z^{2}=n
  25. 2 x 2 + y 2 + 32 z 2 = n 2x^{2}+y^{2}+32z^{2}=n
  26. y 2 = x 3 - n 2 x y^{2}=x^{3}-n^{2}x
  27. y 2 = x 3 + a x + b y^{2}=x^{3}+ax+b
  28. L ( E ( 𝐐 ) , s ) = n > 0 a ( n ) n - s L(E(\mathbf{Q}),s)=\sum_{n>0}a(n)n^{-s}
  29. a ( n ) q n , q = exp ( 2 π i z ) \sum a(n)q^{n},\qquad q=\exp(2\pi iz)
  30. y 2 - y = x 3 - x y^{2}-y=x^{3}-x
  31. f ( z ) = q - 2 q 2 - 3 q 3 + 2 q 4 - 2 q 5 + 6 q 6 + , q = exp ( 2 π i z ) f(z)=q-2q^{2}-3q^{3}+2q^{4}-2q^{5}+6q^{6}+\cdots,\qquad q=\exp(2\pi iz)
  32. y 2 - y = x 3 - x y^{2}-y=x^{3}-x
  33. x ( z ) = q - 2 + 2 q - 1 + 5 + 9 q + 18 q 2 + 29 q 3 + 51 q 4 + y ( z ) = q - 3 + 3 q - 2 + 9 q - 1 + 21 + 46 q + 92 q 2 + 180 q 3 + \begin{aligned}\displaystyle x(z)&\displaystyle=q^{-2}+2q^{-1}+5+9q+18q^{2}+29% q^{3}+51q^{4}+\ldots\\ \displaystyle y(z)&\displaystyle=q^{-3}+3q^{-2}+9q^{-1}+21+46q+92q^{2}+180q^{3% }+\ldots\end{aligned}
  34. x ( a z + b c z + d ) = x ( z ) x\left(\frac{az+b}{cz+d}\right)=x(z)
  35. a p + b p = c p a^{p}+b^{p}=c^{p}
  36. y 2 = x ( x - a p ) ( x + b p ) y^{2}=x(x-a^{p})(x+b^{p})
  37. Δ = 1 256 ( a b c ) 2 p \Delta=\frac{1}{256}(abc)^{2p}
  38. max ( | x | , | y | ) < exp ( [ 10 6 H ] 10 6 ) \max(|x|,|y|)<\exp\left(\left[10^{6}H\right]^{{10}^{6}}\right)
  39. p < d 3 d 2 p<d^{3d^{2}}
  40. y 2 = x 3 - p x - q y^{2}=x^{3}-px-q
  41. y 2 = 4 x 3 + b 2 x 2 + 2 b 4 x + b 6 y^{2}=4x^{3}+b_{2}x^{2}+2b_{4}x+b_{6}
  42. y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}
  43. | card E ( K ) - ( q + 1 ) | 2 q |\mathrm{card}E(K)-(q+1)|\leq 2\sqrt{q}
  44. y 2 = x 3 - x y^{2}=x^{3}-x
  45. Z ( E ( K ) , T ) exp ( n = 1 card [ E ( K n ) ] T n n ) Z(E(K),T)\equiv\exp\left(\sum_{n=1}^{\infty}\mathrm{card}\left[E(K_{n})\right]% {T^{n}\over n}\right)
  46. Z ( E ( K ) , T ) = 1 - a T + q T 2 ( 1 - q T ) ( 1 - T ) Z(E(K),T)=\frac{1-aT+qT^{2}}{(1-qT)(1-T)}
  47. Z ( E ( K ) , 1 q T ) = Z ( E ( K ) , T ) ( 1 - a T + q T 2 ) = ( 1 - α T ) ( 1 - β T ) \begin{aligned}\displaystyle Z\left(E(K),\frac{1}{qT}\right)&\displaystyle=Z(E% (K),T)\\ \displaystyle\left(1-aT+qT^{2}\right)&\displaystyle=(1-\alpha T)(1-\beta T)% \end{aligned}
  48. q \scriptstyle\sqrt{q}
  49. 1 + 2 T 2 ( 1 - T ) ( 1 - 2 T ) \frac{1+2T^{2}}{(1-T)(1-2T)}
  50. | E ( 𝐅 2 r ) | = { 2 r + 1 r odd 2 r + 1 - 2 ( - 2 ) r 2 r even \left|E(\mathbf{F}_{2^{r}})\right|=\begin{cases}2^{r}+1&r\,\text{ odd}\\ 2^{r}+1-2(-2)^{\frac{r}{2}}&r\,\text{ even}\end{cases}
  51. 2 q \scriptstyle 2\sqrt{q}

Elliptic_curve_cryptography.html

  1. y 2 = x 3 + a x + b , y^{2}=x^{3}+ax+b,\,
  2. ( p ) × (\mathbb{Z}_{p})^{\times}
  3. n G = nG=\infty
  4. E ( 𝔽 p ) E(\mathbb{F}_{p})
  5. h = 1 n | E ( 𝔽 p ) | h=\frac{1}{n}|E(\mathbb{F}_{p})|
  6. h 4 h\leq 4
  7. h = 1 h=1
  8. ( p , a , b , G , n , h ) (p,a,b,G,n,h)
  9. ( m , f , a , b , G , n , h ) (m,f,a,b,G,n,h)
  10. 𝔽 2 m \mathbb{F}_{2^{m}}
  11. p B - 1 p^{B}-1
  12. 2 2
  13. 𝔽 p \mathbb{F}_{p}
  14. 𝔽 p B \mathbb{F}_{p^{B}}
  15. E ( 𝔽 q ) E(\mathbb{F}_{q})
  16. | E ( 𝔽 q ) | = q |E(\mathbb{F}_{q})|=q
  17. 𝔽 q \mathbb{F}_{q}
  18. O ( n ) O(\sqrt{n})
  19. 𝔽 q \mathbb{F}_{q}
  20. q 2 256 q\approx 2^{256}
  21. 𝔽 q \mathbb{F}_{q}
  22. x 𝔽 q x\in\mathbb{F}_{q}
  23. y 𝔽 q y\in\mathbb{F}_{q}
  24. x y = 1 xy=1
  25. ( X , Y , Z ) (X,Y,Z)
  26. x = X Z x=\frac{X}{Z}
  27. y = Y Z y=\frac{Y}{Z}
  28. ( X , Y , Z ) (X,Y,Z)
  29. x = X Z 2 x=\frac{X}{Z^{2}}
  30. y = Y Z 3 y=\frac{Y}{Z^{3}}
  31. x = X Z x=\frac{X}{Z}
  32. y = Y Z 2 y=\frac{Y}{Z^{2}}
  33. ( X , Y , Z , a Z 4 ) (X,Y,Z,aZ^{4})
  34. ( X , Y , Z , Z 2 , Z 3 ) (X,Y,Z,Z^{2},Z^{3})
  35. p 2 d p\approx 2^{d}
  36. p = 2 521 - 1 p=2^{521}-1
  37. p = 2 256 - 2 32 - 2 9 - 2 8 - 2 7 - 2 6 - 2 4 - 1. p=2^{256}-2^{32}-2^{9}-2^{8}-2^{7}-2^{6}-2^{4}-1.
  38. 𝔽 p \mathbb{F}_{p}
  39. 𝔽 p \mathbb{F}_{p}
  40. 𝔽 2 m \mathbb{F}_{2^{m}}
  41. P = Q P=Q
  42. P Q P\neq Q

Elliptic_function.html

  1. f f
  2. \mathbb{C}
  3. ω 1 \omega_{1}
  4. ω 2 \omega_{2}
  5. ω 1 ω 2 \frac{\omega_{1}}{\omega_{2}}\notin\mathbb{R}
  6. f ( z ) = f ( z + ω 1 ) f(z)=f(z+\omega_{1})
  7. f ( z ) = f ( z + ω 2 ) f(z)=f(z+\omega_{2})
  8. z z\in\mathbb{C}
  9. Λ = { m ω 1 + n ω 2 m , n } \Lambda=\left\{m\omega_{1}+n\omega_{2}\mid m,n\in\mathbb{Z}\right\}
  10. f ( z ) = f ( z + ω ) f(z)=f(z+\omega)
  11. ω Λ \omega\in\Lambda
  12. ( z ) \wp\left(z\right)
  13. Λ \Lambda
  14. ( z ) = 1 z 2 + ω Λ { 0 } ( 1 ( z - ω ) 2 - 1 ω 2 ) \wp\left(z\right)=\frac{1}{z^{2}}+\sum_{\omega\in\Lambda\smallsetminus\left\{0% \right\}}\left(\frac{1}{\left(z-\omega\right)^{2}}-\frac{1}{\omega^{2}}\right)
  15. z z + ω z\mapsto z+\omega
  16. ω Λ \omega\in\Lambda
  17. - 1 ω 2 -\frac{1}{\omega^{2}}
  18. | z | R \left|z\right|\leq R
  19. | ω | > 2 R \left|\omega\right|>2R
  20. | 1 ( z - ω ) 2 - 1 ω 2 | = | 2 ω z - z 2 ω 2 ( ω - z ) 2 | = | z ( 2 - z ω ) ω 3 ( 1 - z ω ) 2 | 10 R | ω | 3 \left|\frac{1}{\left(z-\omega\right)^{2}}-\frac{1}{\omega^{2}}\right|=\left|% \frac{2\omega z-z^{2}}{\omega^{2}\left(\omega-z\right)^{2}}\right|=\left|\frac% {z\left(2-\frac{z}{\omega}\right)}{\omega^{3}\left(1-\frac{z}{\omega}\right)^{% 2}}\right|\leq\frac{10R}{\left|\omega\right|^{3}}
  21. ω 0 1 | ω | 3 \sum_{\omega\neq 0}\frac{1}{\left|\omega\right|^{3}}
  22. Λ \Lambda
  23. \wp
  24. ( ( z ) ) 2 = 4 ( ( z ) ) 3 - g 2 ( z ) - g 3 \left(\wp^{\prime}\left(z\right)\right)^{2}=4\left(\wp\left(z\right)\right)^{3% }-g_{2}\wp\left(z\right)-g_{3}
  25. g 2 = 60 ω Λ { 0 } 1 ω 4 g_{2}=60\sum_{\omega\in\Lambda\smallsetminus\left\{0\right\}}\frac{1}{\omega^{% 4}}
  26. g 3 = 140 ω Λ { 0 } 1 ω 6 . g_{3}=140\sum_{\omega\in\Lambda\smallsetminus\left\{0\right\}}\frac{1}{\omega^% {6}}.
  27. ( , ) \left(\wp,\wp^{\prime}\right)
  28. \wp
  29. Λ \Lambda
  30. Λ \Lambda
  31. ω 1 = 1 \omega_{1}=1
  32. ω 2 = τ \omega_{2}=\tau
  33. Im ( τ ) > 0 \operatorname{Im}\left(\tau\right)>0
  34. H = { z | I m ( z ) > 0 } H=\left\{z\in\mathbb{C}|Im\left(z\right)>0\right\}
  35. h : H h:H\rightarrow\mathbb{C}
  36. h ( τ ) = h ( a τ + b c τ + d ) h\left(\tau\right)=h\left(\frac{a\tau+b}{c\tau+d}\right)
  37. ( a c b d ) SL 2 ( ) \left(\begin{matrix}a&c\\ b&d\end{matrix}\right)\in\,\text{SL}_{2}\left(\mathbb{Z}\right)
  38. j ( τ ) = 1728 g 2 3 g 2 3 - 27 g 3 2 j\left(\tau\right)=\frac{1728g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}
  39. g 2 g_{2}
  40. g 3 g_{3}

Elliptic_integral.html

  1. f f
  2. f ( x ) = c x R ( t , P ( t ) ) d t , f(x)=\int_{c}^{x}R\left(t,\sqrt{P(t)}\right)\,dt,
  3. R R
  4. P P
  5. c c
  6. P P
  7. R ( x , y ) R(x,y)
  8. y y
  9. α α
  10. k = s i n α k=sinα
  11. φ φ
  12. x x
  13. u u
  14. x = s i n φ = s n u x=sinφ=snu
  15. s n sn
  16. u u
  17. m m
  18. cos φ = cn u , and 1 - m sin 2 φ = dn u . \cos\varphi=\textrm{cn}\;u,\qquad\textrm{and}\qquad\sqrt{1-m\sin^{2}\varphi}=% \textrm{dn}\;u.
  19. Δ ( φ ) = d n u Δ(φ)=dnu
  20. F F
  21. F ( φ , k ) = F ( φ | k 2 ) = F ( sin φ ; k ) = 0 φ d θ 1 - k 2 sin 2 θ . F(\varphi,k)=F(\varphi\,|\,k^{2})=F(\sin\varphi;k)=\int_{0}^{\varphi}\frac{d% \theta}{\sqrt{1-k^{2}\sin^{2}\theta}}.
  22. t = sin θ , x = sin φ t=\sin\theta,x=\sin\varphi
  23. F ( x ; k ) = 0 x d t ( 1 - t 2 ) ( 1 - k 2 t 2 ) . F(x;k)=\int_{0}^{x}\frac{dt}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}}.
  24. F ( φ α ) = F ( φ , sin α ) = 0 φ d θ 1 - ( sin θ sin α ) 2 . F(\varphi\setminus\alpha)=F(\varphi,\sin\alpha)=\int_{0}^{\varphi}\frac{d% \theta}{\sqrt{1-(\sin\theta\sin\alpha)^{2}}}.
  25. F ( φ , sin α ) = F ( φ | sin 2 α ) = F ( φ α ) = F ( sin φ ; sin α ) . F(\varphi,\sin\alpha)=F(\varphi\,|\,\sin^{2}\alpha)=F(\varphi\setminus\alpha)=% F(\sin\varphi;\sin\alpha).
  26. x = sn ( u , k ) x=\operatorname{sn}(u,k)
  27. F ( x ; k ) = u ; F(x;k)=u;
  28. F ( k , φ ) F(k,φ)
  29. E ( k , φ ) E(k,φ)
  30. F ( φ , k ) F(φ,k)
  31. φ φ
  32. k k
  33. K ( k ) K(k)
  34. Π ( φ , n , k ) Π(φ,n,k)
  35. φ φ
  36. n n
  37. K ( m ) = 0 π / 2 d θ 1 - m sin 2 θ K(m)=\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-m\sin^{2}\theta}}
  38. E E
  39. E ( φ , k ) = E ( φ | k 2 ) = E ( sin φ ; k ) = 0 φ 1 - k 2 sin 2 θ d θ . E(\varphi,k)=E(\varphi\,|\,k^{2})=E(\sin\varphi;k)=\int_{0}^{\varphi}\sqrt{1-k% ^{2}\sin^{2}\theta}\,d\theta.
  40. t = sin θ and x = sin φ t=\sin\theta\;\,\text{and}\;x=\sin\varphi
  41. E ( x ; k ) = 0 x 1 - k 2 t 2 1 - t 2 d t . E(x;k)=\int_{0}^{x}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\,dt.
  42. E ( φ α ) = E ( φ , sin α ) = 0 φ 1 - ( sin θ sin α ) 2 d θ . E(\varphi\setminus\alpha)=E(\varphi,\sin\alpha)=\int_{0}^{\varphi}\sqrt{1-(% \sin\theta\sin\alpha)^{2}}\,d\theta.
  43. E ( sn ( u ; k ) ; k ) = 0 u dn 2 ( w ; k ) d w = u - k 2 0 u sn 2 ( w ; k ) d w = ( 1 - k 2 ) u + k 2 0 u cn 2 ( w ; k ) d w . E(\mathrm{sn}(u;k);k)=\int_{0}^{u}\mathrm{dn}^{2}(w;k)\,dw=u-k^{2}\int_{0}^{u}% \mathrm{sn}^{2}(w;k)\,dw=(1-k^{2})u+k^{2}\int_{0}^{u}\mathrm{cn}^{2}(w;k)\,dw.
  44. φ \varphi\,\!
  45. E E
  46. m ( φ ) = a ( E ( φ , e ) + d 2 d φ 2 E ( φ , e ) ) , m(\varphi)=a\left(E(\varphi,e)+\frac{d^{2}}{d\varphi^{2}}E(\varphi,e)\right),
  47. Π Π
  48. Π ( n ; φ α ) = 0 φ 1 1 - n sin 2 θ d θ 1 - ( sin θ sin α ) 2 \Pi(n;\varphi\setminus\alpha)=\int_{0}^{\varphi}\frac{1}{1-n\sin^{2}\theta}% \frac{d\theta}{\sqrt{1-(\sin\theta\sin\alpha)^{2}}}
  49. Π ( n ; φ | m ) = 0 sin φ 1 1 - n t 2 d t ( 1 - m t 2 ) ( 1 - t 2 ) . \Pi(n;\varphi\,|\,m)=\int_{0}^{\sin\varphi}\frac{1}{1-nt^{2}}\frac{dt}{\sqrt{(% 1-mt^{2})(1-t^{2})}}.
  50. n n
  51. Π ( 1 ; π 2 | m ) \Pi(1;\tfrac{\pi}{2}\,|\,m)\,\!
  52. m m
  53. Π ( n ; am ( u ; k ) ; k ) = 0 u d w 1 - n sn 2 ( w ; k ) . \Pi(n;\,\mathrm{am}(u;k);\,k)=\int_{0}^{u}\frac{dw}{1-n\,\mathrm{sn}^{2}(w;k)}.
  54. φ \varphi\,\!
  55. Π Π
  56. m ( φ ) = a ( 1 - e 2 ) Π ( e 2 ; φ | e 2 ) . m(\varphi)=a(1-e^{2})\Pi(e^{2};\varphi\,|\,e^{2}).
  57. φ = π / 2 φ=π/2
  58. x = 1 x=1
  59. K K
  60. K ( k ) = 0 π / 2 d θ 1 - k 2 sin 2 θ = 0 1 d t ( 1 - t 2 ) ( 1 - k 2 t 2 ) , K(k)=\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}=\int_{0}^{1}% \frac{dt}{\sqrt{(1-t^{2})(1-k^{2}t^{2})}},
  61. K ( k ) = F ( π 2 , k ) = F ( π 2 | k 2 ) = F ( 1 ; k ) . K(k)=F(\tfrac{\pi}{2},k)=F(\tfrac{\pi}{2}\,|\,k^{2})=F(1;k).
  62. K ( k ) = π 2 n = 0 [ ( 2 n ) ! 2 2 n ( n ! ) 2 ] 2 k 2 n = π 2 n = 0 [ P 2 n ( 0 ) ] 2 k 2 n , K(k)=\frac{\pi}{2}\sum_{n=0}^{\infty}\left[\frac{(2n)!}{2^{2n}(n!)^{2}}\right]% ^{2}k^{2n}=\frac{\pi}{2}\sum_{n=0}^{\infty}[P_{2n}(0)]^{2}k^{2n},
  63. K ( k ) = π 2 { 1 + ( 1 2 ) 2 k 2 + ( 1 3 2 4 ) 2 k 4 + + [ ( 2 n - 1 ) ! ! ( 2 n ) ! ! ] 2 k 2 n + } , K(k)=\frac{\pi}{2}\left\{1+\left(\frac{1}{2}\right)^{2}k^{2}+\left(\frac{1% \cdot 3}{2\cdot 4}\right)^{2}k^{4}+\cdots+\left[\frac{\left(2n-1\right)!!}{% \left(2n\right)!!}\right]^{2}k^{2n}+\cdots\right\},
  64. n ! ! n!!
  65. K ( k ) = π 2 F 1 2 ( 1 2 , 1 2 ; 1 ; k 2 ) . K(k)=\tfrac{\pi}{2}\,{}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right).
  66. K ( k ) = π / 2 agm ( 1 - k , 1 + k ) . K(k)=\frac{\pi/2}{\mathrm{agm}(1-k,1+k)}.
  67. K ( 0 ) = π 2 K ( 2 2 ) = 1 4 π Γ ( 1 4 ) 2 K ( 1 4 ( 6 - 2 ) ) = 3 1 4 2 7 3 π Γ ( 1 3 ) 3 K ( 1 4 ( 6 + 2 ) ) = 3 3 4 2 7 3 π Γ ( 1 3 ) 3 K ( 2 - 4 - 3 2 ) = ( 2 - 2 ) π 3 2 4 Γ ( 3 4 ) 2 \begin{aligned}\displaystyle K(0)&\displaystyle=\frac{\pi}{2}\\ \displaystyle K\left(\frac{\sqrt{2}}{2}\right)&\displaystyle=\frac{1}{4\sqrt{% \pi}}\;\Gamma\left(\frac{1}{4}\right)^{2}\\ \displaystyle K\left(\frac{1}{4}\left(\sqrt{6}-\sqrt{2}\right)\right)&% \displaystyle=\frac{3^{\frac{1}{4}}}{2^{\frac{7}{3}}\pi}\Gamma\left(\frac{1}{3% }\right)^{3}\\ \displaystyle K\left(\frac{1}{4}\left(\sqrt{6}+\sqrt{2}\right)\right)&% \displaystyle=\frac{3^{\frac{3}{4}}}{2^{\frac{7}{3}}\pi}\Gamma\left(\frac{1}{3% }\right)^{3}\\ \displaystyle K\left(2\,\sqrt{-4-3\,\sqrt{2}}\right)&\displaystyle=\frac{\left% (2-\sqrt{2}\right)\pi^{\frac{3}{2}}}{4\,\Gamma\left(\frac{3}{4}\right)^{2}}% \end{aligned}
  68. K ( k ) = π 2 θ 3 2 ( q ) , K(k)=\frac{\pi}{2}\theta_{3}^{2}(q),
  69. q ( k ) = exp ( - π K ( k ) K ( k ) ) . q(k)=\exp\left(-\pi\frac{K^{\prime}(k)}{K(k)}\right).
  70. K ( k 2 ) π 2 + π 8 k 2 1 - k 2 - π 16 k 4 1 - k 2 K(k^{2})\approx\frac{\pi}{2}+\frac{\pi}{8}\frac{k^{2}}{1-k^{2}}-\frac{\pi}{16}% \frac{k^{4}}{1-k^{2}}
  71. d d k [ k ( 1 - k 2 ) d K ( k ) d k ] = k K ( k ) \frac{\mathrm{d}}{\mathrm{d}k}\left[k(1-k^{2})\frac{\mathrm{d}K(k)}{\mathrm{d}% k}\right]=kK(k)
  72. K ( 1 - k 2 ) K(\sqrt{1-k^{2}})
  73. K ( 1 - k 2 ) = E ( k ) k ( 1 - k 2 ) - K ( k ) k K(\sqrt{1-k^{2}})=\frac{E(k)}{k(1-k^{2})}-\frac{K(k)}{k}
  74. E ( k ) E(k)
  75. E E
  76. E ( k ) = 0 π / 2 1 - k 2 sin 2 θ d θ = 0 1 1 - k 2 t 2 1 - t 2 d t , E(k)=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\theta}\ d\theta=\int_{0}^{1}\frac{% \sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}dt,
  77. E ( ϕ , k ) E(\phi,k)
  78. E ( k ) = E ( π 2 , k ) = E ( 1 ; k ) . E(k)=E\left(\frac{\pi}{2},k\right)=E(1;k).
  79. a a
  80. b b
  81. e = 1 - b 2 / a 2 e=\sqrt{1-b^{2}/a^{2}}
  82. E ( e ) E(e)
  83. c c
  84. a a
  85. c = 4 a E ( e ) . c=4aE(e).
  86. E ( k ) = π 2 n = 0 [ ( 2 n ) ! 2 2 n ( n ! ) 2 ] 2 k 2 n 1 - 2 n , E(k)=\frac{\pi}{2}\sum_{n=0}^{\infty}\left[\frac{(2n)!}{2^{2n}(n!)^{2}}\right]% ^{2}\frac{k^{2n}}{1-2n},
  87. E ( k ) = π 2 { 1 - ( 1 2 ) 2 k 2 1 - ( 1 3 2 4 ) 2 k 4 3 - - [ ( 2 n - 1 ) ! ! ( 2 n ) ! ! ] 2 k 2 n 2 n - 1 - } . E(k)=\frac{\pi}{2}\left\{1-\left(\frac{1}{2}\right)^{2}\frac{k^{2}}{1}-\left(% \frac{1\cdot 3}{2\cdot 4}\right)^{2}\frac{k^{4}}{3}-\cdots-\left[\frac{\left(2% n-1\right)!!}{\left(2n\right)!!}\right]^{2}\frac{k^{2n}}{2n-1}-\cdots\right\}.
  88. E ( k ) = π 2 F 1 2 ( 1 2 , - 1 2 ; 1 ; k 2 ) . E(k)=\tfrac{\pi}{2}\,{}_{2}F_{1}\left(\tfrac{1}{2},-\tfrac{1}{2};1;k^{2}\right).
  89. E ( 0 ) = π 2 E(0)=\tfrac{\pi}{2}
  90. E ( 1 ) = 1 E(1)=1\,\!
  91. E ( 2 2 ) = π 3 2 Γ ( 1 4 ) - 2 + 1 8 π Γ ( 1 4 ) 2 E\left(\tfrac{\sqrt{2}}{2}\right)=\pi^{\frac{3}{2}}\Gamma\left(\tfrac{1}{4}% \right)^{-2}+\tfrac{1}{8\sqrt{\pi}}\Gamma\left(\tfrac{1}{4}\right)^{2}
  92. E ( 1 4 ( 6 - 2 ) ) = 2 1 3 3 - 3 4 π 2 Γ ( 1 3 ) - 3 + 2 - 10 3 3 - 1 4 π - 1 ( 3 + 1 ) Γ ( 1 3 ) 3 E\left(\tfrac{1}{4}\left(\sqrt{6}-\sqrt{2}\right)\right)=2^{\frac{1}{3}}3^{-% \frac{3}{4}}\pi^{2}\Gamma\left(\tfrac{1}{3}\right)^{-3}+2^{-\frac{10}{3}}3^{-% \frac{1}{4}}\pi^{-1}\left(\sqrt{3}+1\right)\Gamma\left(\tfrac{1}{3}\right)^{3}
  93. E ( 1 4 ( 6 + 2 ) ) = 2 1 3 3 - 1 4 π 2 Γ ( 1 3 ) - 3 + 2 - 10 3 3 1 4 π - 1 ( 3 - 1 ) Γ ( 1 3 ) 3 E\left(\tfrac{1}{4}\left(\sqrt{6}+\sqrt{2}\right)\right)=2^{\frac{1}{3}}3^{-% \frac{1}{4}}\pi^{2}\Gamma\left(\tfrac{1}{3}\right)^{-3}+2^{-\frac{10}{3}}3^{% \frac{1}{4}}\pi^{-1}\left(\sqrt{3}-1\right)\Gamma\left(\tfrac{1}{3}\right)^{3}
  94. E ( 2 - 4 - 3 2 ) ) = ( 2 + 2 ) ( π 2 + 4 Γ ( 3 4 ) 4 ) 4 π Γ ( 3 4 ) 2 E\left(2\,\sqrt{-4-3\,\sqrt{2}})\right)=\frac{\left(2+\sqrt{2}\right)\left(\pi% ^{2}+4\,\Gamma\left(\frac{3}{4}\right)^{4}\right)}{4\,\sqrt{\pi}\,\Gamma\left(% \frac{3}{4}\right)^{2}}
  95. d E ( k ) d k = E ( k ) - K ( k ) k \frac{\mathrm{d}E(k)}{\mathrm{d}k}=\frac{E(k)-K(k)}{k}
  96. ( k 2 - 1 ) d d k [ k d E ( k ) d k ] = k E ( k ) (k^{2}-1)\frac{\mathrm{d}}{\mathrm{d}k}\left[k\;\frac{\mathrm{d}E(k)}{\mathrm{% d}k}\right]=kE(k)
  97. E ( 1 - k 2 ) - K ( 1 - k 2 ) E(\sqrt{1-k^{2}})-K(\sqrt{1-k^{2}})
  98. Π Π
  99. Π ( n , k ) = 0 π / 2 d θ ( 1 - n sin 2 θ ) 1 - k 2 sin 2 θ . \Pi(n,k)=\int_{0}^{\pi/2}\frac{d\theta}{(1-n\sin^{2}\theta)\sqrt{1-k^{2}\sin^{% 2}\theta}}.
  100. n n
  101. Π ( n , k ) = 0 π / 2 d θ ( 1 + n sin 2 θ ) 1 - k 2 sin 2 θ . \Pi^{\prime}(n,k)=\int_{0}^{\pi/2}\frac{d\theta}{(1+n\sin^{2}\theta)\sqrt{1-k^% {2}\sin^{2}\theta}}.
  102. Π ( n , k ) n = 1 2 ( k 2 - n ) ( n - 1 ) ( E ( k ) + 1 n ( k 2 - n ) K ( k ) + 1 n ( n 2 - k 2 ) Π ( n , k ) ) \frac{\partial\Pi(n,k)}{\partial n}=\frac{1}{2(k^{2}-n)(n-1)}\left(E(k)+\frac{% 1}{n}(k^{2}-n)K(k)+\frac{1}{n}(n^{2}-k^{2})\Pi(n,k)\right)
  103. Π ( n , k ) k = k n - k 2 ( E ( k ) k 2 - 1 + Π ( n , k ) ) \frac{\partial\Pi(n,k)}{\partial k}=\frac{k}{n-k^{2}}\left(\frac{E(k)}{k^{2}-1% }+\Pi(n,k)\right)
  104. K ( k ) E ( 1 - k 2 ) + E ( k ) K ( 1 - k 2 ) - K ( k ) K ( 1 - k 2 ) = π 2 . K(k)E\left(\sqrt{1-k^{2}}\right)+E(k)K\left(\sqrt{1-k^{2}}\right)-K(k)K\left(% \sqrt{1-k^{2}}\right)=\frac{\pi}{2}.

Elliptical_polarization.html

  1. 𝐄 ( 𝐫 , t ) = 𝐄 Re { | ψ exp [ i ( k z - ω t ) ] } \mathbf{E}(\mathbf{r},t)=\mid\mathbf{E}\mid\mathrm{Re}\left\{|\psi\rangle\exp% \left[i\left(kz-\omega t\right)\right]\right\}
  2. 𝐁 ( 𝐫 , t ) = 𝐳 ^ × 𝐄 ( 𝐫 , t ) \mathbf{B}(\mathbf{r},t)=\hat{\mathbf{z}}\times\mathbf{E}(\mathbf{r},t)
  3. ω = c k \omega=ck
  4. c c
  5. 𝐄 \mid\mathbf{E}\mid
  6. | ψ = def ( ψ x ψ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) |\psi\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}
  7. 𝐄 \mathbf{E}
  8. a = | 𝐄 | 1 + 1 - sin 2 ( 2 θ ) sin 2 β 2 a=|\mathbf{E}|\sqrt{\frac{1+\sqrt{1-\sin^{2}(2\theta)\sin^{2}\beta}}{2}}
  9. b = | 𝐄 | 1 - 1 - sin 2 ( 2 θ ) sin 2 β 2 b=|\mathbf{E}|\sqrt{\frac{1-\sqrt{1-\sin^{2}(2\theta)\sin^{2}\beta}}{2}}
  10. β = α y - α x \beta=\alpha_{y}-\alpha_{x}
  11. ϕ \phi
  12. tan 2 ϕ = tan 2 θ cos β \tan 2\phi=\tan 2\theta\cos\beta
  13. β = 0 \beta=0
  14. ( a = | 𝐄 | , b = 0 (a=|\mathbf{E}|,b=0
  15. ϕ = θ \phi=\theta
  16. | 𝐄 | cos θ |\mathbf{E}|\cos\theta
  17. | 𝐄 | sin θ |\mathbf{E}|\sin\theta
  18. β \beta
  19. ϕ θ \phi\neq\theta
  20. β \beta
  21. β = ± π / 2 \beta=\pm\pi/2
  22. θ = π / 4 \theta=\pi/4
  23. a = b = | 𝐄 | / 2 a=b=|\mathbf{E}|/\sqrt{2}
  24. β = π / 2 \beta=\pi/2
  25. β = - π / 2 \beta=-\pi/2

Embedding.html

  1. f : X Y . f:X\hookrightarrow Y.
  2. f : X Y f:X\to Y
  3. X X
  4. Y Y
  5. f f
  6. X X
  7. f ( X ) f(X)
  8. f ( X ) f(X)
  9. Y Y
  10. f : X Y f:X\to Y
  11. X X
  12. Y Y
  13. f ( X ) f(X)
  14. Y Y
  15. X X
  16. X Y X\to Y
  17. X X
  18. M M
  19. N N
  20. f : M N f:M\to N
  21. f f
  22. x M x\in M
  23. x U M x\in U\subset M
  24. f : U N f:U\to N
  25. N = n N=\mathbb{R}^{n}
  26. n n
  27. m m
  28. M M
  29. n = 2 m n=2m
  30. m m
  31. n = 2 m n=2m
  32. 3 \mathbb{R}^{3}
  33. f : X Y f:X\rightarrow Y
  34. f ( X ) = f ( X ) Y f(\partial X)=f(X)\cap\partial Y
  35. f ( X ) f(X)
  36. Y \partial Y
  37. f ( X ) f(\partial X)
  38. f ( X ) Y f(\partial X)\subseteq\partial Y
  39. f ( X X ) Y Y f(X\setminus\partial X)\subseteq Y\setminus\partial Y
  40. v , w T x ( M ) v,w\in T_{x}(M)
  41. g ( v , w ) = h ( d f ( v ) , d f ( w ) ) . g(v,w)=h(df(v),df(w)).\,
  42. A , B A,B
  43. h : A B h:A\to B
  44. h h
  45. n n
  46. f σ f\in\sigma
  47. a 1 , , a n A n , a_{1},\ldots,a_{n}\in A^{n},
  48. h ( f A ( a 1 , , a n ) ) = f B ( h ( a 1 ) , , h ( a n ) ) h(f^{A}(a_{1},\ldots,a_{n}))=f^{B}(h(a_{1}),\ldots,h(a_{n}))
  49. n n
  50. R σ R\in\sigma
  51. a 1 , , a n A n , a_{1},\ldots,a_{n}\in A^{n},
  52. A R ( a 1 , , a n ) A\models R(a_{1},\ldots,a_{n})
  53. B R ( h ( a 1 ) , , h ( a n ) ) . B\models R(h(a_{1}),\ldots,h(a_{n})).
  54. A R ( a 1 , , a n ) A\models R(a_{1},\ldots,a_{n})
  55. ( a 1 , , a n ) R A (a_{1},\ldots,a_{n})\in R^{A}
  56. x 1 , x 2 X : x 1 x 2 F ( x 1 ) F ( x 2 ) \forall x_{1},x_{2}\in X:x_{1}\leq x_{2}\Leftrightarrow F(x_{1})\leq F(x_{2})
  57. y Y : { x : F ( x ) y } \forall y\in Y:\{x:F(x)\leq y\}
  58. ϕ : X Y \phi:X\to Y
  59. C > 0 C>0
  60. L d X ( x , y ) d Y ( ϕ ( x ) , ϕ ( y ) ) C L d X ( x , y ) Ld_{X}(x,y)\leq d_{Y}(\phi(x),\phi(y))\leq CLd_{X}(x,y)
  61. L > 0 L>0
  62. ( X , ) (X,\|\cdot\|)
  63. k k
  64. 2 k \ell_{2}^{k}
  65. X X

Empirical_formula.html

  1. ( 48.64 g C 1 ) ( 1 mol 12.01 g C ) = 4.049 mol \left(\frac{48.64\mbox{ g C}~{}}{1}\right)\left(\frac{1\mbox{ mol }~{}}{12.01% \mbox{ g C}~{}}\right)=4.049\ \,\text{mol}
  2. ( 8.16 g H 1 ) ( 1 mol 1.008 g H ) = 8.095 mol \left(\frac{8.16\mbox{ g H}~{}}{1}\right)\left(\frac{1\mbox{ mol }~{}}{1.008% \mbox{ g H}~{}}\right)=8.095\ \,\text{mol}
  3. ( 43.20 g O 1 ) ( 1 mol 16.00 g O ) = 2.7 mol \left(\frac{43.20\mbox{ g O}~{}}{1}\right)\left(\frac{1\mbox{ mol }~{}}{16.00% \mbox{ g O}~{}}\right)=2.7\ \,\text{mol}
  4. 4.049 mol 2.7 mol = 1.5 \frac{4.049\mbox{ mol }~{}}{2.7\mbox{ mol }~{}}=1.5
  5. 8.095 mol 2.7 mol = 3 \frac{8.095\mbox{ mol }~{}}{2.7\mbox{ mol }~{}}=3
  6. 2.7 mol 2.7 mol = 1 \frac{2.7\mbox{ mol }~{}}{2.7\mbox{ mol }~{}}=1
  7. 1.5 × 2 = 3 1.5\times 2=3
  8. 3 × 2 = 6 3\times 2=6
  9. 1 × 2 = 2 1\times 2=2

Empty_set.html

  1. \emptyset
  2. A : A \forall A:\emptyset\subseteq A
  3. A : A = A \forall A:A\cup\emptyset=A
  4. A : A = \forall A:A\cap\emptyset=\emptyset
  5. A : A × = \forall A:A\times\emptyset=\emptyset
  6. A : A A = \forall A:A\subseteq\emptyset\Rightarrow A=\emptyset
  7. 2 = { } 2^{\emptyset}=\{\emptyset\}
  8. card ( ) = 0 \mathrm{card}(\emptyset)=0
  9. \emptyset
  10. \emptyset
  11. V = V=\emptyset
  12. \emptyset
  13. \emptyset
  14. \emptyset
  15. \emptyset
  16. \emptyset
  17. \emptyset
  18. \emptyset
  19. \emptyset
  20. - , -\infty\!\,,
  21. + , +\infty\!\,,
  22. sup = min ( { - , + } ) = - , \sup\emptyset=\min(\{-\infty,+\infty\}\cup\mathbb{R})=-\infty,
  23. inf = max ( { - , + } ) = + . \inf\emptyset=\max(\{-\infty,+\infty\}\cup\mathbb{R})=+\infty.
  24. \emptyset
  25. \emptyset

Endomorphism_ring.html

  1. E n d ( 2 × 2 , + ) M 2 ( 2 ) End(\mathbb{Z}_{2}\times\mathbb{Z}_{2},+)\cong M_{2}(\mathbb{Z}_{2})
  2. ( 2 × 2 , + ) (\mathbb{Z}_{2}\times\mathbb{Z}_{2},+)
  3. 2 × 2 2\times 2
  4. 2 \mathbb{Z}_{2}

Energy.html

  1. W = C 𝐅 d 𝐬 W=\int_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{s}
  2. W W
  3. E = h ν E=h\nu
  4. h h
  5. ν \nu
  6. E = m c 2 E=mc^{2}
  7. E p E_{p}
  8. E k E_{k}
  9. E p i + E k i = E p F + E k F E_{pi}+E_{ki}=E_{pF}+E_{kF}
  10. E p = m g h E_{p}=mgh
  11. E k = 1 2 m v 2 E_{k}=\frac{1}{2}mv^{2}
  12. E p + E k = E t o t a l E_{p}+E_{k}=E_{total}
  13. c 2 c^{2}
  14. 9 × 10 16 9\times 10^{16}
  15. Δ E Δ t 2 \Delta E\Delta t\geq\frac{\hbar}{2}
  16. Δ E = W + Q \Delta{}E=W+Q
  17. E E
  18. W W
  19. Q Q
  20. Q Q
  21. Δ E = W \Delta{}E=W
  22. E E
  23. Δ E = W + Q + E \Delta{}E=W+Q+E
  24. E E
  25. d E = T d S - P d V \mathrm{d}E=T\mathrm{d}S-P\mathrm{d}V\,
  26. d E = δ Q + δ W \mathrm{d}E=\delta Q+\delta W
  27. δ Q \delta Q
  28. δ W \delta W

Energy_level.html

  1. n = n=∞
  2. n n
  3. E n = - h c R Z 2 n 2 E_{n}=-hcR_{\infty}\frac{Z^{2}}{n^{2}}
  4. Z Z
  5. n n
  6. h h
  7. c c
  8. n n
  9. E = h ν = h c / λ E=h ν=h c / λ
  10. n n
  11. 1 λ = R Z 2 ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda}=RZ^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)
  12. Z Z
  13. Z Z
  14. E n , l = - h c R Z eff 2 n 2 E_{n,l}=-hcR_{\infty}\frac{{Z_{\rm eff}}^{2}}{n^{2}}
  15. [ u e l l ] [u^{\prime}ell^{\prime}]
  16. L L
  17. U = - s y m b o l μ L 𝐁 U=-symbol{\mu}_{L}\cdot\mathbf{B}
  18. - s y m b o l μ L = e 2 m 𝐋 = μ B 𝐋 -symbol{\mu}_{L}=\dfrac{e\hbar}{2m}\mathbf{L}=\mu_{B}\mathbf{L}
  19. - s y m b o l μ S = - μ B g S 𝐒 -symbol{\mu}_{S}=-\mu_{B}g_{S}\mathbf{S}
  20. μ \mathbf{μ}
  21. s y m b o l μ = s y m b o l μ L + s y m b o l μ S symbol{\mu}=symbol{\mu}_{L}+symbol{\mu}_{S}
  22. U B = - s y m b o l μ 𝐁 = μ B B ( M L + g S M S ) U_{B}=-symbol{\mu}\cdot\mathbf{B}=\mu_{B}B(M_{L}+g_{S}M_{S})
  23. E = E electronic + E vibrational + E rotational + E nuclear + E translational E=E_{\rm{electronic}}+E_{\rm{vibrational}}+E_{\rm{rotational}}+E_{\rm{nuclear}% }+E_{\rm{translational}}\,
  24. E < s u b > e l e c t r o n i c E<sub>electronic

Engine.html

  1. F F
  2. τ = | r × F | = r F sin ( r , F ) \tau=|r\times F|=rF\sin(r,F)
  3. r r
  4. F F
  5. 𝐫 × 𝐅 \mathbf{r}×\mathbf{F}
  6. P = d W d t P=\frac{\mathrm{d}W}{\mathrm{d}t}
  7. P = F v P=F\cdot v
  8. P = τ ω P=\tau\omega

Enhanced_Interior_Gateway_Routing_Protocol.html

  1. [ ( K 1 Bandwidth E + K 2 Bandwidth E 256 - Load + K 3 Delay E ) K 5 K 4 + Reliability ] 256 \bigg[\bigg(K_{1}\cdot{\,\text{Bandwidth}}_{E}+\frac{K_{2}\cdot{\,\text{% Bandwidth}}_{E}}{256-\,\text{Load}}+K_{3}\cdot{\,\text{Delay}}_{E}\bigg)\cdot% \frac{K_{5}}{K_{4}+\,\text{Reliability}}\bigg]\cdot 256
  2. K 1 K_{1}
  3. K 5 K_{5}
  4. K 5 K_{5}
  5. K 5 K 4 + Reliability \tfrac{K_{5}}{K_{4}+\,\text{Reliability}}
  6. K 1 K_{1}
  7. K 3 K_{3}
  8. ( Bandwidth E + Delay E ) 256 ({\,\text{Bandwidth}}_{E}+\,\text{Delay}_{E})\cdot 256
  9. Bandwidth E {\,\text{Bandwidth}}_{E}
  10. Delay E {\,\text{Delay}}_{E}

Enthalpy.html

  1. H = U + p V H=U+pV
  2. H = U + p V H=U+pV\,
  3. H = Σ k H k H=\Sigma_{k}H_{k}
  4. H = ρ h d V , H=\int\rho h\mathrm{d}V,
  5. d U = δ Q - δ W . \mathrm{d}U=\delta Q-\delta W.
  6. d U = T d S - p d V . \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V.
  7. d U + d ( p V ) = T d S - p d V + d ( p V ) \mathrm{d}U+\mathrm{d}(pV)=T\mathrm{d}S-p\mathrm{d}V+\mathrm{d}(pV)
  8. d ( U + p V ) = T d S + V d p . \mathrm{d}(U+pV)=T\mathrm{d}S+V\mathrm{d}p.
  9. d H ( S , p ) = T d S + V d p . \mathrm{d}H(S,p)=T\mathrm{d}S+V\mathrm{d}p.
  10. d H = C p d T + V ( 1 - α T ) d p . \mathrm{d}H=C_{p}\mathrm{d}T+V(1-\alpha T)\mathrm{d}p.
  11. α = 1 V ( V T ) p . \alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}.
  12. α T = 1 \alpha T=1
  13. d H = C p d T \mathrm{d}H=C_{p}\mathrm{d}T
  14. d H = T d S + V d p + i μ i d N i \mathrm{d}H=T\mathrm{d}S+V\mathrm{d}p+\sum_{i}\mu_{i}\mathrm{d}N_{i}
  15. d U = δ Q - p d V - δ W \mathrm{d}U=\delta Q-p\mathrm{d}V-\delta W^{\prime}
  16. d H = δ Q + V d p - δ W . \mathrm{d}H=\delta Q+V\mathrm{d}p-\delta W^{\prime}.
  17. d H = δ Q \mathrm{d}H=\delta Q
  18. Δ \Delta
  19. H H
  20. = =
  21. H f H_{f}
  22. - -
  23. H i H_{i}
  24. Δ H \Delta H
  25. H f H_{f}
  26. H f H_{f}
  27. H i H_{i}
  28. H i H_{i}
  29. d U = δ Q + d U i n - d U o u t - δ W \mathrm{d}U=\mathrm{\delta}Q+\mathrm{d}U_{in}-\mathrm{d}U_{out}-\mathrm{\delta}W
  30. U i n U_{in}
  31. U o u t U_{out}
  32. δ W = d ( p o u t V o u t ) - d ( p i n V i n ) + δ W s h a f t \mathrm{\delta}W=\mathrm{d}(p_{out}V_{out})-\mathrm{d}(p_{in}V_{in})+\mathrm{% \delta}W_{shaft}
  33. d U c v = δ Q + d U i n + d ( p i n V i n ) - d U o u t - d ( p o u t V o u t ) - δ W s h a f t \mathrm{d}U_{cv}=\mathrm{\delta}Q+\mathrm{d}U_{in}+\mathrm{d}(p_{in}V_{in})-% \mathrm{d}U_{out}-\mathrm{d}(p_{out}V_{out})-\mathrm{\delta}W_{shaft}
  34. H H
  35. p V pV
  36. d U c v = δ Q + d H i n - d H o u t - δ W s h a f t \mathrm{d}U_{cv}=\mathrm{\delta}Q+\mathrm{d}H_{in}-\mathrm{d}H_{out}-\mathrm{% \delta}W_{shaft}
  37. d U d t = Σ k Q ˙ k + Σ k H ˙ k - Σ k p k d V k d t - P , \frac{\mathrm{d}U}{\mathrm{d}t}=\Sigma_{k}\dot{Q}_{k}+\Sigma_{k}\dot{H}_{k}-% \Sigma_{k}p_{k}\frac{\mathrm{d}V_{k}}{\mathrm{d}t}-P,
  38. Σ \Sigma
  39. H ˙ k \dot{H}_{k}
  40. H ˙ k = h k m ˙ k = H m n ˙ k \dot{H}_{k}=h_{k}\dot{m}_{k}=H_{m}\dot{n}_{k}
  41. m ˙ k \dot{m}_{k}
  42. n ˙ k \dot{n}_{k}
  43. P = Σ k Q ˙ k + Σ k H ˙ k - Σ k p k d V k d t P=\Sigma_{k}\left\langle\dot{Q}_{k}\right\rangle+\Sigma_{k}\left\langle\dot{H}% _{k}\right\rangle-\Sigma_{k}\left\langle p_{k}\frac{\mathrm{d}V_{k}}{\mathrm{d% }t}\right\rangle
  44. m ˙ \dot{m}
  45. Q ˙ \dot{Q}
  46. 0 = m ˙ h 1 - m ˙ h 2 . 0=\dot{m}h_{1}-\dot{m}h_{2}.
  47. h 1 = h 2 h_{1}=h_{2}
  48. h f = x f h g + ( 1 - x f ) h h . h_{f}=x_{f}h_{g}+(1-x_{f})h_{h}.
  49. Q ˙ \dot{Q}
  50. 0 = - Q ˙ + m ˙ h 1 - m ˙ h 2 + P . 0=-\dot{Q}+\dot{m}h_{1}-\dot{m}h_{2}+P.
  51. 0 = - Q ˙ T a + m ˙ s 1 - m ˙ s 2 . 0=-\frac{\dot{Q}}{T_{a}}+\dot{m}s_{1}-\dot{m}s_{2}.
  52. Q ˙ \dot{Q}
  53. P min m ˙ = h 2 - h 1 - T a ( s 2 - s 1 ) . \frac{P_{\rm min}}{\dot{m}}=h_{2}-h_{1}-T_{a}(s_{2}-s_{1}).
  54. P min m ˙ = 1 2 ( d h - T a d s ) . \frac{P_{\rm min}}{\dot{m}}=\int_{1}^{2}(\mathrm{d}h-T_{a}\mathrm{d}s).
  55. P min m ˙ = 1 2 v d p . \frac{P_{\rm min}}{\dot{m}}=\int_{1}^{2}v\mathrm{d}p.
  56. α T = T V ( ( n R T / P ) T ) p = n R T P V = 1 \alpha T=\frac{T}{V}\left(\frac{\partial(nRT/P)}{\partial T}\right)_{p}=\frac{% nRT}{PV}=1

Enthalpy_of_vaporization.html

  1. Δ H vap = Δ U vap + p Δ V \Delta{}H_{\mathrm{vap}}=\Delta{}U_{\mathrm{vap}}+p\Delta\,V
  2. Δ v S = S g a s - S l i q u i d = Δ v H / T b \Delta\,_{v}S=S_{gas}-S_{liquid}=\Delta\,_{v}H/T_{b}
  3. Δ v S \Delta\,_{v}S
  4. Δ G = Δ H - T Δ S \Delta\,G=\Delta\,H-T\Delta\,S

Entire_function.html

  1. f ( z ) = n = 0 a n z n f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}
  2. lim n | a n | 1 n = 0 \lim_{n\to\infty}\left|a_{n}\right|^{\frac{1}{n}}=0
  3. lim n ln | a n | n = - . \lim_{n\to\infty}\frac{\ln|a_{n}|}{n}=-\infty.
  4. Re a n = 1 n ! d n d r n Re f ( r ) at r = 0 \operatorname{Re}a_{n}=\frac{1}{n!}\frac{d^{n}}{dr^{n}}\operatorname{Re}f(r)\ % \,\text{ at }r=0
  5. Im a n = 1 n ! d n d r n Re f ( r e - i π / ( 2 n ) ) at r = 0. \operatorname{Im}a_{n}=\frac{1}{n!}\frac{d^{n}}{dr^{n}}\operatorname{Re}f(re^{% -i\pi/(2n)})\ \,\text{ at }r=0.
  6. lim m | z m | = \lim_{m\to\infty}|z_{m}|=\infty
  7. lim m f ( z m ) = w \lim_{m\to\infty}f(z_{m})=w
  8. | f ( z ) | M | z | n |f(z)|\leq M|z|^{n}
  9. | z | R |z|\geq R
  10. M | z | n | f ( z ) | M|z|^{n}\leq|f(z)|
  11. | z | R |z|\geq R
  12. F * ( z ) F^{*}(z)
  13. F ¯ ( z ¯ ) ) . \bar{F}(\bar{z})).
  14. f ( z ) = c + k = 1 ( z k ) n k f(z)=c+\sum_{k=1}^{\infty}\left(\frac{z}{k}\right)^{n_{k}}
  15. n k := 2 k ln g ( k + 2 ) n_{k}:=2\big\lceil k\ln g(k+2)\big\rceil\,
  16. ρ = lim sup r ln ( ln f , B r ) ln r , \rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln\|f\|_{\infty,B_{r}})}{\ln\,r},
  17. f , B r \|f\|_{\infty,\,B_{r}}
  18. exp ( 2 z 2 ) \exp(2z^{2})
  19. f ( z ) = n = 0 a n z n , f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},
  20. ρ = lim sup n n ln n - ln | a n | , \rho=\limsup_{n\to\infty}\frac{n\ln n}{-\ln|a_{n}|},
  21. ( e ρ σ ) 1 / ρ = lim sup n n 1 / ρ | a n | 1 / n . (e\rho\sigma)^{1/\rho}=\limsup_{n\to\infty}n^{1/\rho}|a_{n}|^{1/n}.
  22. ρ = lim sup n n ln n n ln n - ln | f ( n ) ( z 0 ) | = ( 1 - lim sup n ln | f ( n ) ( z 0 ) | n ln n ) - 1 , \rho=\limsup_{n\to\infty}\frac{n\ln n}{n\ln n-\ln|f^{(n)}(z_{0})|}=\left(1-% \limsup_{n\to\infty}\frac{\ln|f^{(n)}(z_{0})|}{n\ln n}\right)^{-1},
  23. ( ρ σ ) 1 / ρ = e 1 - 1 / ρ lim sup n | f ( n ) ( z 0 ) | 1 / n n 1 - 1 / ρ . (\rho\sigma)^{1/\rho}=e^{1-1/\rho}\limsup_{n\to\infty}\frac{|f^{(n)}(z_{0})|^{% 1/n}}{n^{1-1/\rho}}.
  24. f ( z ) = n = 1 ( e ρ σ / n ) n / ρ z n f(z)=\sum_{n=1}^{\infty}(e\rho\sigma/n)^{n/\rho}z^{n}
  25. n = 0 2 - n 2 z n \sum_{n=0}^{\infty}2^{-n^{2}}z^{n}
  26. f ( z 4 ) where f ( u ) = cos ( u ) + cosh ( u ) f(\sqrt[4]{z})\,\text{ where }f(u)=\cos(u)+\cosh(u)
  27. f ( z 3 ) where f ( u ) = e u + e ω u + e ω 2 u = e u + 2 e - u / 2 cos ( 3 u / 2 ) , with ω a complex cube root of 1 f(\sqrt[3]{z})\,\text{ where }f(u)=e^{u}+e^{\omega u}+e^{\omega^{2}u}=e^{u}+2e% ^{-u/2}\cos(\sqrt{3}u/2),\,\text{with }\omega\,\text{ a complex cube root of 1}
  28. cos ( a z ) \cos(a\sqrt{z})
  29. n = 2 z n / ( n ln n ) n . ( σ = 0 ) \sum_{n=2}^{\infty}z^{n}/(n\ln n)^{n}.\ (\sigma=0)
  30. f ( z ) = z m e P ( z ) n = 1 ( 1 - z z n ) exp ( z z n + + 1 p ( z z n ) p ) , f(z)=z^{m}e^{P(z)}\prod_{n=1}^{\infty}\left(1-\frac{z}{z_{n}}\right)\exp\left(% \frac{z}{z_{n}}+\ldots+\frac{1}{p}\left(\frac{z}{z_{n}}\right)^{p}\right),
  31. n = 1 1 | z n | p + 1 \sum_{n=1}^{\infty}\frac{1}{|z_{n}|^{p+1}}
  32. cos ( z ) \cos(\sqrt{z})
  33. ( 1 - ( z - d ) 2 / n ) n (1-(z-d)^{2}/n)^{n}
  34. ( ( 1 + i z / n ) n + ( 1 - i z / n ) n ) / 2 ((1+iz/n)^{n}+(1-iz/n)^{n})/2
  35. m = 1 n ( 1 - z 2 ( ( m - 1 2 ) π ) 2 ) \prod_{m=1}^{n}\left(1-\frac{z^{2}}{((m-\frac{1}{2})\pi)^{2}}\right)
  36. p ( z ) = k = 0 n a k z k \textstyle p(z)=\sum_{k=0}^{n}a_{k}z^{k}
  37. | p ( z ) | ( k = 0 n | a k | ) | z | n \textstyle|p(z)|\leq\left(\sum_{k=0}^{n}|a_{k}|\right)|z|^{n}

Entropy.html

  1. Δ S = d Q rev T \Delta S=\int\frac{dQ\text{rev}}{T}
  2. T T
  3. d Q dQ
  4. d Q dQ
  5. Q H Q_{H}
  6. T H T_{H}
  7. Q C Q_{C}
  8. T C T_{C}
  9. Q H Q_{H}
  10. Q H Q_{H}
  11. Q C Q_{C}
  12. Q H Q_{H}
  13. Q C Q_{C}
  14. Q H > Q C Q_{H}>Q_{C}
  15. W = ( T H - T C T H ) Q H = ( 1 - T C T H ) Q H W=\left(\frac{T_{H}-T_{C}}{T_{H}}\right)Q_{H}=\left(1-\frac{T_{C}}{T_{H}}% \right)Q_{H}
  16. 1 - T C T H 1-\frac{T_{C}}{T_{H}}
  17. W = Q H - Q C W=Q_{H}-Q_{C}
  18. Q H T H - Q C T C = 0 \frac{Q_{H}}{T_{H}}-\frac{Q_{C}}{T_{C}}=0
  19. Q H T H = Q C T C \frac{Q_{H}}{T_{H}}=\frac{Q_{C}}{T_{C}}
  20. W < ( 1 - T C T H ) Q H W<\left(1-\frac{T_{C}}{T_{H}}\right)Q_{H}
  21. Q H - Q C < ( 1 - T C T H ) Q H Q_{H}-Q_{C}<\left(1-\frac{T_{C}}{T_{H}}\right)Q_{H}
  22. Q C > T C T H Q H Q_{C}>\frac{T_{C}}{T_{H}}Q_{H}
  23. S i = Q i / T i S_{i}=Q_{i}/T_{i}
  24. S H - S C < 0 S_{H}-S_{C}<0
  25. S H < S C S_{H}<S_{C}
  26. δ Q rev T = 0. \oint\frac{\delta Q\text{rev}}{T}=0.
  27. L δ Q rev T \int_{L}\frac{\delta Q\text{rev}}{T}
  28. d S = δ Q rev T . dS=\frac{\delta Q\text{rev}}{T}.
  29. S = - k B i p i ln p i , S=-k_{\mathrm{B}}\sum_{i}p_{i}\ln p_{i},
  30. S = - k B Tr ( ρ ^ ln ( ρ ^ ) ) , S=-k_{\mathrm{B}}\,\text{Tr}\,(\widehat{\rho}\ln(\widehat{\rho})),
  31. ρ ^ \widehat{\rho}
  32. Tr \,\text{Tr}
  33. ln \ln
  34. S = k B ln Ω . S=k_{\mathrm{B}}\ln\Omega.
  35. d U = T d S - P d V dU=TdS-PdV
  36. Q ˙ \dot{Q}
  37. W ˙ S \dot{W}_{S}
  38. Q ˙ / T , \dot{Q}/T,
  39. d S d t = k = 1 K M ˙ k S ^ k + Q ˙ T + S ˙ g e n \frac{dS}{dt}=\sum_{k=1}^{K}\dot{M}_{k}\hat{S}_{k}+\frac{\dot{Q}}{T}+\dot{S}_{gen}
  40. k = 1 K M ˙ k S ^ k \sum_{k=1}^{K}\dot{M}_{k}\hat{S}_{k}
  41. S ^ \hat{S}
  42. Q ˙ T \frac{\dot{Q}}{T}
  43. S ˙ g e n \dot{S}_{gen}
  44. Q ˙ / T \dot{Q}/T
  45. Q ˙ j / T j , \sum\dot{Q}_{j}/T_{j},
  46. Q ˙ j \dot{Q}_{j}
  47. T j T_{j}
  48. V 0 V_{0}
  49. P 0 P_{0}
  50. V V
  51. P P
  52. Δ S = n R ln V V 0 = - n R ln P P 0 . \Delta S=nR\ln\frac{V}{V_{0}}=-nR\ln\frac{P}{P_{0}}.
  53. n n
  54. R R
  55. T 0 T_{0}
  56. T T
  57. Δ S = n C P ln T T 0 \Delta S=nC_{P}\ln\frac{T}{T_{0}}
  58. Δ S = n C v ln T T 0 \Delta S=nC_{v}\ln\frac{T}{T_{0}}
  59. Δ S = n C v ln T T 0 + n R ln V V 0 \Delta S=nC_{v}\ln\frac{T}{T_{0}}+nR\ln\frac{V}{V_{0}}
  60. Δ S = n C P ln T T 0 - n R ln P P 0 \Delta S=nC_{P}\ln\frac{T}{T_{0}}-nR\ln\frac{P}{P_{0}}
  61. Δ S fus = Δ H fus T m . \Delta S\text{fus}=\frac{\Delta H\text{fus}}{T\text{m}}.
  62. Δ S vap = Δ H vap T b . \Delta S\text{vap}=\frac{\Delta H\text{vap}}{T\text{b}}.
  63. Disorder = C D C I . \mbox{Disorder}~{}={C_{D}\over C_{I}}.\,
  64. Order = 1 - C O C I . \mbox{Order}~{}=1-{C_{O}\over C_{I}}.\,
  65. X 0 X_{0}
  66. X 1 X_{1}
  67. X X
  68. λ \lambda
  69. X X
  70. λ \lambda
  71. X 1 X_{1}
  72. ( 1 - λ ) (1-\lambda)
  73. X 0 X_{0}
  74. S = - k B Tr ( ρ log ρ ) S=-k_{\mathrm{B}}\mathrm{Tr}(\rho\log\rho)\!
  75. S = - k B i p i log p i S=-k_{\mathrm{B}}\sum_{i}p_{i}\,\log\,p_{i}
  76. H ( X ) = - i = 1 n p ( x i ) log p ( x i ) . H(X)=-\sum_{i=1}^{n}p(x_{i})\log p(x_{i}).
  77. H = - i = 1 W p log ( p ) = log ( W ) H=-\sum_{i=1}^{W}p\log(p)=\log(W)
  78. H = k log ( W ) H=k\,\log(W)

Entropy_(information_theory).html

  1. \Eta ( X ) = E [ I ( X ) ] = E [ - ln ( P ( X ) ) ] . \Eta(X)=\mathrm{E}[\mathrm{I}(X)]=\mathrm{E}[-\ln(\mathrm{P}(X))].
  2. \Eta ( X ) = i P ( x i ) I ( x i ) = - i P ( x i ) log b P ( x i ) , \Eta(X)=\sum_{i}{\mathrm{P}(x_{i})\,\mathrm{I}(x_{i})}=-\sum_{i}{\mathrm{P}(x_% {i})\log_{b}\mathrm{P}(x_{i})},
  3. e e
  4. e e
  5. lim p 0 + p log ( p ) = 0. \lim_{p\to 0^{+}}p\log(p)=0.
  6. \Eta ( X | Y ) = i , j p ( x i , y j ) log p ( y j ) p ( x i , y j ) \Eta(X|Y)=\sum_{i,j}p(x_{i},y_{j})\log\frac{p(y_{j})}{p(x_{i},y_{j})}
  7. p i log 1 p i \sum{p_{i}\log{1\over p_{i}}}
  8. p i p_{i}
  9. I ( p ) = log ( 1 / p ) \mathrm{I}(p)=\log(1/p)
  10. n i = N p i n_{i}=Np_{i}
  11. i n i I ( p i ) = N p i log ( 1 / p i ) \sum_{i}{n_{i}\mathrm{I}(p_{i})}=\sum{Np_{i}\log(1/p_{i})}
  12. i p i log 1 p i . \sum_{i}{p_{i}\log{1\over p_{i}}}.
  13. S = - k B p i ln p i S=-k_{\mathrm{B}}\sum p_{i}\ln p_{i}\,
  14. S = - k B Tr ( ρ ln ρ ) S=-k_{\mathrm{B}}\,{\rm Tr}(\rho\ln\rho)\,
  15. 2 128 - 1 2^{128-1}
  16. 2 127 - 1 2^{127-1}
  17. 2 126 2^{126}
  18. 2 65 - 1 2^{65-1}
  19. \Eta ( 𝒮 ) = - p i log 2 p i , \Eta(\mathcal{S})=-\sum p_{i}\log_{2}p_{i},\,\!
  20. \Eta ( 𝒮 ) = - i p i j p i ( j ) log 2 p i ( j ) , \Eta(\mathcal{S})=-\sum_{i}p_{i}\sum_{j}\ p_{i}(j)\log_{2}p_{i}(j),\,\!
  21. p i ( j ) p_{i}(j)
  22. \Eta ( 𝒮 ) = - i p i j p i ( j ) k p i , j ( k ) log 2 p i , j ( k ) . \Eta(\mathcal{S})=-\sum_{i}p_{i}\sum_{j}p_{i}(j)\sum_{k}p_{i,j}(k)\ \log_{2}\ % p_{i,j}(k).\,\!
  23. 𝒮 \mathcal{S}
  24. \Eta b ( 𝒮 ) = - i = 1 n p i log b p i , \Eta_{b}(\mathcal{S})=-\sum_{i=1}^{n}p_{i}\log_{b}p_{i},\,\!
  25. η ( X ) = - i = 1 n p ( x i ) log b ( p ( x i ) ) log b ( n ) \eta(X)=-\sum_{i=1}^{n}\frac{p(x_{i})\log_{b}(p(x_{i}))}{\log_{b}(n)}
  26. log b ( n ) {\log_{b}(n)}
  27. - K i = 1 n p i log ( p i ) -K\sum_{i=1}^{n}p_{i}\log(p_{i})
  28. \Eta n ( p 1 , , p n ) = \Eta ( X ) \Eta_{n}(p_{1},\ldots,p_{n})=\Eta(X)
  29. \Eta n ( p 1 , p 2 , ) = \Eta n ( p 2 , p 1 , ) \Eta_{n}\left(p_{1},p_{2},\ldots\right)=\Eta_{n}\left(p_{2},p_{1},\ldots\right)
  30. \Eta n ( p 1 , , p n ) \Eta n ( 1 n , , 1 n ) = log b ( n ) . \Eta_{n}(p_{1},\ldots,p_{n})\leq\Eta_{n}\left(\frac{1}{n},\ldots,\frac{1}{n}% \right)=\log_{b}(n).
  31. \Eta n ( 1 n , , 1 n n ) = log b ( n ) < log b ( n + 1 ) = \Eta n + 1 ( 1 n + 1 , , 1 n + 1 n + 1 ) . \Eta_{n}\bigg(\underbrace{\frac{1}{n},\ldots,\frac{1}{n}}_{n}\bigg)=\log_{b}(n% )<\log_{b}(n+1)=\Eta_{n+1}\bigg(\underbrace{\frac{1}{n+1},\ldots,\frac{1}{n+1}% }_{n+1}\bigg).
  32. \Eta n ( 1 n , , 1 n ) = \Eta k ( b 1 n , , b k n ) + i = 1 k b i n \Eta b i ( 1 b i , , 1 b i ) . \Eta_{n}\left(\frac{1}{n},\ldots,\frac{1}{n}\right)=\Eta_{k}\left(\frac{b_{1}}% {n},\ldots,\frac{b_{k}}{n}\right)+\sum_{i=1}^{k}\frac{b_{i}}{n}\,\Eta_{b_{i}}% \left(\frac{1}{b_{i}},\ldots,\frac{1}{b_{i}}\right).
  33. \Eta n + 1 ( p 1 , , p n , 0 ) = \Eta n ( p 1 , , p n ) \Eta_{n+1}(p_{1},\ldots,p_{n},0)=\Eta_{n}(p_{1},\ldots,p_{n})
  34. \Eta ( X ) = E [ log b ( 1 p ( X ) ) ] log b ( E [ 1 p ( X ) ] ) = log b ( n ) \Eta(X)=\operatorname{E}\left[\log_{b}\left(\frac{1}{p(X)}\right)\right]\leq% \log_{b}\left(\operatorname{E}\left[\frac{1}{p(X)}\right]\right)=\log_{b}(n)
  35. \Eta ( X , Y ) = \Eta ( X | Y ) + \Eta ( Y ) = \Eta ( Y | X ) + \Eta ( X ) . \Eta(X,Y)=\Eta(X|Y)+\Eta(Y)=\Eta(Y|X)+\Eta(X).
  36. \Eta ( X ) + \Eta ( f ( X ) | X ) = \Eta ( f ( X ) ) + \Eta ( X | f ( X ) ) , \Eta(X)+\Eta(f(X)|X)=\Eta(f(X))+\Eta(X|f(X)),
  37. \Eta ( X | Y ) = \Eta ( X ) . \Eta(X|Y)=\Eta(X).
  38. \Eta ( X , Y ) \Eta ( X ) + \Eta ( Y ) . \Eta(X,Y)\leq\Eta(X)+\Eta(Y).
  39. 𝕏 \mathbb{X}
  40. h [ f ] = E [ - ln ( f ( x ) ) ] = - 𝕏 f ( x ) ln ( f ( x ) ) d x . h[f]=\operatorname{E}[-\ln(f(x))]=-\int_{\mathbb{X}}f(x)\ln(f(x))\,dx.
  41. Δ \Delta
  42. f ( x i ) Δ = i Δ ( i + 1 ) Δ f ( x ) d x f(x_{i})\Delta=\int_{i\Delta}^{(i+1)\Delta}f(x)\,dx
  43. - f ( x ) d x = lim Δ 0 i = - f ( x i ) Δ \int_{-\infty}^{\infty}f(x)\,dx=\lim_{\Delta\to 0}\sum_{i=-\infty}^{\infty}f(x% _{i})\Delta
  44. H Δ := - i = - f ( x i ) Δ log ( f ( x i ) Δ ) H^{\Delta}:=-\sum_{i=-\infty}^{\infty}f(x_{i})\Delta\log\left(f(x_{i})\Delta\right)
  45. H Δ = - i = - f ( x i ) Δ log ( f ( x i ) ) - i = - f ( x i ) Δ log ( Δ ) . H^{\Delta}=-\sum_{i=-\infty}^{\infty}f(x_{i})\Delta\log(f(x_{i}))-\sum_{i=-% \infty}^{\infty}f(x_{i})\Delta\log(\Delta).
  46. i = - f ( x i ) Δ - f ( x ) d x = 1 i = - f ( x i ) Δ log ( f ( x i ) ) - f ( x ) log f ( x ) d x . \begin{aligned}\displaystyle\sum_{i=-\infty}^{\infty}f(x_{i})\Delta&% \displaystyle\to\int_{-\infty}^{\infty}f(x)\,dx=1\\ \displaystyle\sum_{i=-\infty}^{\infty}f(x_{i})\Delta\log(f(x_{i}))&% \displaystyle\to\int_{-\infty}^{\infty}f(x)\log f(x)\,dx.\end{aligned}
  47. h [ f ] = lim Δ 0 ( H Δ + log Δ ) = - - f ( x ) log f ( x ) d x , h[f]=\lim_{\Delta\to 0}\left(H^{\Delta}+\log\Delta\right)=-\int_{-\infty}^{% \infty}f(x)\log f(x)\,dx,
  48. D KL ( p m ) = log ( f ( x ) ) p ( d x ) = f ( x ) log ( f ( x ) ) m ( d x ) . D_{\mathrm{KL}}(p\|m)=\int\log(f(x))p(dx)=\int f(x)\log(f(x))m(dx).
  49. | A | d - 1 i = 1 d | P i ( A ) | |A|^{d-1}\leq\prod_{i=1}^{d}|P_{i}(A)|
  50. P i ( A ) = { ( x 1 , , x i - 1 , x i + 1 , , x d ) : ( x 1 , , x d ) A } . P_{i}(A)=\{(x_{1},...,x_{i-1},x_{i+1},...,x_{d}):(x_{1},...,x_{d})\in A\}.
  51. \Eta [ ( X 1 , , X d ) ] 1 r i = 1 n H [ ( X j ) j S i ] \Eta[(X_{1},...,X_{d})]\leq\frac{1}{r}\sum_{i=1}^{n}H[(X_{j})_{j\in S_{i}}]
  52. ( X j ) j S i (X_{j})_{j\in S_{i}}
  53. | A | |A|
  54. ( X j ) j S i (X_{j})_{j\in S_{i}}
  55. \Eta [ ( X j ) j S i ] log | P i ( A ) | \Eta[(X_{j})_{j\in S_{i}}]\leq\log|P_{i}(A)|
  56. \Eta ( q ) = - q log 2 ( q ) - ( 1 - q ) log 2 ( 1 - q ) . \Eta(q)=-q\log_{2}(q)-(1-q)\log_{2}(1-q).
  57. ( n k ) q q n ( 1 - q ) n - n q {\textstyle\left({{n}\atop{k}}\right)}q^{qn}(1-q)^{n-nq}
  58. i = 0 n ( n i ) q i ( 1 - q ) n - i = ( q + ( 1 - q ) ) n = 1. \sum_{i=0}^{n}{\textstyle\left({{n}\atop{i}}\right)}q^{i}(1-q)^{n-i}=(q+(1-q))% ^{n}=1.
  59. ( n k ) q q n ( 1 - q ) n - n q 1 n + 1 {\textstyle\left({{n}\atop{k}}\right)}q^{qn}(1-q)^{n-nq}\geq\tfrac{1}{n+1}
  60. 2 n \Eta ( k / n ) 2^{n\Eta(k/n)}

Enzyme.html

  1. CO 2 + H 2 O Carbonic anhydrase H 2 CO 3 \mathrm{CO_{2}+H_{2}O\xrightarrow{Carbonic\ anhydrase}H_{2}CO_{3}}
  2. H 2 CO 3 Carbonic anhydrase CO 2 + H 2 O \mathrm{H_{2}CO_{3}\xrightarrow{Carbonic\ anhydrase}CO_{2}+H_{2}O}

Ephemeris_time.html

  1. δ t = + 24 s .349 + 72 s .3165 T + 29 s .949 T 2 + 1.821 B \delta t=+24^{s}.349+72^{s}.3165T+29^{s}.949T^{2}+1.821B

Epimorphism.html

  1. g 1 f = g 2 f g 1 = g 2 . g_{1}\circ f=g_{2}\circ f\Rightarrow g_{1}=g_{2}.
  2. Hom ( Y , Z ) Hom ( X , Z ) g g f \begin{matrix}\operatorname{Hom}(Y,Z)&\rightarrow&\operatorname{Hom}(X,Z)\\ g&\mapsto&gf\end{matrix}
  3. Hom ( Y , - ) Hom ( X , - ) \begin{matrix}\operatorname{Hom}(Y,-)&\rightarrow&\operatorname{Hom}(X,-)\end{matrix}

EPR_paradox.html

  1. | Φ + = 1 2 ( | 00 + | 11 ) |\Phi^{+}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)
  2. 3 / 2 {3}/{2}
  3. S x = 2 [ 0 1 1 0 ] , S y = 2 [ 0 - i i 0 ] , S z = 2 [ 1 0 0 - 1 ] S_{x}=\frac{\hbar}{2}\begin{bmatrix}0&1\\ 1&0\end{bmatrix},\quad S_{y}=\frac{\hbar}{2}\begin{bmatrix}0&-i\\ i&0\end{bmatrix},\quad S_{z}=\frac{\hbar}{2}\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  4. \hbar
  5. | + z [ 1 0 ] , | - z [ 0 1 ] \left|+z\right\rangle\leftrightarrow\begin{bmatrix}1\\ 0\end{bmatrix},\quad\left|-z\right\rangle\leftrightarrow\begin{bmatrix}0\\ 1\end{bmatrix}
  6. | + x 1 2 [ 1 1 ] , | - x 1 2 [ 1 - 1 ] \left|+x\right\rangle\leftrightarrow\frac{1}{\sqrt{2}}\begin{bmatrix}1\\ 1\end{bmatrix},\quad\left|-x\right\rangle\leftrightarrow\frac{1}{\sqrt{2}}% \begin{bmatrix}1\\ -1\end{bmatrix}
  7. V V V\otimes V
  8. | ψ = 1 2 ( | + z | - z - | - z | + z ) \left|\psi\right\rangle=\frac{1}{\sqrt{2}}\bigg(\left|+z\right\rangle\otimes% \left|-z\right\rangle-\left|-z\right\rangle\otimes\left|+z\right\rangle\bigg)
  9. | ψ = - 1 2 ( | + x | - x - | - x | + x ) \left|\psi\right\rangle=-\frac{1}{\sqrt{2}}\bigg(\left|+x\right\rangle\otimes% \left|-x\right\rangle-\left|-x\right\rangle\otimes\left|+x\right\rangle\bigg)
  10. | + z | ϕ ϕ V \left|+z\right\rangle\otimes\left|\phi\right\rangle\quad\phi\in V
  11. | + z | - z \left|+z\right\rangle\otimes\left|-z\right\rangle
  12. | - z | ϕ ϕ V \left|-z\right\rangle\otimes\left|\phi\right\rangle\quad\phi\in V
  13. | - z | + z \left|-z\right\rangle\otimes\left|+z\right\rangle
  14. [ S x , S z ] = - i S y 0 \left[S_{x},S_{z}\right]=-i\hbar S_{y}\neq 0
  15. Δ S x 2 Δ S z 2 1 4 | [ S x , S z ] | 2 \left\langle{\Delta S_{x}}^{2}\right\rangle\left\langle{\Delta S_{z}}^{2}% \right\rangle\geq\frac{1}{4}\left|\left\langle\left[S_{x},S_{z}\right]\right% \rangle\right|^{2}

Epsilon.html

  1. ϵ \epsilon\!
  2. ε \varepsilon\!
  3. ϵ x . ϕ \epsilon x.\phi

Equal_temperament.html

  1. 2 12 = 1.059463094359295264561825 \sqrt[12]{2}=1.059463094359295264561825
  2. 2 24 \sqrt[24]{2}
  3. ( 1.059463094359295264561825 ) 12 1.99999999999988 (1.059463094359295264561825)^{12}\approx 1.99999999999988
  4. ( 1.059463094359295264561825 ) 84 127.999999999946 (1.059463094359295264561825)^{84}\approx 127.999999999946
  5. r n = p r^{n}=p
  6. r = p n r=\sqrt[n]{p}
  7. c = w n c=\frac{w}{n}
  8. 2 12 = 2 1 12 1.059463 \sqrt[12]{2}=2^{\frac{1}{12}}\approx 1.059463
  9. P n = P a ( 2 12 ) ( n - a ) P_{n}=P_{a}(\sqrt[12]{2})^{(n-a)}
  10. P 40 = 440 ( 2 12 ) ( 40 - 49 ) 261.626 Hz P_{40}=440(\sqrt[12]{2})^{(40-49)}\approx 261.626~{}\,\text{Hz}
  11. 2 0 / 12 = 1 2^{0/12}=1
  12. 1 1 \begin{matrix}\frac{1}{1}\end{matrix}
  13. 2 1 / 12 = 2 12 2^{1/12}=\sqrt[12]{2}
  14. 16 15 \begin{matrix}\frac{16}{15}\end{matrix}
  15. 2 2 / 12 = 2 6 2^{2/12}=\sqrt[6]{2}
  16. 9 8 \begin{matrix}\frac{9}{8}\end{matrix}
  17. 2 3 / 12 = 2 4 2^{3/12}=\sqrt[4]{2}
  18. 6 5 \begin{matrix}\frac{6}{5}\end{matrix}
  19. 2 4 / 12 = 2 3 2^{4/12}=\sqrt[3]{2}
  20. 5 4 \begin{matrix}\frac{5}{4}\end{matrix}
  21. 2 5 / 12 = 32 12 2^{5/12}=\sqrt[12]{32}
  22. 4 3 \begin{matrix}\frac{4}{3}\end{matrix}
  23. 2 6 / 12 = 2 2^{6/12}=\sqrt{2}
  24. 7 5 \begin{matrix}\frac{7}{5}\end{matrix}
  25. 2 7 / 12 = 128 12 2^{7/12}=\sqrt[12]{128}
  26. 3 2 \begin{matrix}\frac{3}{2}\end{matrix}
  27. 2 8 / 12 = 4 3 2^{8/12}=\sqrt[3]{4}
  28. 8 5 \begin{matrix}\frac{8}{5}\end{matrix}
  29. 2 9 / 12 = 8 4 2^{9/12}=\sqrt[4]{8}
  30. 5 3 \begin{matrix}\frac{5}{3}\end{matrix}
  31. 2 10 / 12 = 32 6 2^{10/12}=\sqrt[6]{32}
  32. 16 9 \begin{matrix}\frac{16}{9}\end{matrix}
  33. 2 11 / 12 = 2048 12 2^{11/12}=\sqrt[12]{2048}
  34. 15 8 \begin{matrix}\frac{15}{8}\end{matrix}
  35. 2 12 / 12 = 2 2^{12/12}={2}
  36. 2 1 \begin{matrix}\frac{2}{1}\end{matrix}
  37. 3 / 2 7 \sqrt[7]{3/2}
  38. log 2 ( 3 ) \log_{2}(3)
  39. 3 13 \sqrt[13]{3}
  40. 3 / 2 9 \sqrt[9]{3/2}
  41. 3 / 2 11 \sqrt[11]{3/2}
  42. 3 / 2 20 \sqrt[20]{3/2}

Equation.html

  1. 3 x + 5 y = 2 5 x + 8 y = 3 \begin{aligned}\displaystyle 3x+5y&\displaystyle=2\\ \displaystyle 5x+8y&\displaystyle=3\end{aligned}
  2. x 2 - y 2 = ( x + y ) ( x - y ) x^{2}-y^{2}=(x+y)(x-y)
  3. sin 2 ( θ ) + cos 2 ( θ ) = 1 \sin^{2}(\theta)+\cos^{2}(\theta)=1
  4. sin ( 2 θ ) = 2 sin ( θ ) cos ( θ ) \sin(2\theta)=2\sin(\theta)\cos(\theta)
  5. 3 sin ( θ ) cos ( θ ) = 1 , 3\sin(\theta)\cos(\theta)=1\,,
  6. 3 2 sin ( 2 θ ) = 1 , \frac{3}{2}\sin(2\theta)=1\,,
  7. θ = 1 2 arcsin ( 2 3 ) 20.9 . \theta=\frac{1}{2}\arcsin\left(\frac{2}{3}\right)\approx 20.9^{\circ}.
  8. x = 1 x=1
  9. x = 1. x=1.
  10. f ( s ) = s 2 f(s)=s^{2}
  11. x 2 = 1 x^{2}=1
  12. x = - 1. x=-1.
  13. P = 0 P=0
  14. P = Q P=Q
  15. x 5 - 3 x + 1 = 0 x^{5}-3x+1=0
  16. y 4 + x y 2 = x 3 3 - x y 2 + y 2 - 1 7 y^{4}+\frac{xy}{2}=\frac{x^{3}}{3}-xy^{2}+y^{2}-\frac{1}{7}
  17. 3 x + 2 y - z = 1 2 x - 2 y + 4 z = - 2 - x + 1 2 y - z = 0 \begin{aligned}\displaystyle 3x&&\displaystyle\;+&&\displaystyle 2y&&% \displaystyle\;-&&\displaystyle z&&\displaystyle\;=&&\displaystyle 1&\\ \displaystyle 2x&&\displaystyle\;-&&\displaystyle 2y&&\displaystyle\;+&&% \displaystyle 4z&&\displaystyle\;=&&\displaystyle-2&\\ \displaystyle-x&&\displaystyle\;+&&\displaystyle\tfrac{1}{2}y&&\displaystyle\;% -&&\displaystyle z&&\displaystyle\;=&&\displaystyle 0&\end{aligned}
  18. x , y , z x,y,z
  19. x = 1 y = - 2 z = - 2 \begin{aligned}\displaystyle x&\displaystyle\,=&\displaystyle 1\\ \displaystyle y&\displaystyle\,=&\displaystyle-2\\ \displaystyle z&\displaystyle\,=&\displaystyle-2\end{aligned}
  20. a x + b y + c z + d = 0 ax+by+cz+d=0
  21. a , b , c a,b,c
  22. d d
  23. x , y , z x,y,z
  24. a , b , c a,b,c
  25. 𝟚 \mathbb{R^{2}}
  26. \mathbb{R}
  27. x 2 + y 2 = z 2 x^{2}+y^{2}=z^{2}
  28. x = cos t y = sin t \begin{aligned}\displaystyle x&\displaystyle=\cos t\\ \displaystyle y&\displaystyle=\sin t\end{aligned}
  29. π \pi

Equation_of_state.html

  1. p V = constant . pV=\mathrm{constant}.\,\!
  2. V 1 T 1 = V 2 T 2 . \frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}.
  3. p total = p 1 + p 2 + + p n = p total = i = 1 n p i . p\text{total}=p_{1}+p_{2}+\cdots+p_{n}=p\text{total}=\sum_{i=1}^{n}p_{i}.
  4. p V m = R ( T C + 273.15 ) . \ pV_{m}=R(T_{C}+273.15).
  5. f ( p , V , T ) = 0. {\ f(p,V,T)=0}.
  6. p \ p
  7. V \ V
  8. n \ n
  9. V m \ V_{m}
  10. V n \frac{V}{n}
  11. T \ T
  12. R \ R
  13. p c \ p_{c}
  14. V c \ V_{c}
  15. T c \ T_{c}
  16. p V = n R T . {\ pV=nRT}.
  17. f ( p , V , T ) = p V - n R T = 0 f(p,V,T)=pV-nRT=0
  18. p = ρ ( γ - 1 ) e {\ p=\rho(\gamma-1)e}
  19. ρ \rho
  20. γ = C p / C v \gamma=C_{p}/C_{v}
  21. e = C v T e=C_{v}T
  22. C v C_{v}
  23. C p C_{p}
  24. ( p + a V m 2 ) ( V m - b ) = R T {\left(p+\frac{a}{V_{m}^{2}}\right)\left(V_{m}-b\right)=RT}
  25. V m V_{m}
  26. a a
  27. b b
  28. p c , T c p_{c},T_{c}
  29. V c V_{c}
  30. V c V_{c}
  31. a = 3 p c V c 2 a=3p_{c}\,V_{c}^{2}
  32. b = V c 3 . b=\frac{V_{c}}{3}.
  33. a = 27 ( R T c ) 2 64 p c a=\frac{27(R\,T_{c})^{2}}{64p_{c}}
  34. b = R T c 8 p c . b=\frac{R\,T_{c}}{8p_{c}}.
  35. a a
  36. b b
  37. V m V_{m}
  38. V m - b V_{m}-b
  39. V m V_{m}
  40. p + p+
  41. p p
  42. ρ \rho
  43. ρ 2 \rho^{2}
  44. 1 V m 2 \frac{1}{V_{m}^{2}}
  45. ( P r + 3 V r 2 ) ( 3 V r - 1 ) = 8 T r {\left(P_{r}+\frac{3}{V_{r}^{2}}\right)\left(3V_{r}-1\right)=8T_{r}}
  46. V r 3 - ( 1 3 + 8 T r 3 P r ) V r 2 + 3 V r P r - 1 P r = 0 {V_{r}^{3}-\left(\frac{1}{3}+\frac{8T_{r}}{3P_{r}}\right)V_{r}^{2}+\frac{3V_{r% }}{P_{r}}-\frac{1}{P_{r}}=0}
  47. a = 0.42748 R 2 T c 2.5 p c a=\frac{0.42748\,R^{2}\,T_{c}^{\,2.5}}{p_{c}}
  48. b = 0.08662 R T c p c b=\frac{0.08662\,R\,T_{c}}{p_{c}}
  49. p p c < T 2 T c . \frac{p}{p_{c}}<\frac{T}{2T_{c}}.
  50. p = R T V m - b - a α V m ( V m + b ) p=\frac{R\,T}{V_{m}-b}-\frac{a\,\alpha}{V_{m}\left(V_{m}+b\right)}
  51. a = 0.427 R 2 T c 2 P c a=\frac{0.427\,R^{2}\,T_{c}^{2}}{P_{c}}
  52. b = 0.08664 R T c P c b=\frac{0.08664\,R\,T_{c}}{P_{c}}
  53. α = ( 1 + ( 0.48508 + 1.55171 ω - 0.15613 ω 2 ) ( 1 - T r 0.5 ) ) 2 \alpha=\left(1+\left(0.48508+1.55171\,\omega-0.15613\,\omega^{2}\right)\left(1% -T_{r}^{\,0.5}\right)\right)^{2}
  54. T r = T T c T_{r}=\frac{T}{T_{c}}
  55. α \alpha
  56. α = ( 1 + ( 0.48 + 1.574 ω - 0.176 ω 2 ) ( 1 - T r 0.5 ) ) 2 \alpha=\left(1+\left(0.48+1.574\,\omega-0.176\,\omega^{2}\right)\left(1-T_{r}^% {\,0.5}\right)\right)^{2}
  57. α = 1.202 exp ( - 0.30288 T r ) . \alpha=1.202\exp\left(-0.30288\,T_{r}\right).
  58. A = a α P R 2 T 2 A=\frac{a\,\alpha\,P}{R^{2}\,T^{2}}
  59. B = b P R T B=\frac{b\,P}{R\,T}
  60. 0 = Z 3 - Z 2 + Z ( A - B - B 2 ) - A B , 0=Z^{3}-Z^{2}+Z\left(A-B-B^{2}\right)-AB\;,
  61. R R
  62. T c T_{c}
  63. p = R T V m - b - a α V m 2 + 2 b V m - b 2 p=\frac{R\,T}{V_{m}-b}-\frac{a\,\alpha}{V_{m}^{2}+2bV_{m}-b^{2}}
  64. a = 0.457235 R 2 T c 2 p c a=\frac{0.457235\,R^{2}\,T_{c}^{2}}{p_{c}}
  65. b = 0.077796 R T c p c b=\frac{0.077796\,R\,T_{c}}{p_{c}}
  66. α = ( 1 + κ ( 1 - T r 0.5 ) ) 2 \alpha=\left(1+\kappa\left(1-T_{r}^{\,0.5}\right)\right)^{2}
  67. κ = 0.37464 + 1.54226 ω - 0.26992 ω 2 \kappa=0.37464+1.54226\,\omega-0.26992\,\omega^{2}
  68. T r = T T c T_{r}=\frac{T}{T_{c}}
  69. A = a α p R 2 T 2 A=\frac{a\alpha p}{R^{2}\,T^{2}}
  70. B = b p R T B=\frac{bp}{RT}
  71. Z 3 - ( 1 - B ) Z 2 + ( A - 2 B - 3 B 2 ) Z - ( A B - B 2 - B 3 ) = 0 , Z^{3}-(1-B)\ Z^{2}+(A-2B-3B^{2})\ Z-(AB-B^{2}-B^{3})=0\;,
  72. ω \omega
  73. R R
  74. κ = κ 0 + κ 1 ( 1 + T r 0.5 ) ( 0.7 - T r ) \kappa=\kappa_{0}+\kappa_{1}\left(1+T_{r}^{0.5}\right)\left(0.7-T_{r}\right)
  75. κ 0 = 0.378893 + 1.4897153 ω - 0.17131848 ω 2 + 0.0196554 ω 3 \kappa_{0}=0.378893+1.4897153\,\omega-0.17131848\,\omega^{2}+0.0196554\,\omega% ^{3}
  76. κ 1 \,\kappa_{1}
  77. κ 1 = 0 \,\kappa_{1}=0
  78. κ = κ 0 \kappa=\kappa_{0}
  79. κ 1 \,\kappa_{1}
  80. κ = κ 0 + [ κ 1 + κ 2 ( κ 3 - T r ) ( 1 - T r 0.5 ) ] ( 1 + T r 0.5 ) ( 0.7 - T r ) \kappa=\kappa_{0}+\left[\kappa_{1}+\kappa_{2}\left(\kappa_{3}-T_{r}\right)% \left(1-T_{r}^{0.5}\right)\right]\left(1+T_{r}^{0.5}\right)\left(0.7-T_{r}\right)
  81. κ 0 = 0.378893 + 1.4897153 ω - 0.17131848 ω 2 + 0.0196554 ω 3 \kappa_{0}=0.378893+1.4897153\,\omega-0.17131848\,\omega^{2}+0.0196554\,\omega% ^{3}
  82. κ 1 \,\kappa_{1}
  83. κ 2 \,\kappa_{2}
  84. κ 3 \,\kappa_{3}
  85. p V m R T = Z = 1 + Z rep + Z att \frac{pV_{m}}{RT}=Z=1+Z^{\rm{rep}}+Z^{\rm{att}}
  86. Z rep = 4 c η 1 - 1.9 η Z^{\rm{rep}}=\frac{4c\eta}{1-1.9\eta}
  87. Z att = - z m q η Y 1 + k 1 η Y Z^{\rm{att}}=-\frac{z_{m}q\eta Y}{1+k_{1}\eta Y}
  88. c c
  89. c = 1 c=1
  90. c = 1 + 3.535 ω + 0.533 ω 2 c=1+3.535\omega+0.533\omega^{2}
  91. ω \omega
  92. η \eta
  93. η = v * n V \eta=\frac{v^{*}n}{V}
  94. v * v^{*}
  95. n n
  96. V V
  97. c c
  98. v * = k T c P c Φ v^{*}=\frac{kT_{c}}{P_{c}}\Phi
  99. Φ = 0.0312 + 0.087 ( c - 1 ) + 0.008 ( c - 1 ) 2 1.000 + 2.455 ( c - 1 ) + 0.732 ( c - 1 ) 2 \Phi=\frac{0.0312+0.087(c-1)+0.008(c-1)^{2}}{1.000+2.455(c-1)+0.732(c-1)^{2}}
  100. k k
  101. η \eta
  102. η = R T c P c Φ 1 V m . \eta=\frac{RT_{c}}{P_{c}}\Phi\frac{1}{V_{m}}.
  103. q q
  104. Y Y
  105. q = 1 + k 3 ( c - 1 ) q=1+k_{3}(c-1)
  106. Y = exp ( ϵ k T ) - k 2 Y=\exp\left(\frac{\epsilon}{kT}\right)-k_{2}
  107. ϵ \epsilon
  108. ϵ k = 1.000 + 0.945 ( c - 1 ) + 0.134 ( c - 1 ) 2 1.023 + 2.225 ( c - 1 ) + 0.478 ( c - 1 ) 2 \frac{\epsilon}{k}=\frac{1.000+0.945(c-1)+0.134(c-1)^{2}}{1.023+2.225(c-1)+0.4% 78(c-1)^{2}}
  109. z m z_{m}
  110. k 1 k_{1}
  111. k 2 k_{2}
  112. k 3 k_{3}
  113. z m = 9.49 z_{m}=9.49
  114. k 1 = 1.7745 k_{1}=1.7745
  115. k 2 = 1.0617 k_{2}=1.0617
  116. k 3 = 1.90476. k_{3}=1.90476.
  117. p ( V - b ) = R T e - a / R T V \ p(V-b)=RTe^{-a/RTV}
  118. T c = a 4 R b , p c = a 4 b 2 e 2 , V c = 2 b . \ T_{c}=\frac{a}{4Rb},\ p_{c}=\frac{a}{4b^{2}e^{2}},\ V_{c}=2b.
  119. p V m R T = 1 + B V m + C V m 2 + D V m 3 + \frac{pV_{m}}{RT}=1+\frac{B}{V_{m}}+\frac{C}{V_{m}^{2}}+\frac{D}{V_{m}^{3}}+\dots
  120. B = - V c B=-V_{c}\,
  121. C = V c 2 9 C=\frac{V_{c}^{2}}{9}
  122. B = b - a R T . B=b-\frac{a}{RT}.
  123. p = ρ R T + ( B 0 R T - A 0 - C 0 T 2 + D 0 T 3 - E 0 T 4 ) ρ 2 + ( b R T - a - d T ) ρ 3 + α ( a + d T ) ρ 6 + c ρ 3 T 2 ( 1 + γ ρ 2 ) exp ( - γ ρ 2 ) p=\rho RT+\left(B_{0}RT-A_{0}-\frac{C_{0}}{T^{2}}+\frac{D_{0}}{T^{3}}-\frac{E_% {0}}{T^{4}}\right)\rho^{2}+\left(bRT-a-\frac{d}{T}\right)\rho^{3}+\alpha\left(% a+\frac{d}{T}\right)\rho^{6}+\frac{c\rho^{3}}{T^{2}}\left(1+\gamma\rho^{2}% \right)\exp\left(-\gamma\rho^{2}\right)
  124. a ( T , ρ ) R T = a o ( T , ρ ) + a r ( T , ρ ) R T = α o ( τ , δ ) + α r ( τ , δ ) \frac{a(T,\rho)}{RT}=\frac{a^{o}(T,\rho)+a^{r}(T,\rho)}{RT}=\alpha^{o}(\tau,% \delta)+\alpha^{r}(\tau,\delta)
  125. τ = T r T , δ = ρ ρ r \tau=\frac{T_{r}}{T},\delta=\frac{\rho}{\rho_{r}}
  126. p = ρ ( γ - 1 ) e - γ p 0 p=\rho(\gamma-1)e-\gamma p^{0}\,
  127. e e
  128. γ \gamma
  129. p 0 p^{0}
  130. c 2 = γ ( p + p 0 ) / ρ c^{2}=\gamma(p+p^{0})/\rho
  131. p = ρ m c s 2 p=\rho_{m}c_{s}^{2}
  132. p p
  133. ρ m \rho_{m}
  134. c s c_{s}
  135. p V m = R T Li α + 1 ( z ) ζ ( α ) ( T T c ) α pV_{m}=RT~{}\frac{\textrm{Li}_{\alpha+1}(z)}{\zeta(\alpha)}\left(\frac{T}{T_{c% }}\right)^{\alpha}
  136. p = A ( 1 - ω R 1 V ) exp ( - R 1 V ) + B ( 1 - ω R 2 V ) exp ( - R 2 V ) + ω e 0 V p=A\cdot\left(1-\frac{\omega}{R_{1}\cdot V}\right)\cdot\exp(-R_{1}\cdot V)+B% \cdot\left(1-\frac{\omega}{R_{2}\cdot V}\right)\cdot\exp(-R_{2}\cdot V)+\frac{% \omega\cdot e_{0}}{V}
  137. V = ρ e / ρ V=\rho_{e}/\rho
  138. ρ e \rho_{e}
  139. ρ \rho
  140. A A
  141. B B
  142. R 1 R_{1}
  143. R 2 R_{2}
  144. ω \omega
  145. ρ 0 \rho_{0}
  146. V D V_{D}
  147. P C J P_{CJ}
  148. e 0 e_{0}
  149. ρ 0 \rho_{0}\,
  150. v D v_{D}\,
  151. p C J p_{CJ}\,
  152. A A\,
  153. B B\,
  154. R 1 R_{1}\,
  155. R 2 R_{2}\,
  156. ω \omega\,
  157. e 0 e_{0}\,
  158. p ( V ) = 3 K 0 ( 1 - η η 2 ) exp [ 3 2 ( K 0 - 1 ) ( 1 - η ) ] , η := ( V / V 0 ) 1 / 3 , K 0 := d K 0 d p p(V)=3K_{0}\left(\frac{1-\eta}{\eta^{2}}\right)\exp\left[\tfrac{3}{2}\left(K_{% 0}^{\prime}-1\right)(1-\eta)\right]~{},~{}~{}\eta:=(V/V_{0})^{1/3}~{},~{}~{}K_% {0}^{\prime}:=\frac{dK_{0}}{dp}
  159. p ( V ) = K 0 K 0 [ ( V V 0 ) - K 0 - 1 ] . p(V)=\frac{K_{0}}{K_{0}^{\prime}}\left[\left(\frac{V}{V_{0}}\right)^{-K_{0}^{% \prime}}-1\right]\,.
  160. p ( V ) = 3 K 0 2 [ ( V 0 V ) 7 3 - ( V 0 V ) 5 3 ] { 1 + 3 4 ( K 0 - 4 ) [ ( V 0 V ) 2 3 - 1 ] } . p(V)=\frac{3K_{0}}{2}\left[\left(\frac{V_{0}}{V}\right)^{\frac{7}{3}}-\left(% \frac{V_{0}}{V}\right)^{\frac{5}{3}}\right]\left\{1+\frac{3}{4}\left(K_{0}^{% \prime}-4\right)\left[\left(\frac{V_{0}}{V}\right)^{\frac{2}{3}}-1\right]% \right\}.
  161. p ( V ) = { k 1 ξ + k 2 ξ 2 + k 3 ξ 3 + Δ p Compression k 1 ξ Tension ; ξ := V 0 V - 1 p(V)=\begin{cases}k_{1}~{}\xi+k_{2}~{}\xi^{2}+k_{3}~{}\xi^{3}+\Delta p&\qquad% \,\text{Compression}\\ k_{1}~{}\xi&\qquad\,\text{Tension}\end{cases}~{};~{}~{}\xi:=\cfrac{V_{0}}{V}-1
  162. p ( V ) - p 0 = Γ V ( e - e 0 ) p(V)-p_{0}=\frac{\Gamma}{V}(e-e_{0})
  163. p ( V ) = - β ( V V 0 ) n ln ( V V 0 ) p(V)=-\beta\left(\frac{V}{V_{0}}\right)^{n}\ln\left(\frac{V}{V_{0}}\right)
  164. β = K 0 \beta=K_{0}
  165. V 0 V_{0}
  166. n n
  167. n = - 1 6 - γ G n=-\frac{1}{6}-\gamma_{G}

Equations_of_motion.html

  1. M [ 𝐫 ( t ) , 𝐫 ˙ ( t ) , 𝐫 ¨ ( t ) , t ] = 0 M\left[\mathbf{r}(t),\mathbf{\dot{r}}(t),\mathbf{\ddot{r}}(t),t\right]=0
  2. 𝐫 ( 0 ) , 𝐫 ˙ ( 0 ) . \mathbf{r}(0),\quad\mathbf{\dot{r}}(0).
  3. M ~ [ 𝐩 ( t ) , 𝐩 ˙ ( t ) , 𝐩 ¨ ( t ) , t ] = 0 \tilde{M}\left[\mathbf{p}(t),\mathbf{\dot{p}}(t),\mathbf{\ddot{p}}(t),t\right]=0
  4. 𝐩 ( 0 ) , 𝐩 ˙ ( 0 ) . \mathbf{p}(0),\quad\mathbf{\dot{p}}(0).
  5. M [ 𝐪 ( t ) , 𝐪 ˙ ( t ) , t ] = 0 , M ~ [ 𝐩 ( t ) , 𝐩 ˙ ( t ) , t ] = 0 M\left[\mathbf{q}(t),\mathbf{\dot{q}}(t),t\right]=0,\quad\tilde{M}\left[% \mathbf{p}(t),\mathbf{\dot{p}}(t),t\right]=0
  6. 𝐯 = d 𝐫 d t , 𝐚 = d 𝐯 d t = d 2 𝐫 d t 2 \mathbf{v}=\frac{d\mathbf{r}}{dt},\quad\mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac% {d^{2}\mathbf{r}}{dt^{2}}\,\!
  7. s y m b o l ω = 𝐧 ^ d θ d t , s y m b o l α = d ω d t = 𝐧 ^ d 2 θ d t 2 symbol{\omega}=\mathbf{\hat{n}}\frac{d\theta}{dt},\quad symbol{\alpha}=\frac{d% \mathbf{\omega}}{dt}=\mathbf{\hat{n}}\frac{d^{2}\theta}{dt^{2}}\,\!
  8. 𝐧 ^ = 𝐞 ^ r × 𝐞 ^ θ \mathbf{\hat{n}}=\mathbf{\hat{e}}_{r}\times\mathbf{\hat{e}}_{\theta}\,\!
  9. 𝐞 ^ r \scriptstyle\mathbf{\hat{e}}_{r}\,\!
  10. 𝐞 ^ θ \scriptstyle\mathbf{\hat{e}}_{\theta}\,\!
  11. 𝐯 = s y m b o l ω × 𝐫 \mathbf{v}=symbol{\omega}\times\mathbf{r}\,\!
  12. 𝐚 = s y m b o l α × 𝐫 + s y m b o l ω × 𝐯 \mathbf{a}=symbol{\alpha}\times\mathbf{r}+symbol{\omega}\times\mathbf{v}\,\!
  13. v = a t + v 0 [ 1 ] \begin{aligned}\displaystyle v&\displaystyle=at+v_{0}\quad[1]\\ \end{aligned}
  14. r = r 0 + v 0 t + a t 2 2 [ 2 ] \begin{aligned}\displaystyle r&\displaystyle=r_{0}+v_{0}t+\frac{{a}t^{2}}{2}% \quad[2]\\ \end{aligned}
  15. r = r 0 + ( v + v 0 2 ) t [ 3 ] v 2 = v 0 2 + 2 a ( r - r 0 ) [ 4 ] r = r 0 + v t - a t 2 2 [ 5 ] \begin{aligned}\displaystyle r&\displaystyle=r_{0}+\left(\frac{v+v_{0}}{2}% \right)t\quad[3]\\ \displaystyle v^{2}&\displaystyle=v_{0}^{2}+2a\left(r-r_{0}\right)\quad[4]\\ \displaystyle r&\displaystyle=r_{0}+vt-\frac{{a}t^{2}}{2}\quad[5]\\ \end{aligned}
  16. r 0 r_{0}
  17. r r
  18. v 0 v_{0}
  19. v v
  20. a a
  21. t t
  22. 𝐯 = 𝐚 d t = 𝐚 t + 𝐯 0 [ 1 ] 𝐫 = ( 𝐚 t + 𝐯 0 ) d t = 𝐚 t 2 2 + 𝐯 0 t + 𝐫 0 [ 2 ] \begin{aligned}\displaystyle\mathbf{v}&\displaystyle=\int\mathbf{a}dt=\mathbf{% a}t+\mathbf{v}_{0}\quad[1]\\ \displaystyle\mathbf{r}&\displaystyle=\int(\mathbf{a}t+\mathbf{v}_{0})dt=\frac% {\mathbf{a}t^{2}}{2}+\mathbf{v}_{0}t+\mathbf{r}_{0}\quad[2]\\ \end{aligned}
  23. v \displaystyle v
  24. 𝐯 + 𝐯 0 2 \frac{\mathbf{v}+\mathbf{v}_{0}}{2}
  25. 𝐫 = ( 𝐯 + 𝐯 0 2 ) t [ 3 ] \mathbf{r}=\left(\frac{\mathbf{v}+\mathbf{v}_{0}}{2}\right)t\quad[3]\,\!
  26. r = r 0 + ( v + v 0 2 ) t [ 3 ] r=r_{0}+\left(\frac{v+v_{0}}{2}\right)t\quad[3]\,\!
  27. t = ( r - r 0 ) ( 2 v + v 0 ) t=\left(r-r_{0}\right)\left(\frac{2}{v+v_{0}}\right)\,\!
  28. v \displaystyle v
  29. 2 ( r - r 0 ) - v t = v 0 t 2\left(r-r_{0}\right)-vt=v_{0}t\,\!
  30. r = a t 2 2 + 2 r - 2 r 0 - v t + r 0 0 = a t 2 2 + r - r 0 - v t r = r 0 + v t - a t 2 2 [ 5 ] \begin{aligned}\displaystyle r&\displaystyle=\frac{{a}t^{2}}{2}+2r-2r_{0}-vt+r% _{0}\\ \displaystyle 0&\displaystyle=\frac{{a}t^{2}}{2}+r-r_{0}-vt\\ \displaystyle r&\displaystyle=r_{0}+vt-\frac{{a}t^{2}}{2}\quad[5]\end{aligned}\,\!
  31. v \displaystyle v
  32. 𝐯 \displaystyle\mathbf{v}
  33. v 2 = 𝐯 𝐯 = ( 𝐯 0 + 𝐚 t ) ( 𝐯 0 + 𝐚 t ) = v 0 2 + 2 t ( 𝐚 𝐯 0 ) + a 2 t 2 v^{2}=\mathbf{v}\cdot\mathbf{v}=(\mathbf{v}_{0}+\mathbf{a}t)\cdot(\mathbf{v}_{% 0}+\mathbf{a}t)=v_{0}^{2}+2t(\mathbf{a}\cdot\mathbf{v}_{0})+a^{2}t^{2}
  34. ( 2 𝐚 ) ( 𝐫 - 𝐫 0 ) = ( 2 𝐚 ) ( 𝐯 0 t + 1 2 𝐚 t 2 ) = 2 t ( 𝐚 𝐯 0 ) + a 2 t 2 = v 2 - v 0 2 (2\mathbf{a})\cdot(\mathbf{r}-\mathbf{r}_{0})=(2\mathbf{a})\cdot\left(\mathbf{% v}_{0}t+\frac{1}{2}\mathbf{a}t^{2}\right)=2t(\mathbf{a}\cdot\mathbf{v}_{0})+a^% {2}t^{2}=v^{2}-v_{0}^{2}
  35. v 2 = v 0 2 + 2 ( 𝐚 ( 𝐫 - 𝐫 0 ) ) \therefore v^{2}=v_{0}^{2}+2(\mathbf{a}\cdot(\mathbf{r}-\mathbf{r}_{0}))
  36. s = v 2 - u 2 - 2 g . s=\frac{v^{2}-u^{2}}{-2g}.
  37. s = u 2 2 g . s=\frac{u^{2}}{2g}.
  38. ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 θ = θ 0 + 1 2 ( ω 0 + ω ) t ω 2 = ω 0 2 + 2 α ( θ - θ 0 ) θ = θ 0 + ω t - 1 2 α t 2 \begin{aligned}\displaystyle\omega&\displaystyle=\omega_{0}+\alpha t\\ \displaystyle\theta&\displaystyle=\theta_{0}+\omega_{0}t+\tfrac{1}{2}\alpha t^% {2}\\ \displaystyle\theta&\displaystyle=\theta_{0}+\tfrac{1}{2}(\omega_{0}+\omega)t% \\ \displaystyle\omega^{2}&\displaystyle=\omega_{0}^{2}+2\alpha(\theta-\theta_{0}% )\\ \displaystyle\theta&\displaystyle=\theta_{0}+\omega t-\tfrac{1}{2}\alpha t^{2}% \\ \end{aligned}\,\!
  39. 𝐫 = 𝐫 ( r ( t ) , θ ( t ) ) = r 𝐞 ^ r 𝐯 = 𝐞 ^ r d r d t + r ω 𝐞 ^ θ 𝐚 = ( d 2 r d t 2 - r ω 2 ) 𝐞 ^ r + ( r α + 2 ω d r d t ) 𝐞 ^ θ \begin{aligned}\displaystyle\mathbf{r}&\displaystyle=\mathbf{r}\left(r(t),% \theta(t)\right)=r\mathbf{\hat{e}}_{r}\\ \displaystyle\mathbf{v}&\displaystyle=\mathbf{\hat{e}}_{r}\frac{dr}{dt}+r% \omega\mathbf{\hat{e}}_{\theta}\\ \displaystyle\mathbf{a}&\displaystyle=\left(\frac{d^{2}r}{dt^{2}}-r\omega^{2}% \right)\mathbf{\hat{e}}_{r}+\left(r\alpha+2\omega\frac{dr}{dt}\right)\mathbf{% \hat{e}}_{\theta}\end{aligned}\,\!
  40. 𝐞 ^ r , 𝐞 ^ θ , \scriptstyle\mathbf{\hat{e}}_{r},\mathbf{\hat{e}}_{\theta},\,\!
  41. 𝐞 ^ r , 𝐞 ^ θ , 𝐞 ^ ϕ \scriptstyle\mathbf{\hat{e}}_{r},\mathbf{\hat{e}}_{\theta},\mathbf{\hat{e}}_{% \phi}\,\!
  42. 𝐫 = 𝐫 ( t ) = r 𝐞 ^ r 𝐯 = v 𝐞 ^ r + r d θ d t 𝐞 ^ θ + r d ϕ d t sin θ 𝐞 ^ ϕ 𝐚 = ( a - r ( d θ d t ) 2 - r ( d ϕ d t ) 2 sin 2 θ ) 𝐞 ^ r + ( r d 2 θ d t 2 + 2 v d θ d t - r ( d ϕ d t ) 2 sin θ cos θ ) 𝐞 ^ θ + ( r d 2 ϕ d t 2 sin θ + 2 v d ϕ d t sin θ + 2 r d θ d t d ϕ d t cos θ ) 𝐞 ^ ϕ \begin{aligned}\displaystyle\mathbf{r}&\displaystyle=\mathbf{r}\left(t\right)=% r\mathbf{\hat{e}}_{r}\\ \displaystyle\mathbf{v}&\displaystyle=v\mathbf{\hat{e}}_{r}+r\,\frac{d\theta}{% dt}\mathbf{\hat{e}}_{\theta}+r\,\frac{d\phi}{dt}\,\sin\theta\mathbf{\hat{e}}_{% \phi}\\ \displaystyle\mathbf{a}&\displaystyle=\left(a-r\left(\frac{d\theta}{dt}\right)% ^{2}-r\left(\frac{d\phi}{dt}\right)^{2}\sin^{2}\theta\right)\mathbf{\hat{e}}_{% r}\\ &\displaystyle+\left(r\frac{d^{2}\theta}{dt^{2}}+2v\frac{d\theta}{dt}-r\left(% \frac{d\phi}{dt}\right)^{2}\sin\theta\cos\theta\right)\mathbf{\hat{e}}_{\theta% }\\ &\displaystyle+\left(r\frac{d^{2}\phi}{dt^{2}}\,\sin\theta+2v\,\frac{d\phi}{dt% }\,\sin\theta+2r\,\frac{d\theta}{dt}\,\frac{d\phi}{dt}\,\cos\theta\right)% \mathbf{\hat{e}}_{\phi}\end{aligned}\,\!
  43. d 2 x d t 2 = - ω 2 x \frac{d^{2}x}{dt^{2}}=-\omega^{2}x
  44. ω = 2 π f = 2 π / T \omega=2\pi f=2\pi/T
  45. d 2 x d t 2 = - n ω n 2 x \frac{d^{2}x}{dt^{2}}=-\sum_{n}\omega_{n}^{2}x
  46. d 2 𝐫 d t 2 = - n ω n 2 𝐫 n \frac{d^{2}\mathbf{r}}{dt^{2}}=-\sum_{n}\omega_{n}^{2}\mathbf{r}_{n}
  47. d 2 θ d t 2 = - ω 2 θ \frac{d^{2}\theta}{dt^{2}}=-\omega^{2}\theta
  48. d 2 θ d t 2 = - n ω n 2 θ \frac{d^{2}\theta}{dt^{2}}=-\sum_{n}\omega_{n}^{2}\theta
  49. 𝐅 = d 𝐩 d t \mathbf{F}=\frac{d\mathbf{p}}{dt}
  50. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  51. d 𝐩 i d t = 𝐅 E + i j 𝐅 i j \frac{d\mathbf{p}_{i}}{dt}=\mathbf{F}_{E}+\sum_{i\neq j}\mathbf{F}_{ij}\,\!
  52. τ = d 𝐋 d t \mathbf{\tau}=\frac{d\mathbf{L}}{dt}\,\!
  53. τ = 𝐈 s y m b o l α \mathbf{\tau}=\mathbf{I}\cdot symbol{\alpha}
  54. d 𝐋 i d t = τ E + i j τ i j \frac{d\mathbf{L}_{i}}{dt}=\mathbf{\tau}_{E}+\sum_{i\neq j}\mathbf{\tau}_{ij}\,\!
  55. - m g sin θ = m d 2 ( θ ) d t 2 d 2 θ d t 2 = - g sin θ -mg\sin\theta=m\frac{d^{2}(\ell\theta)}{dt^{2}}\quad\Rightarrow\quad\frac{d^{2% }\theta}{dt^{2}}=-\frac{g}{\ell}\sin\theta\,\!
  56. F 0 sin ( ω t ) = m ( d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x ) , F_{0}\sin(\omega t)=m\left(\frac{d^{2}x}{dt^{2}}+2\zeta\omega_{0}\frac{dx}{dt}% +\omega_{0}^{2}x\right),
  57. - G m M | 𝐫 | 2 𝐞 ^ r + 𝐑 = m d 2 𝐫 d t 2 + 0 d 2 𝐫 d t 2 = - G M | 𝐫 | 2 𝐞 ^ r + 𝐀 -\frac{GmM}{|\mathbf{r}|^{2}}\mathbf{\hat{e}}_{r}+\mathbf{R}=m\frac{d^{2}% \mathbf{r}}{dt^{2}}+0\quad\Rightarrow\quad\frac{d^{2}\mathbf{r}}{dt^{2}}=-% \frac{GM}{|\mathbf{r}|^{2}}\mathbf{\hat{e}}_{r}+\mathbf{A}\,\!
  58. 𝐪 ˙ = d 𝐪 / d t \mathbf{\dot{q}}=d\mathbf{q}/dt
  59. 𝐩 = L / 𝐪 ˙ = S / 𝐪 \mathbf{p}=\partial L/\partial\mathbf{\dot{q}}=\partial S/\partial\mathbf{q}
  60. L = L [ 𝐪 ( t ) , 𝐪 ˙ ( t ) , t ] L=L\left[\mathbf{q}(t),\mathbf{\dot{q}}(t),t\right]
  61. H = H [ 𝐪 ( t ) , 𝐩 ( t ) , t ] H=H\left[\mathbf{q}(t),\mathbf{p}(t),t\right]
  62. S [ 𝐪 , t ] = t 1 t 2 L ( 𝐪 , 𝐪 ˙ , t ) d t S[\mathbf{q},t]=\int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)dt
  63. δ S = 0 \delta S=0
  64. d d t ( L 𝐪 ˙ ) = L 𝐪 \frac{d}{dt}\left(\frac{\partial L}{\partial\mathbf{\dot{q}}}\right)=\frac{% \partial L}{\partial\mathbf{q}}
  65. 𝐩 ˙ = - H 𝐪 𝐪 ˙ = + H 𝐩 \mathbf{\dot{p}}=-\frac{\partial H}{\partial\mathbf{q}}\quad\mathbf{\dot{q}}=+% \frac{\partial H}{\partial\mathbf{p}}
  66. 𝐩 𝐪 , H - H . \mathbf{p}\rightleftharpoons\mathbf{q},\quad H\rightarrow-H.
  67. - S ( 𝐪 , t ) t = H ( 𝐪 , 𝐩 , t ) -\frac{\partial S(\mathbf{q},t)}{\partial t}=H\left(\mathbf{q},\mathbf{p},t\right)
  68. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\,\!
  69. m d 2 𝐫 d t 2 = q ( 𝐄 + d 𝐫 d t × 𝐁 ) m\frac{d^{2}\mathbf{r}}{dt^{2}}=q\left(\mathbf{E}+\frac{d\mathbf{r}}{dt}\times% \mathbf{B}\right)\,\!
  70. d 𝐩 d t = q ( 𝐄 + 𝐩 × 𝐁 m ) \frac{d\mathbf{p}}{dt}=q\left(\mathbf{E}+\frac{\mathbf{p}\times\mathbf{B}}{m}% \right)\,\!
  71. L = m 2 𝐫 ˙ 𝐫 ˙ + q 𝐀 𝐫 ˙ - q ϕ L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{% \dot{r}}-q\phi
  72. 𝐏 = L 𝐫 ˙ = m 𝐫 ˙ + q 𝐀 \mathbf{P}=\frac{\partial L}{\partial\mathbf{\dot{r}}}=m\mathbf{\dot{r}}+q% \mathbf{A}
  73. H = ( 𝐏 - q 𝐀 ) 2 2 m - q ϕ H=\frac{\left(\mathbf{P}-q\mathbf{A}\right)^{2}}{2m}-q\phi\,\!
  74. d s = g α β d x α d x β ds=\sqrt{g_{\alpha\beta}dx^{\alpha}dx^{\beta}}
  75. d 2 x μ d s 2 = - Γ μ d x α d s α β d x β d s \frac{d^{2}x^{\mu}}{ds^{2}}=-\Gamma^{\mu}{}_{\alpha\beta}\frac{dx^{\alpha}}{ds% }\frac{dx^{\beta}}{ds}
  76. d 2 ξ α d s 2 = - R α d x α d s β γ δ ξ γ d x δ d s \frac{d^{2}\xi^{\alpha}}{ds^{2}}=-R^{\alpha}{}_{\beta\gamma\delta}\frac{dx^{% \alpha}}{ds}\xi^{\gamma}\frac{dx^{\delta}}{ds}
  77. 1 v 2 2 X t 2 = 2 X \frac{1}{v^{2}}\frac{\partial^{2}X}{\partial t^{2}}=\nabla^{2}X
  78. i Ψ t = H ^ Ψ , i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi\,,
  79. H ^ \hat{H}

Equatorial_coordinate_system.html

  1. δ δ
  2. α α
  3. x x
  4. y y
  5. z z
  6. z z
  7. x x
  8. y y
  9. z z
  10. X X
  11. Y Y
  12. Z Z
  13. R R
  14. ( = [ u r a d i c a l , u X , u \xb 2 + , u Y , u \xb 2 + , u Z , u \xb 2 ] ) (=[u^{\prime}radical^{\prime},u^{\prime}X^{\prime},u^{\prime}\xb 2+^{\prime},u% ^{\prime}Y^{\prime},u^{\prime}\xb 2+^{\prime},u^{\prime}Z^{\prime},u^{\prime}% \xb 2^{\prime}])
  15. ξ ξ
  16. η η
  17. ζ ζ
  18. Δ Δ
  19. ( = [ u r a d i c a l , u 3 b e , u \xb 2 + , u 3 b 7 , u \xb 2 + , u 3 b 6 , u \xb 2 ] ) (=[u^{\prime}radical^{\prime},u^{\prime}\u{0}3be^{\prime},u^{\prime}\xb 2+^{% \prime},u^{\prime}\u{0}3b7^{\prime},u^{\prime}\xb 2+^{\prime},u^{\prime}\u{0}3% b6^{\prime},u^{\prime}\xb 2^{\prime}])
  20. X R {X\over R}
  21. ξ Δ = cos δ cos α {\xi\over\mathit{\Delta}}=\cos\delta\cos\alpha
  22. Y R {Y\over R}
  23. η Δ = cos δ sin α {\eta\over\mathit{\Delta}}=\cos\delta\sin\alpha
  24. Z R {Z\over R}
  25. ζ Δ = sin δ {\zeta\over\mathit{\Delta}}=\sin\delta
  26. x x
  27. y y
  28. z z
  29. I I
  30. J J
  31. K K
  32. I ^ \hat{I}
  33. J ^ \hat{J}
  34. K ^ \hat{K}
  35. α \alpha
  36. δ \delta
  37. Δ \mathit{\Delta}
  38. ξ \xi
  39. η \eta
  40. ζ \zeta
  41. X X
  42. Y Y
  43. Z Z
  44. x x
  45. y y
  46. z z
  47. x x
  48. y y
  49. z z
  50. x x
  51. y y
  52. z z
  53. ξ ξ
  54. η η
  55. ζ ζ
  56. ξ = x + X \xi=x+X
  57. η = y + Y \eta=y+Y
  58. ζ = z + Z \zeta=z+Z

Equivalence_class.html

  1. X X
  2. ~{}
  3. X X
  4. a a
  5. X X
  6. X X
  7. a a
  8. X X
  9. X X
  10. ~{}
  11. X / X/~{}
  12. X X
  13. ~{}
  14. a a
  15. X X
  16. a a a~{}a
  17. a a
  18. b b
  19. X X
  20. a b a~{}b
  21. b a b~{}a
  22. a a
  23. b b
  24. c c
  25. X X
  26. a b a~{}b
  27. b c b~{}c
  28. a c a~{}c
  29. a a
  30. a a aa
  31. [ a ] = { x X a x } [a]=\{x\in X\mid a\sim x\}
  32. a a
  33. ~{}
  34. a a
  35. R R
  36. R R
  37. a a
  38. X X
  39. R R
  40. X / R X/R
  41. X X
  42. R R
  43. X X
  44. R R
  45. x [ x ] x\mapsto[x]
  46. X X
  47. X / R X/R
  48. s s
  49. s s ( c ) = c ss(c)=c
  50. c c
  51. s ( c ) s(c)
  52. c c
  53. a b a~{}b
  54. a b a−b
  55. n n
  56. n n
  57. a m o d n amodn
  58. a a
  59. n n
  60. X X
  61. ~{}
  62. X / X/~{}
  63. X / X/~{}
  64. X X
  65. ~{}
  66. 𝐙 \mathbf{Z}
  67. x y x~{}y
  68. x y x−y
  69. 77 77
  70. 99 99
  71. 11 11
  72. 𝐙 / \mathbf{Z}/~{}
  73. X X
  74. ( a , b ) (a,b)
  75. b b
  76. ~{}
  77. X X
  78. ( a , b ) ( c , d ) (a,b)~{}(c,d)
  79. a d = b c ad=bc
  80. ( a , b ) (a,b)
  81. a / b a/b
  82. X X
  83. x x
  84. X X
  85. x x xx
  86. x x xx
  87. y y yy
  88. X X
  89. X X
  90. X X
  91. X X
  92. x y x~{}y
  93. x x
  94. y y
  95. x y x~{}y
  96. x x = y y xx=yy
  97. ~{}
  98. X X
  99. x x
  100. y y
  101. X X
  102. x y x\sim y
  103. [ x ] = [ y ] [x]=[y]
  104. [ x ] [ y ] . [x]\cap[y]\neq\emptyset.
  105. ~{}
  106. X X
  107. X X
  108. s s
  109. t t
  110. s t s~{}t
  111. ~{}
  112. X X
  113. P ( x ) P(x)
  114. X X
  115. x y x~{}y
  116. P ( x ) P(x)
  117. P ( y ) P(y)
  118. P P
  119. ~{}
  120. ~{}
  121. f f
  122. X X
  123. Y Y
  124. f ( x < s u b > 1 ) = f ( x 2 ) f(x<sub>1)=f(x_{2})

Equivalence_relation.html

  1. [ a ] = { b X a b } [a]=\{b\in X\mid a\sim b\}
  2. { a , b , c } \{a,b,c\}
  3. { ( a , a ) , ( b , b ) , ( c , c ) , ( b , c ) , ( c , b ) } \{(a,a),(b,b),(c,c),(b,c),(c,b)\}
  4. [ a ] = { a } , [ b ] = [ c ] = { b , c } [a]=\{a\},~{}~{}~{}~{}[b]=[c]=\{b,c\}
  5. { { a } , { b , c } } \{\{a\},\{b,c\}\}
  6. a , b X a,b\in X
  7. [ a ] := { x X a x } [a]:=\{x\in X\mid a\sim x\}
  8. X / := { [ x ] x X } X/\mathord{\sim}:=\{[x]\mid x\in X\}
  9. π : X X / \pi:X\to X/\mathord{\sim}
  10. π ( x ) = [ x ] \pi(x)=[x]
  11. x y f ( x ) = f ( y ) x\sim y\iff f(x)=f(y)
  12. B n = 1 e k = 0 k n k ! , B_{n}=\frac{1}{e}\sum_{k=0}^{\infty}\frac{k^{n}}{k!},
  13. \square
  14. {}_{\Box}