P=2pieG/3*P^2*R^2
\frac{\partial v_x}{\partial x} =-\frac{2x^2}{(x^2+y^2+z^2)^2}+\frac{1}{x^2+y^2+z^2} = \frac{-2x^2}{r^4} + \frac{1}{r^2}
\frac{\partial v_y}{\partial y} =-\frac{2y^2}{r^4}+\frac{1}{r^2}
=\frac{-2x^2-2y^2-2z^2}{r^4}+\frac{3}{r^2}=\frac{-2r^2}{r^4}+\frac{3}{r^2}=\frac{1}{r^2}
f(x) = \frac{ \int_{-\infty}^\infty \sin(x^2)\,dx}{\Gamma(x)}
{\lfloor}\int_{4}^{4+4} cosh (\theta + \theta + \theta){\rfloor}
\begin{align*} \nabla \times \vec E = -\frac{\partial \vec B}{\partial t} \\ \nabla \times \vec H = \vec J +\frac{\partial \vec D}{\partial t} \end{align*}
t= \frac {\frac{2\frac{2\sqrt{2}Vo}{2}\frac{2\sqrt{2}Vo}{2} \frac{2\sqrt{2}Vo}{2}\sqrt{2}Vo}{2}\frac{2\sqrt{2} Vo}{2}\frac{2\sqrt{2}Vo}{2}\frac{2\sqrt{2}Vo}{2} }{g}
t= \frac{Vo\sqrt{2}}{g}
\begin{array}{r l r l} u &= \ln x \quad & dv &= x\,dx \\ du &= \frac{1}{x}\,dx & v &= \frac{1}{2} x^2 \end{array}
\langle x^1 | x^2 \rangle = \langle 0 | \varphi (x^1) \varphi ^{\dagger} (x^2) | 0 \rangle = \delta (x^1 - x^2)
\langle p_1 | p_2 \rangle = \langle 0 | a(p_1) a^{\dagger} (p_2) | 0 \rangle = \delta (p_1 - p_2)
(AB)C=([a_{ij}]\times[b_{ij}])\times[c_{ij}]=[\sum_{k=1}^{n}a_{ik}b_{kj}]\times[c_{ij}]=[\sum_{l=1}^{p}(\sum_{k=1}^{n}a_{ik}b_{kl})c_{lj}]=[\sum_{l=1}^{p}(\sum_{k=1}^{n}a_{ik}b_{kl}c_{lj})]
=[\sum_{k=1}^{n}(\sum_{l=1}^{p}a_{ik}b_{kl}c_{lj})]=[\sum_{k=1}^{n}a_{ik}(\sum_{l=1}^{p}b_{kl}c_{lj})]=[a_{ik}]\times[\sum_{l=1}^{p}b_{kl}c_{lj}]=[a_{ik}]\times([b_{ik}]\times[c_{ik}])=A(BC)
\int_{1}^{\sqrt[3]{3}}z^2dz \times \cos{\frac{3\pi}{9}}=\ln{\sqrt[3]{e}}
\int \frac{3}{4+3x^2}dx=\frac{1}{2}\sqrt{3}\arctan(\frac{x\sqrt{3}}{2})+C
\begin{equation*}\begin{split}\mbox{Variable} \quad &\mbox{Meaning}\\ Q[i,k] \quad &\mbox{At time i the read-write head blah blah blah}\\ &\mbox{blah blah blah}\\ H[i,j] \quad &\mbox{At time i}\end{split}\end{equation*}
\begin{array}{ll}\mbox{Variable} & \mbox{Meaning}\\ Q[i,k] &\mbox{At time i, M is in state}\, q_k\\ H[i,j] &\mbox{At time i, the read-write head}\\ &\mbox{is scanning tape square j}\\ S[i,j,k] &\mbox{At time i, the contents of}\\ &\mbox{tape square j is symbol}\, s_k\end{array}
\begin{array}P=A(1+\frac{r}{100})^n \\ \sqrt[n]{\frac{P}{A}}=\frac{r+100}{100} \\ r=100\sqrt[n]{\frac{P}{A}}-100\end{array}
I = \sqrt {[\frac {22.6T + 0.088t -490.58} {5 * 10^{-5} * t}] * t_{0}} Amp
A=60+(S+\Delta P) \times \frac{L_2}{L_1}
dt = \frac{da}{H_0 \left(\frac{\Omega_{m,0}}{a} + a^2 \Omega_{\Lambda,0}\right)^{\frac{1}{2}}}
\begin{gathered} ax^2 + bx + c = 0,\\ x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }} {{2a}} \hfill \\ \end{gathered}
{\left\{ \begin{gathered} x = r\left( {\theta - \sin \theta } \right) \hfill \\ y = r\left( {1 - \cos \theta } \right) \hfill \\ \end{gathered} \right\}}
\left(\begin{array}{cc}\frac{\partial u}{\partial x}&\frac{\partial \nu}{\partial x}\\ \frac{\partial \nu}{\partial x}&\frac{\partial u}{\partial x}\end{array}\right)
\phi(x,y)
\frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial \nu^2} = 0
w = u(x,y) + i\nu(x,y)
\frac{\partial \phi}{\partial x} = \frac{\partial \phi}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial \phi}{\partial \nu} \frac{\partial \nu}{\partial x}
\frac{\partial \phi}{\partial y} = \frac{\partial \phi}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial \phi}{\partial \nu} \frac{\partial \nu}{\partial y}
\frac{\partial^2 \phi}{\partial x^2} = \frac{\partial \phi}{\partial u} \frac{\partial^2 u}{\partial x^2} + \left ( \frac{\partial^2 \phi}{\partial u^2} \frac{\partial u}{\partial x} + \frac{\partial^2 \phi}{\partial \nu \partial u} \frac{\partial \nu}{\partial x} \right ) \frac{\partial u}{\partial x} + \frac{\partial \phi}{\partial \nu} \frac{\partial^2 \nu}{\partial x^2} + \left ( \frac{\partial^2 \phi}{\partial u \partial \nu} \frac{\partial u}{\partial x} + \frac{\partial^2 \phi}{\partial \nu} \frac{\partial \nu}{\partial x} \right ) \frac{\partial \nu}{\partial x}
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = \frac{\partial \phi}{\partial u} \left ( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right ) + \frac{\partial^2 \phi}{\partial u^2} \left [ \left ( \frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial u}{\partial y} \right )^2 \right ] + 2 \frac{\partial^2 \phi}{\partial u \partial \nu} \left ( \frac{\partial u}{\partial x} \frac{\partial \nu}{\partial y} \right ) + \frac{\partial \phi}{\partial \nu} \left ( \frac{\partial^2 \nu}{\partial x^2} + \frac{\partial^2 \nu}{\partial y^2} \right ) + \frac{\partial^2 \phi}{\partial \nu^2} \left [ \left ( \frac{\partial \nu}{\partial x} \right )^2 + \left ( \frac{\partial \nu}{\partial y} \right )^2 \right ]
\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = \frac{\partial^2 \phi}{\partial u^2} \left [ \left ( \frac{\partial \nu}{\partial x} \right )^2 + \left ( -\frac{\partial \nu}{\partial x} \right )^2 \right ] + \frac{\partial^2 \phi}{\partial \nu^2} \left [ \left ( \frac{\partial \nu}{\partial x} \right )^2 + \left ( \frac{\partial u}{\partial x} \right )^2 \right ] = \left [ \left ( \frac{\partial u}{\partial x} \right )^2 + \left ( \frac{\partial \nu}{\partial x} \right )^2 \right ] \left ( \frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial \nu} \right ) = |f'(z)|^2 \left ( \frac{\partial^2 \phi}{\partial u^2} + \frac{\partial^2 \phi}{\partial \nu^2} \right )
\int_{0}^{\frac{\pi}{2}} \log \cos \Omega \ d \Omega
f(x) = \sin x, \mbox{if} \ x \ \epsilon \ (0, \frac{\pi}{2})
\begin{array}{1|c|c|c|c|c|} \text{Fixed Point}&\text{Jacobian}&\text{Eigenvalues}&\text{Ei genvectors}&\text{type}\\ \hline (0,0)& \left(\begin{array}{cc}1.5 & 0 \\ 0 & 2 \end{array}\right)& 2,1.5 & \left(\begin{array}{ca}0 \\ 1 \end{array}\right) \left(\begin{array}{ca}1 \\ 0 \end{array}\right)& \text{Source}\\ \hline (0,2)& \left(\begin{array}{cc}0.5 & 0 \\ -1.5 & -2 \end{array}\right)& -2,0.5 & \left(\begin{array}{ca}0 \\ 1 \end{array}\right) \left(\begin{array}{ca}0.86 \\ -0.51 \end{array}\right)& \text{Saddle}\\ \hline (1.5,0)& \left(\begin{array}{cc}-1.5 & -0.75 \\ 0 & 0.88 \end{array}\right)& -1.5,0.88 & \left(\begin{array}{ca}1 \\ 0 \end{array}\right) \left(\begin{array}{ca}-0.3 \\ 0.95 \end{array}\right)& \text{Saddle}\\ \hline (4/5,7/5)& \left(\begin{array}{cc}-0.8 & -0.4 \\ -1.05 & -1.4 \end{array}\right)& -1.8,-0.38 & \left(\begin{array}{ca}0.37 \\ 0.93 \end{array}\right) \left(\begin{array}{ca}0.69 \\ -0.72 \end{array}\right)& \text{Sink}\\ \hline \end{array}