User:Mikhail Klassen/sandbox

$$

\frac{V_{max}}{V_{min}} = \frac{1+|T|^2}{1-|T|^2}

$$

$$

E = E_0 e^{-i(\omega t - k_z z)} + E_0 \Gamma e^{-i(\omega t + k_z z)}

$$

$$

\Delta \phi = k_0 d \left(\sqrt{\frac{\epsilon}{\epsilon_0}} - 1\right)

$$

$$

k = \frac{\omega}{c}\sqrt{\frac{\epsilon}{\epsilon_0}}

$$

$$

\begin{align}

\omega^2 & = c^2(k_x^2 + k_y^2 + k_z^2)\\ & = c^2 \left( \frac{\pi^2}{a^2} + k_z^2 \right)\\ & = \omega_{x}^2 + c^2 k_z^2

\end{align}

$$

$$ \begin{align}

\mathbf{v} & = \nabla \times \boldsymbol{\psi} \\ & = \mathbf{\hat{z}} \times \nabla \boldsymbol{\psi} \\ & = \left\langle - \frac{\partial \psi}{\partial y}, \frac{\partial \psi}{\partial x}, 0 \right \rangle

\end{align} $$

$$ \nabla \cdot \mathbf{v} = 0 $$

$$ \begin{align}

\frac{\partial N}{\partial t} + \nabla \cdot \left(\mathbf{v}N\right) & = \frac{\partial N}{\partial t} + \left(\mathbf{v} \cdot \nabla\right) N + N \nabla \cdot \mathbf{v} \\ & = \frac{\partial N}{\partial t} + \left(\mathbf{v} \cdot \nabla\right) N \\ & = \frac{\partial N}{\partial t} + v_x \frac{\partial N}{\partial x} + v_y \frac{\partial N}{\partial y} \\ & = \frac{\partial N}{\partial t} - \frac{\partial \psi}{\partial y} \frac{\partial N}{\partial x} + \frac{\partial \psi}{\partial x} \frac{\partial N}{\partial y} = 0

\end{align} $$

$$

\psi(x,y) = \sum_{k_x,k_y} A_{m,n} sin(k_x x)sin(k_y y)

$$

$$ \begin{align} k_x & = \frac{m \pi}{a} \\ k_y & = \frac{n \pi}{a} \end{align} $$

$$ \frac{\partial \psi}{\partial y} \frac{\partial N}{\partial x} - \frac{\partial \psi}{\partial x} \frac{\partial N}{\partial y} = \frac{\partial N}{\partial t} $$

$$ \begin{align} \frac{\partial \psi}{\partial x} & = A_{11}(\pi/a)cos(\pi x/a)sin(\pi y/a) + A_{12}(\pi/a)cos(\pi x/a)sin(2 \pi y/a)\\ & + A_{21}(2 \pi/a)cos(2 \pi x/a)sin(\pi y/a) + A_{22}(2 \pi/a)cos(2 \pi x/a)sin(2 \pi y/a) \end{align} $$

$$ \begin{align} \frac{\partial \psi}{\partial y} & = A_{11}(\pi/a)sin(\pi x/a)cos(\pi y/a) + A_{12}(2 \pi/a)sin(\pi x/a)cos(2 \pi y/a)\\ & + A_{21}(\pi/a)sin(2 \pi x/a)cos(\pi y/a) + A_{22}(2 \pi/a)sin(2 \pi x/a)cos(2 \pi y/a) \end{align} $$

$$ \left[ \begin{smallmatrix} (\pi/a)sin(\pi x/a)cos(\pi y/a)N_x - (\pi/a)cos(\pi x/a)sin(\pi y/a)N_y \\ (2 \pi/a)sin(\pi x/a)cos(2 \pi y/a)N_x - (\pi/a)cos(\pi x/a)sin(2 \pi y/a)N_y \\ (\pi/a)sin(2 \pi x/a)cos(\pi y/a)N_x - (2 \pi/a)cos(2 \pi x/a)sin(\pi y/a)N_y \\ (2 \pi/a)sin(2 \pi x/a)cos(2 \pi y/a)N_x - (2 \pi/a)cos(2 \pi x/a)sin(2 \pi y/a)N_y \end{smallmatrix} \right]^T \cdot \left( \begin{smallmatrix} A_{11} \\  A_{12} \\  A_{21} \\  A_{22} \end{smallmatrix} \right) = \begin{smallmatrix} N_t \end{smallmatrix} $$

$$ \begin{align} N_x & = N_x(x,y)\\ N_y & = N_y(x,y)\\ N_t & = N_t(x,y) \end{align} $$

$$ C = \begin{bmatrix} 8 &  1  &  6 \\     3  &  5  &  7 \\     4  &  9  &  2 \end{bmatrix} $$

$$ C' = \begin{bmatrix} 8.00000000000001 &        1.00000000000001  &        5.99999999999999 \\          3.00000000000001  &        4.99999999999997  &        7.00000000000002 \\          4.00000000000001  &        9.00000000000001  &                       2 \end{bmatrix} $$ $$

C = \begin{bmatrix} 4 &                     0.5  &                       3 \\                       1.5   &                    2.5   &                    3.5 \\                         2    &                   4.5    &                     1 \end{bmatrix} $$

$$

C' = \begin{bmatrix} 4.00000000000001 &       0.500000000000004 &                        3 \\          1.50000000000001  &        2.49999999999999  &        3.50000000000001 \\                         2   &       4.50000000000001   &                      1 \end{bmatrix} $$

$$ v = v_s \frac{R_2}{R_1 + R_2} $$

$$ \frac{\partial}{\partial R_1}\frac{\partial v}{\partial R_2} = 0 $$

$$ \frac{\partial}{\partial R_1}\frac{\partial v}{\partial R_2} = v_s \frac{R_2^2-R_1^2}{\left(R_1+R_2\right)^4} $$

$$ R_1 = R_2 $$