User:Arksine/sandbox

$$\nabla = \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix}$$

gradient: $$\nabla\varphi = \begin{bmatrix} \frac{\partial\varphi}{\partial x} \\ \frac{\partial\varphi}{\partial y} \\ \frac{\partial\varphi}{\partial z} \end{bmatrix}$$

divergence: $$\nabla\cdot\mathbf A = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$

curl: $$\nabla\times\mathbf A = \begin{vmatrix} \mathbf i&\mathbf j&\mathbf k \\ \frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z} \\ A_x&A_y&A_z \end{vmatrix} = \begin{bmatrix} \frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z} \\ \frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x} \\ \frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y} \end{bmatrix}$$

Laplacian: $$\nabla^2\varphi = \nabla\cdot(\nabla\varphi) = \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix} \cdot \begin{bmatrix} \frac{\partial\varphi}{\partial x} \\ \frac{\partial\varphi}{\partial y} \\ \frac{\partial\varphi}{\partial z} \end{bmatrix} = \frac{\partial^2\varphi}{\partial^2x} + \frac{\partial^2\varphi}{\partial^2y} + \frac{\partial^2\varphi}{\partial^2z}$$

vector Laplacian: $$\nabla^2\mathbf A = \nabla(\nabla\cdot\mathbf A) - \nabla\times(\nabla\times\mathbf A) = \nabla \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \right) - \nabla\times \begin{bmatrix} \frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z} \\ \frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x} \\ \frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y} \end{bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \right) \\ \frac{\partial}{\partial y} \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \right) \\ \frac{\partial}{\partial z} \left( \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \right) \end{bmatrix} -$$

$$\partial^\mu = \begin{bmatrix} \frac{1}{c}\frac{\partial}{\partial t} \\ -\frac{\partial}{\partial x} \\ -\frac{\partial}{\partial y} \\ -\frac{\partial}{\partial z} \end{bmatrix}$$

$$\partial_\mu = \begin{bmatrix} \frac{1}{c}\frac{\partial}{\partial t} \\ \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix}$$

$$A^\mu = \begin{bmatrix} A^t \\ A^x \\ A^y \\ A^z \end{bmatrix}$$

divergence: $$\partial_\mu A^\mu$$

d'Alembert: $$\Box = \partial^2 = \partial^\mu\partial_\mu$$

Maxwell's equations in terms of potentials: $$\partial_\mu J^\mu = 0$$ and $$\partial^2 A^\mu = \mu_0 J^\mu$$ where $$A^t = \frac{1}{c}\phi$$ and $$J^t =c\rho$$ with the Lorenz gauge $$\partial_\mu A^\mu = 0$$

$$x^\mu = \begin{bmatrix}x^0\\x^1\\x^2\\x^3\end{bmatrix} = \begin{bmatrix}ct\\x\\y\\z\end{bmatrix}$$

$$\mathrm{d}x^\mu = \begin{bmatrix}\mathrm{d}x^0\\\mathrm{d}x^1\\\mathrm{d}x^2\\\mathrm{d}x^3\end{bmatrix}$$

$$\mathrm{d}x_\mu = \begin{bmatrix}\mathrm{d}x_0&\mathrm{d}x_1&\mathrm{d}x_2&\mathrm{d}x_3\end{bmatrix} = \begin{bmatrix}\mathrm{d}x^0&-\mathrm{d}x^1&-\mathrm{d}x^2&-\mathrm{d}x^3\end{bmatrix}$$

$$\mathrm{d}s^2 = \mathrm{d}x_\mu\mathrm{d}x^\mu$$

$$\eta_{\mu\nu} = \eta^{\mu\nu} = \begin{bmatrix} 1&0&0&0 \\ 0&-1&0&0 \\ 0&0&-1&0 \\ 0&0&0&-1 \end{bmatrix}$$

hint: ISO standards dictate that the "d" in differentials should be actually be in upright font

contravariant: $$A = A^i e_i = \begin{bmatrix} \hat{e} & \hat{i} & \hat{j} & \hat{k} \end{bmatrix} \begin{bmatrix} A^t \\ A^x \\ A^y \\ A^z \end{bmatrix}$$

covariant: $$B = A_i e^i = \begin{bmatrix} B_t & B_x & B_y & B_z \end{bmatrix} \begin{bmatrix} \hat{e} \\ \hat{i} \\ \hat{j} \\ \hat{k} \end{bmatrix}$$

$$J^\mu= \begin{bmatrix} J_t \\ J_x \\ J_y \\ J_z \end{bmatrix} = \begin{bmatrix} c\rho \\ J_x \\ J_y \\ J_z \end{bmatrix} $$ so $$\rho=\frac{J_t}{c}$$

$$\nabla\cdot\mathbf {E}=\frac{\rho}{\epsilon_0}=\frac{1}{\epsilon_0}\frac{J_t}{c}={\mu_0}c^2\frac{J_t}{c}=c{\mu_0}J_t$$

$$\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z} = c{\mu_0}J_t$$

$$\nabla\times\mathbf {B}-\frac{1}{c^2}\frac{\partial\mathbf {E}}{\partial t} = \mu_0 \mathbf {J}$$

$$\begin{bmatrix} \frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z} \\ \frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x} \\ \frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y} \end{bmatrix} -\frac{1}{c^2}\frac{\partial}{\partial t} \begin{bmatrix} E_x \\ E_y \\ E_z \end{bmatrix} = \mu_0 \begin{bmatrix} J_x \\ J_y \\ J_z \end{bmatrix}$$