# EEMAGS Equation Sheet

## Curl and Divergence Identities
1. The Laplacian: can operate on a scalar or vector field
$$\nabla \cdot (\nabla f) = \nabla^2 f$$
$$\frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}$$
2. The curl of a gradient is $0$
$$\nabla \times (\nabla f) = 0$$
3. The gradient of divergence is a scalar
$$\nabla (\nabla \cdot \vec{f})$$
4. The divergence of curl is $0$
$$\nabla \cdot (\nabla \times \vec{v}) = 0$$
5. Curl of curl
$$\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}$$

## Maxwell's Equations
Gauss's Law for $\vec{E}$-fields:
$$\Phi_E =\int \vec{E} \cdot d\vec{s} = \frac{Q_{enc}}{\varepsilon_0}$$
$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$

Gauss's Law for $\vec{B}$-fields:
$$\int \vec{B} \cdot d\vec{s} = 0 $$
$$ \nabla \cdot \vec{B} = 0$$

Faraday's Law
$$\int \vec{E} \cdot d\vec{l} = -\frac{d}{dt}\vec{B} $$
$$ \nabla \times \vec{E} = -\frac{d}{dt}\vec{B}$$

Ampere's Law
$$\int \vec{B} \cdot d\vec{l} = \mu_0\left( \vec{J} + \varepsilon_0 \frac{d\vec{E}}{dt}\right) = \mu_0 I_{enc}$$
$$ \nabla \times \vec{B} = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$$

## Work and Voltage
$$\Delta V = -\int \vec{E} \cdot d\vec{l}$$
$$-\nabla V = \vec{E}$$