The vector potential $\mathbf{A}$ and scalar potential $\phi$ define the fields:

$\mathbf{E} = -\nabla\phi - \frac{\partial \mathbf{A}}{\partial t}, \quad \mathbf{B} = \nabla \times \mathbf{A}$

These automatically satisfy $\nabla \cdot \mathbf{B} = 0$ and $\nabla \times \mathbf{E} = -\partial\mathbf{B}/\partial t$ (two of Maxwell's equations). The gauge transformation:

$\mathbf{A}' = \mathbf{A} + \nabla\chi, \quad \phi' = \phi - \frac{\partial\chi}{\partial t}$

leaves $\mathbf{E}$ and $\mathbf{B}$ invariant for any scalar function $\chi(x,t)$. Common gauges include:

• Coulomb gauge: $\nabla \cdot \mathbf{A} = 0$ (convenient for static problems)
• Lorenz gauge: $\nabla \cdot \mathbf{A} + \mu_0\epsilon_0\frac{\partial\phi}{\partial t} = 0$ (manifestly relativistic)

The electromagnetic energy density is:

$u = \frac{\epsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2$

The Poynting vector describes energy flux:

$\mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B}$

Energy conservation follows from:

$\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}$

where $\mathbf{J} \cdot \mathbf{E}$ represents work done on charges. For a plane wave $\mathbf{E} = E_0\cos(kz - \omega t)\hat{\mathbf{x}}$, $\mathbf{B} = (E_0/c)\cos(kz - \omega t)\hat{\mathbf{y}}$, the time-averaged Poynting vector is:

$\langle\mathbf{S}\rangle = \frac{E_0^2}{2\mu_0 c}\hat{\mathbf{z}} = \frac{\epsilon_0 c E_0^2}{2}\hat{\mathbf{z}}$

pointing in the direction of propagation.

For a rigid body with mass distribution $\rho(\mathbf{r})$, the angular momentum $\mathbf{L}$ relates to angular velocity $\boldsymbol{\omega}$ through:

$L_i = I_{ij}\omega_j$

where the inertia tensor is:

$I_{ij} = \int \rho(\mathbf{r})\left(r^2\delta_{ij} - x_i x_j\right)d^3r$

For a uniform sphere of radius $R$ and mass $M$ rotating about the $z$-axis:

$I_{ij} = \frac{2MR^2}{5}\mathrm{diag}(1,1,1)$

The diagonal form reflects spherical symmetry. The rotational kinetic energy is:

$T = \frac{1}{2}I_{ij}\omega_i\omega_j = \frac{1}{2}\boldsymbol{\omega} \cdot \mathbf{I} \cdot \boldsymbol{\omega}$

In an elastic medium, the force per unit area (traction) $\mathbf{t}$ on a surface with normal $\hat{\mathbf{n}}$ is:

$t_i = \sigma_{ij}n_j$

The stress tensor $\sigma_{ij}$ has components:

$\sigma_{ij} = \frac{F_i}{A_j}$

where $F_i$ is the $i$-component of force on a surface perpendicular to the $j$-direction. For hydrostatic pressure $p$:

$\sigma_{ij} = -p\delta_{ij}$

For a general elastic body, the stress-strain relation (Hooke's law) involves the rank-4 elastic tensor $C_{ijkl}$:

$\sigma_{ij} = C_{ijkl}\epsilon_{kl}$

where $\epsilon_{kl} = \frac{1}{2}(\partial_k u_l + \partial_l u_k)$ is the strain tensor and $\mathbf{u}$ is the displacement field.