The Maxwell field tensor is

$$F^{a b}=\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{array}\right)$$

and the 4-current density is $J^{a}=(\rho, \mathbf{j})$. Write down the 3-vector form of Maxwell's equations and the continuity equation, and obtain the equivalent 4-vector equations.

Consider a Lorentz transformation from a frame $\mathcal{F}$ to a frame $\mathcal{F}^{\prime}$ moving with relative (coordinate) velocity $v$ in the $x$-direction

$$L_{b}^{a}=\left(\begin{array}{cccc} \gamma & \gamma v & 0 & 0 \\ \gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

where $\gamma=1 / \sqrt{1-v^{2}}$. Obtain the transformation laws for $\mathbf{E}$ and $\mathbf{B}$. Which quantities, quadratic in $\mathbf{E}$ and $\mathbf{B}$, are Lorentz scalars?