(i) Starting from

$$F^{\mu \nu}=\left(\begin{array}{cccc} 0 & E_{1} / c & E_{2} / c & E_{3} / c \\ -E_{1} / c & 0 & B_{3} & -B_{2} \\ -E_{2} / c & -B_{3} & 0 & B_{1} \\ -E_{3} / c & B_{2} & -B_{1} & 0 \end{array}\right)$$

and performing a Lorentz transformation with $\gamma=1 / \sqrt{1-u^{2} / c^{2}}$, using

$$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma u / c & 0 & 0 \\ -\gamma u / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

show how $\mathbf{E}$ and $\mathbf{B}$ transform under a Lorentz transformation.

(ii) By taking the limit $c \rightarrow \infty$, obtain the behaviour of $\mathbf{E}$ and $\mathbf{B}$ under a Galilei transfomation and verify the invariance under Galilei transformations of the nonrelativistic equation

$$m \frac{d \mathbf{v}}{d t}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B})$$

(iii) Show that Maxwell's equations admit solutions of the form

$$\mathbf{E}=\mathbf{E}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c), \quad \mathbf{B}=\mathbf{B}_{0} f(t-\mathbf{n} \cdot \mathbf{x} / c)$$

where $f$ is an arbitrary function, $\mathbf{n}$ is a unit vector, and the constant vectors $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$ are subject to restrictions which should be stated.

(iv) Perform a Galilei transformation of a solution $(\star)$, with $\mathbf{n}=(1,0,0)$. Show that, by a particular choice of $u$, the solution may brought to the form

$$\tilde{\mathbf{E}}=\tilde{\mathbf{E}}_{0} g(\tilde{x}), \quad \tilde{\mathbf{B}}=\tilde{\mathbf{B}}_{0} g(\tilde{x}),$$

where $g$ is an arbitrary function and $\tilde{x}$ is a spatial coordinate in the rest frame. By showing that $(t)$ is not a solution of Maxwell's equations in the boosted frame, conclude that Maxwell's equations are not invariant under Galilei transformations.