= Solution
Use coordinates $x^\mu=(ct,\mathbf x)$, signature $(+---)$, and the convention $F^{0i}=-E_i/c$, $F^{ij}=-\epsilon_{ijk}B_k$. The antisymmetric <electromagnetic field tensor> combines six field components:
$$
F^{\mu\nu}=
\begin{pmatrix}
0&-E_x/c&-E_y/c&-E_z/c\\
E_x/c&0&-B_z&B_y\\
E_y/c&B_z&0&-B_x\\
E_z/c&-B_y&B_x&0
\end{pmatrix}.
$$
It transforms as $F'^{\mu\nu}=\Lambda^\mu{}_\alpha\Lambda^\nu{}_\beta F^{\alpha\beta}$. Lowering indices changes the time–space signs: $F_{0i}=E_i/c$, while $F_{ij}=F^{ij}$. The <covariant tensor> instead uses the inverse coordinate Jacobian on each lower index.
For a frame moving with <velocity> $\mathbf v$ relative to the original one,
$$
\boxed{\mathbf E'_\parallel=\mathbf E_\parallel,\quad
\mathbf E'_\perp=\gamma(\mathbf E+\mathbf v\times\mathbf B)_\perp},
$$
$$
\boxed{\mathbf B'_\parallel=\mathbf B_\parallel,\quad
\mathbf B'_\perp=\gamma\left(\mathbf B-\frac{\mathbf v\times\mathbf E}{c^2}\right)_\perp}.
$$
Electric and <magnetic fields> therefore mix under a boost; they are frame-dependent parts of one tensor. For example, a purely transverse <electric field> acquires a <magnetic field> in a boosted frame. The <electromagnetic field invariants> $B^2-E^2/c^2$ and $\mathbf E\cdot\mathbf B$ restrict which fields can be eliminated by a frame change. The covariant <force> law $dp^\mu/d\tau=qF^\mu{}_\nu U^\nu$ guarantees consistent particle dynamics in all <inertial frames>.
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