(a) What is meant by an antisymmetric tensor of second rank? Show that if a second rank tensor is antisymmetric in one Cartesian coordinate system, it is antisymmetric in every Cartesian coordinate system.

(b) Consider the vector field $\mathbf{F}=(y, z, x)$ and the second rank tensor defined by $T_{i j}=\partial F_{i} / \partial x_{j}$. Calculate the components of the antisymmetric part of $T_{i j}$ and verify that it equals $-(1 / 2) \epsilon_{i j k} B_{k}$, where $\epsilon_{i j k}$ is the alternating tensor and $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{F}$.