Chapter 10.11, Problem 11MP

Show that the nine quantities $T_{ij}=\partial V_i/\partial x_j$ (which are the Cartesian components of $\nabla V$ where $V$ is a

vector) satisfy the transformation equations (2.14) for a Cartesian $2^{\text{nd }}$-rank tensor. Show that they do not satisfy the general tensor transformation equations as in (10.12).Hint: Differentiate ( 10.9) or (10.10) partially with respect to, say, $x_k^{\prime }.$ You should get the expected terms [as in (10.12)] plus some extra terms; these extraneous terms show that $\partial V_i/\partial x_j$ is not a tensor under general transformations.Comment: It is possible to express the components of $\nabla V$ correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.