Show that if for every vector $V_{j}$, the quantities $U_{1}=\sum T_{u} V_{j}$ are the components of a vector, then the quantities $T_{i j}$ are the components of a second-order tensor. (This fact is an example of the quotient rule). Hint: Use the transformation (11.12) for a vector to show that $T_{i j}$ satisfies the transformation ( $11.13$ ). Similarly, show that $A_{i j k}$ in the equation of Problem $12.10$ is a fourth-order (Cartesian) tensor if, for every second-order (Cartesian) tensor $Q_{\text {uin }}$ $P_{1 y}$ is a second-order (Cartesian) tensor. Generalize the proof to show that if $U_{1}=\sum T_{i j} V_{4}$, where $U_{1}$ is a covariant vector [satisfying $\left.(13.6)\right]$ and $V^{j}$ is an arb?trary contravariant vector [satisfying $(13.7)]$, then $T_{i j}$ is a second-order covariant rensor. Generalize further to tensors of any order and kind.