Let $\mathbf{U}$ and $\mathbf{V}$ be vectors; we form the following array (in each coordinate system) from the components $U_{1}, U_{2}, U_{3}$ and $V_{1}, V_{2}, V_{3}$ of $\mathbf{U}$ and $\mathbf{V}$ (in that coordinate system):
$$\begin{array}{ccc} U_{1} V_{1} & U_{1} V_{2} & U_{1} V_{3} \ U_{2} V_{1} & U_{2} V_{2} & U_{2} V_{3} \ U_{3} V_{1} & U_{3} V_{2} & U_{3} V_{3} \end{array}$$
We can show that these nine quantities are the components of a second-rank tensor which we shall denote by UV. Note that this is not a dot product or a cross product; it is called the direct product of $\mathbf{U}$ and $\mathbf{V}$ (or outer product or tensor product). Since $\mathbf{U}$ and $\mathbf{V}$ are vectors, their components in a rotated coordinate system are, by (2.12):

$$U_{k}^{\prime}=\sum_{i=1}^{3} a_{k i} U_{i}, \quad V_{l}^{\prime}=\sum_{j=1}^{3} a_{l j} V_{j} .$$
Hence the components of the second-rank tensor UV are
$$U_{k}^{\prime} V_{l}^{\prime}=\sum_{i=1}^{3} a_{k i} U_{i} \sum_{j=1}^{3} a_{l j} V_{j}=\sum_{i, j=1}^{3} a_{k i} a_{l j} U_{i} V_{j},$$
which is just (2.14) with $T_{i j}=U_{i} V_{j}$ and $T_{k l}^{\prime}=U_{k}^{\prime} V_{l}^{\prime}$.

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