Rabbits Four

\begin{bmatrix} \dfrac {\partial x'_1} {\partial x_1} & \dfrac {\partial x'_1} {\partial x_2} & \dfrac {\partial x'_1} {\partial x_3} \\ \\ \dfrac {\partial x'_2} {\partial x_1} & \dfrac {\partial x'_2} {\partial x_2} & \dfrac {\partial x'_2} {\partial x_3} \\ \\ \dfrac {\partial x'_3} {\partial x_1} & \dfrac {\partial x'_3} {\partial x_2} & \dfrac {\partial x'_3} {\partial x_3} \end{bmatrix}

You need to think of something like \dfrac {\partial x'_3} {\partial x_1} as being a single entity. It is not defined as one thing divided by another thing. It is a single thing that tells the story of how x_1 affects x’_3.

Appendix A

In one choice of notation, a component is represented by \dfrac {\partial x'_3} {\partial x_1} , and for another, the component is represented by a_{31}.