What does this partial differential equation mean and how to solve it?

Question

Let $u = \frac{yz}{x}, v = \frac{zx}{y}, w = \frac{xy}{z}$ , find:

$$\frac{\partial (u,v,w)}{\partial (x,y,z)}.$$

Answer 1

Whenever I have seen notation like this, it refers to the following matrix:

\begin{equation} \frac{\partial(u,v,w)}{\partial(x,y,z)} \;=\; \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{bmatrix} \end{equation}

This is known as the Jacobian matrix. Note that the determinant of a Jacobian matrix frequently shows up when making a change of variables in a multidimensional integral.