What does the notation $\frac{\partial(x,y)}{\partial(u,v)}$ mean?

Question

Suppose G$(u,v) = (x, y, z)$

In terms of derivatives, what does $\frac{\partial(x,y)}{\partial(u,v)}$ mean?

Answer 1

$$ \frac{\partial(x,y)}{\partial(u,v)} = \det\begin{bmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} \\[4pt] \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} \end{bmatrix}. $$ This expression is called the Jacobian of the function by which $(x,y)$ depends on $(u,v)$.

There is an identity $$ \frac{\partial(x,y)}{\partial(u,v)} \, du\,dv = dx\,dy, $$ used for transforming an integral in the $(x,y)$-plane to an integral in the $(u,v)$-plane when $x$, $y$ are given functions of $u,v$. Sometimes this makes the integral tractable. (In cases where the Jacobian is negative and one is integrating with respect to area but not with respect to oriented area, one takes the absolute value of the Jacobian.)