Jacobian

An other place that determinant is applied is in the change of variable formula for integrals over regions in $R^n$. We restrict ourselves to integrals over regions in $R^2$.

Let $R$ be a region in $xy$-plane and $S$ be a region in $uv$-plane. Let $R$ and $S$ be related by the equations $x=h(u,v)$ and $y=g(u, v)$ such that every pont in $R$ is the image of a unique point in $S$. Let $f$ be a continuous function on $R$. Assume that $h$ and $g$ have continuous partial derivatives on $S$. Also assume that $\frac{ \partial (x,y) }{\partial(u, v)}=\left\vert \begin{array}{rr} \frac{ \pa... ...eta} & \frac{ \partial y }{ \partial \theta} \\ \end{array}\right\vert \ne 0$. It can be shown that

$${\int \int}_R f(x, y) dA = \int \int _S f( h(u,v), g(u,v))\vert\frac{ \partial (x,y) }{\partial (u, v)}\vert dA$$