TRANFORMATIONS OF MULTIPLE INTEGRALS

In evaluating a multiple integral over a region $\cal R$ , it is often convenient to use coordinates other than rectangular, such as the curvilinear coordinates considered in Chapter 5.

If we let $(u, v)$ be curvilinear coordinates of points in a plane, there will be a set of transformation equations $x = f(u, v),\ y = g(u, v)$ mapping points $(x, y)$ of the xy plane into points $(u, v)$ of the uv plane. In such case the region $\cal R$ of the xy plane is mapped into a region ${\cal R}'$ of the uv plane. We then have
$\displaystyle \underset{\cal R}{\iint}F(x, y)dxdy = \underset{{\cal R}'}{\iint} G(u, v)\left|\frac{\partial (x,y)}{\partial (u, v)}\right| dudv \cdots(9)$
where $G(u, v) \equiv F\{f(u,v), g(u,v)\}$ and
$\displaystyle \frac{\partial (x, y)}{\partial (u, v)} \equiv \left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right| \cdots (10)$
is the Jacobian of x and y with respect to u and v.

Similarly if $(u, v, w)$ are curvilinear coordinates in three dimensions, there will be a set of transformation equations $x = f(u, v, w)$ , $y = g(u, v, w)$ , $z = h(u, v, w)$ and we can write
$\displaystyle \underset{\cal R}{\iiint}F(x, y, z)dxdydz = \underset{{\cal R}'}{\iiint} G(u, v, w) \left| \frac{\partial (x, y, z)}{\partial (u, v, w)} \right|dudvdw \cdots(11)$
where $G(u, v, w) \equiv F\{f(u, v, w), g(u, v, w), h(u, v, w)\}$ and
$\displaystyle \frac{\partial (x, y, z)}{\partial (u, v, w)} \equiv \left| \begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{array} \right|\cdots(12)$
is the Jacobian of x, y and z with respect to u, v and w.

The results (9) and (11) correspond to change of variables for double and triple integrals.

Generalizations to higher dimensions are easily made.