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MI.9 Change of Variables in Multiple Integrals

In one dimensional calculus we often use a change of variable (a substitution) to simplify an integral. By reversing the role of and , we can write the Substitution Rule as

where and Another way of writing formula 1 is as follows:

A change of variables can also be useful in double integrals, We have already seen one example of this: conversion to polar coordinates. The new variables r and are related to the old variables x and y by the equations

and the change of variables formula can be written as

where S is the region in the r-plane that corresponds to the region R in the xy-plane.

More generally we consider a change of variables that is given by a transformation T from the uvplane to the xy-plane:

where x and y are related to u and v by the equations

or, as we sometimes write,

We usually assume that T is a transformation, which means that g and h have continuous first order partial derivates. A transformation T is really just a function whose domain and range are both subsets of If then the point is called the image of the point If no two points have the same image, T is called one-to-one. The figure below shows the effect of a transformation T on a region S in the uv-plane. T transforms S into a region T in the xy-plane called the image of S, consisting of the images of all points in S.

picture of figure 1 on 1061.

If T is a one-to-one transformation, then it has an inverse transformation from the xy-plane to the uvplane and it may be possible to solve. the following equations for x and y.

Now lets see how a change of variables affects a double integral. We start with a small rectangle S in the uv-plane whose lower left corner is the point () and whose dimensions are and

figure three on 1062

The image of S is a region R in the xy-plane, one of whose boundary points is The vector

is the position vector of the image of the point (u,v). the equation of the lower side of S is , whose image curve is given by the vector function r The tangent vector at to this image curve is

Similarly, the tangent vector at to the image curve of the left side of S (namely is

We can approximate the image region by parallelogram determined by the secant vectors

these are shown below:

Figure 4 on 1062

However

and so

Similarly

This means that we can approximate R by a parallelogram determined by the vectors and . Therefore, we can approximate the area of R by the area of this parallelogram, which is:

Computing this cross product we get:

The determinant that arises in this calculation is called the Jacobian of the transformation and is given a special notation.

The Jacobian of the transformation T given by and is

With this notation we can use the above equation to five an approximation of the area of R:

where the Jacobian is evaluated at

Next we divide a region S in the uv-plane into rectangles S and call their images on the xy-plane R Applying the approximation to each R, we approximate the double integral of over R as follows:

The figure below can help explain:

figure 6 on pg 1063

where the Jacobian is evaluated at Notice that this double sum is a Riemann sum for the integral

This argument suggest the following theorem is true:

Change of Variables in a Double Integral

Suppose that T is a one-to-one C transformation whose Jacobian is nonzero and that maps a region S in the uv-plane onto a region R in the xy-plane. Suppose that is continuous on R and that R and S are type I or type II pane regions. Then:

Triple Integrals

There is a similar change of variables formula for the triple integrals. Let T be a transformation that maps a region S in uvw-space onto a region R in xyz-space by means of the equations:

The Jacobian of T is the following determinate:

Under hypotheses similar to those used in the double integral, we have the following formula for triple integrals

 

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