Change of Variables

Many instructors use as an introduction to the general setting of this section and other skip this generalization. We do not address how to find these advantageous changes of coordinates which will transform regions from non-rectangular to rectangular (in the new coordinate space), but rather we develop the Jacobian as the key element in these transitions between coordinates. Also of note, the more time you spent on visualizing regions and setting boundary conditions will yield great benefits in the time needed for students to practice the ideas in this section.

In single variable calculus, we encountered the idea of a change of variable in a definite integral through the method of substitution. For example, given the definite integral \[\begin{aligned}\end{aligned}\] we naturally consider the change of variable \(u = x^2+1\). From this substitution, it follows that \(du = 2x \, dx\), and since \(x = 0\) implies \(u = 1\) and \(x = 2\) implies \(u = 5\), we have transformed the original integral in \(x\) into a new integral in \(u\). In particular, \[\begin{aligned}\end{aligned}\]

The latter integral, of course, is far easier to evaluate.

Through our work with polar, cylindrical, and spherical coordinates, we have already seen some of the issues and advantages that arise from using a change of variables with two or three variables. In the following Preview Activity, we will look geometrically at a couple of examples of changes of coordinates involving polar coordinates. In particular, we will explore exactly what properties an advantageous change of coordinates will have. In what follows, we seek to understand the general ideas behind any change of variables in a multiple integral.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

The general idea behind a change of variables is suggested by Preview Activity. There, we saw that in a change of variables from rectangular coordinates to polar coordinates, a polar rectangle \([r_1, r_2] \times [\theta_1, \theta_2]\) gets mapped to a Cartesian rectangle under the transformation \[\begin{aligned}\end{aligned}\]

Furthermore, it follows generally that for an original polar rectangle \(P = [r_1, r_2] \times [\theta_1, \theta_2]\), the area of the transformed rectangle \(P'\) is given by \(\frac{r_2+r_1}{2} \Delta r \Delta \theta\). Therefore, as \(\Delta r\) and \(\Delta \theta\) go to 0, this area becomes the familiar area element \(dA = r \, dr \, d\theta\) in polar coordinates. When we work with other transformations for different changes in coordinates, we will have to understand how the transformation affects area so that we may use the correct area element in the new system of variables.

We first focus on double integrals. As with single integrals, we may be able to simplify a double integral of the form \[\begin{aligned}\end{aligned}\] by making a change of variables (that is, a substitution) of the form \[\begin{aligned}\end{aligned}\] where \(x\) and \(y\) are functions of new variables \(s\) and \(t\). This transformation introduces a correspondence between a problem in the \(xy\)-plane and one in the the \(st\)-plane. The equations \(x=x(s,t)\) and \(y=y(s,t)\) convert \(s\) and \(t\) to \(x\) and \(y\); we call these formulas the change of variable formulas.

To complete the change of our integral to the new \(s,t\) variables, we need to

1. \(dA\)\(s\)\(t\)
2. transform the integrand (with the change of variable formulas)
3. \(st\)

In Activity, we used a clever relationship to convert our corner points from \(xy\)-coordinates to \(st\)-coordinates, but in general you would need to solve a system of equations to find the corresponding \((s,t)\) points. For example, if we have change of coordinate equations given by \(x=3s-2t\) and \(y =s+4t\), then the transformation of the point \((x,y)=(1,2)\), would come from solving the system of equations \(1= 3s-2t\) and \(2 =s+4t\).

If we shift our attention to the area element associated with a change of variables, we will need to look at how the area is stretched by the transformation between \(xy\)-coordinates to \(st\)-coordinates. Conceptually, derivatives will allow us to measure how quickly coordinates are changing in the context of a change of variables. Let's look back at the single variable example stated in the introduction to this section.

We have seen how in a two dimensional setting, we can convert the region of integration (Activity) and the integrand (the function being integrated) using our change of variable equations. As was suggested by the previous example, the partial derivatives of \(x\) and \(y\) with respect to \(s\) or \(t\) will measure how each variable (\(x\) or \(y\)) is stretched with respect to small a small change in the inputs (\(s\) or \(t\)). We will need to use vectors to quantify how these individual coordinate changes will amount to changes in the way we measure area.

The following result summarizes how to find the area element corresponding to a change of variables in general. We have included the details of the vector argument as a proof to this result.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

The details of an argument for the change of variable formula for triple integrals are more complicated the two dimension case we have just discussed and we omit those here. The general process, though, is the same as the two-dimensional case; Namely, we will use partial derivatives and the determinant of a matrix of the partial derivatives to give the appropriate volume element for the new coordinate.

Given a solid \(S'\) in the \(xyz\)-coordinate system in \(\R^3\), a change of variables transformation \(x=x(s,t,u)\), \(y=y(s,t,u)\), and \(z = z(s,t,u)\) transforms \(S'\) into a region \(S\) in \(stu\)-coordinates. Any function \(f = f(x,y,z)\) defined on \(S'\) can be considered as a function \(f = f(x(s,t,u), y(s,t,u), z(s,t,u))\) in \(stu\)-coordinates defined on \(S\). The volume element \(dV\) in \(xyz\)-coordinates cooresponds to a scaled volume element in \(stu\)-coordinates, where the scale factor is given by the absolute value of the Jacobian, \(\frac{\partial(x,y,z)}{\partial(s,t,u)}\), which is the determinant of the \(3 \times 3\) matrix \[\begin{aligned}\end{aligned}\]. (Recall that this determinant was introduced in Section.) That is, \(\frac{\partial(x,y,z)}{\partial(s,t,u)}\) is given by \[\begin{aligned}\end{aligned}\]

Suppose a change of variables \(x = x(s,t,u)\), \(y = y(s,t,u)\), and \(z = z(s,t,u)\) transforms a closed and bounded region \(S\) in \(stu\)-coordinates into a closed and bounded region \(S'\) in \(xyz\)-coordinates. Under modest conditions (that are studied in advanced calculus), the triple integral \(\iiint_{S'} f(x,y,z) \, dV\) is equal to \[\begin{aligned}\end{aligned}\]