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**15.9** Change of Variables in

### Multiple Integrals

### 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 roles of x and u, we can write*

*where x = g(u) and a = g(c), b = g(d). Another way of *
writing Formula 1 is as follows:

3

### Change of Variables in Multiple Integrals

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 *θ are related to
*the old variables x and y by the equations*

*x = r cos *θ *y = r sin *θ

and the change of variables formula can be written as
*f(x, y) dA = f(r cos *θ*, r sin *θ*) r dr d*θ

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

### Change of Variables in Multiple Integrals

More generally, we consider a change of variables that is
**given by a transformation T from the uv-plane to the **

*xy-plane:*

*T(u, v) = (x, y)*

*where x and y are related to u and v by the equations*
*x = g(u, v)* *y = h(u, v)*

or, as we sometimes write,

*x = x(u, v)* *y = y(u, v)*

5

### Change of Variables in Multiple Integrals

**We usually assume that T is a C**^{1}**transformation, which **
*means that g and h have continuous first-order partial *

derivatives.

*A transformation T is really just a function whose domain *
and range are both subsets of

*If T(u*_{1}*, v*_{1}*) = (x*_{1}*, y*_{1}*), then the point (x*_{1}*, y*_{1}) is called the
**image of the point (u**_{1}*, v*_{1}).

*If no two points have the same image, T is called *
**one-to-one.**

### Change of Variables in Multiple Integrals

*Figure 1 shows the effect of a transformation T on a region *
*S in the uv-plane. *

*T transforms S into a region R in the xy-plane called the *
**image of S, consisting of the images of all points in S.**

7

### Change of Variables in Multiple Integrals

**If T is a one-to-one transformation, then it has an inverse ****transformation T**^{–1} *from the xy-plane to the uv-plane and *
*it may be possible to solve Equations 3 for u and v in terms *
*of x and y:*

*u = G(x, y)* *v = H(x, y)*

### Example 1

A transformation is defined by the equations
*x = u*^{2} *– v*^{2 } *y = 2uv*
Find the image of the square

*S = {(u, v) | 0* *≤ u ≤ 1, 0 ≤ v ≤ 1}.*

Solution:

*The transformation maps the boundary of S into the *
boundary of the image.

*So we begin by finding the images of the sides of S.*

9

*Example 1 – Solution*

*The first side, S*_{1}*, is given by v = 0 (0* *≤ u ≤ 1). *

(See Figure 2.)

**Figure 2**

cont’d

*Example 1 – Solution*

*From the given equations we have x = u*^{2}*, y = 0, and so*
0 *≤ x ≤ 1.*

*Thus S*_{1 }is mapped into the line segment from (0, 0) to
*(1, 0) in the xy-plane.*

*The second side, S*_{2}*, is u = 1 (0* *≤ v ≤ 1) and, putting u = 1 *
in the given equations, we get

*x = 1 – v*^{2 } *y = 2v*

cont’d

11

*Example 1 – Solution*

*Eliminating v, we obtain*

*x = 1 –* 0 *≤ x ≤ 1*

which is part of a parabola.

*Similarly, S*_{3 }*is given by v = 1 (0* *≤ u ≤ 1), whose image is*
the parabolic arc

*x = – 1 –1 ≤ x ≤ 0*

cont’d

*Example 1 – Solution*

*Finally, S*_{4} *is given by u = 0 (0 ≤ v ≤ 1) whose image is *
*x = –v*^{2}*, y = 0, that is, –1 ≤ x ≤ 0. (Notice that as we move *
around the square in the counterclockwise direction, we
also move around the parabolic region

in the counterclockwise direction.)

*The image of S is the region R *
(shown in Figure 2) bounded by
*the x-axis and the parabolas *

given by Equations 4 and 5.

cont’d

13

### Change of Variables in Multiple Integrals

Now let’s 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 (u*_{0}*, v*_{0}) and whose

dimensions are ∆u and ∆v. (See Figure 3.)

**Figure 3**

### Change of Variables in Multiple Integrals

*The image of S is a region R in the xy-plane, one of whose *
*boundary points is (x*_{0}*, y*_{0}*) = T(u*_{0}*, v*_{0}).

The vector

**r(u, v) = g(u, v) i + h(u, v) j**

*is the position vector of the image of the point (u, v). *

*The equation of the lower side of S is v = v*_{0}, whose image
* curve is given by the vector function r(u, v* ).

15

### Change of Variables in Multiple Integrals

*The tangent vector at (x*_{0}*, y*_{0}) to this image curve is
**r**_{u}*= g*_{u}*(u*_{0}*, v*_{0}**) i + h**_{u}*(u*_{0}*, v*_{0}**) j**

*Similarly, the tangent vector at (x*_{0}*, y*_{0}) to the image curve of
*the left side of S (namely, u = u*_{0}) is

**r**_{v}*= g*_{v}*(u*_{0}*, v*_{0}**) i + h**_{v}*(u*_{0}*, v*_{0}**) j**

### Change of Variables in Multiple Integrals

*We can approximate the image region R = T(S) by a *
parallelogram determined by the secant vectors

**a = r(u**_{0} + ∆u, v_{0}**) – r(u**_{0}*, v*_{0}**) b = r(u**_{0}*, v*_{0 }+ ∆v) – r(u_{0}*, v*_{0})
shown in Figure 4.

17

### Change of Variables in Multiple Integrals

But

**and so r(u**_{0} + ∆u, v_{0}**) – r(u**_{0}*, v*_{0}) ≈ ∆u r* _{u}*
Similarly

**r(u**_{0}

*, v*

_{0 }+ ∆v) – r(u

_{0}

*, v*

_{0}) ≈ ∆v r

_{v}*This means that we can approximate R by a parallelogram *
determined by the vectors ∆u r* _{u }*and ∆v r

*. (See Figure 5.)*

_{v}**Figure 5**

### Change of Variables in Multiple Integrals

*Therefore we can approximate the area of R by the area*
of this parallelogram, which is

| (∆u r* _{u}*)

**× (∆v r**

_{v}**)| = | r**

_{u }**× r**

*| ∆u ∆v*

_{v}Computing the cross product, we obtain

19

### Change of Variables in Multiple Integrals

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

notation.

With this notation we can use Equation 6 to give an approximation to the area ∆A of R:

*where the Jacobian is evaluated at (u*_{0}*, v*_{0}).

### Change of Variables in Multiple Integrals

*Next we divide a region S in the uv-plane into rectangles S*_{ij}*and call their images in the xy-plane R** _{ij}*. (See Figure 6.)

21

### Change of Variables in Multiple Integrals

*Applying the approximation (8) to each R** _{ij}*, we approximate

*the double integral of f over R as follows:*

*where the Jacobian is evaluated at (u*_{i}*, v** _{j}*). Notice that this
double sum is a Riemann sum for the integral

### Change of Variables in Multiple Integrals

The foregoing argument suggests that the following theorem is true.

*Theorem 9 says that we change from an integral in x and y *
*to an integral in u and v by expressing x and y in terms of *
*u and v and writing*

23

### Change of Variables in Multiple Integrals

Notice the similarity between Theorem 9 and the one-dimensional formula in Equation 2.

*Instead of the derivative dx/du, we have the absolute value *
of the Jacobian, that is, |*∂(x, y)/∂(u, v)|.*

As a first illustration of Theorem 9, we show that the

formula for integration in polar coordinates is just a special case.

### Change of Variables in Multiple Integrals

*Here the transformation T from the r*θ*-plane to the xy-plane *
is given by

*x = g(r, *θ*) = r cos *θ *y = h(r, *θ*) = r sin *θ

and the geometry of the transformation is shown in Figure 7.

*T maps an ordinary rectangle in the r*θ -plane to a polar
*rectangle in the xy-plane.*

25

### Change of Variables in Multiple Integrals

*The Jacobian of T is*

Thus Theorem 9 gives

### Triple Integrals

27

### Triple Integrals

There is a similar change of variables formula for 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

*x = g(u, v, w)* *y = h(u, v, w) z = k(u, v, w)*

### Triple Integrals

**The Jacobian of T is the following 3 × 3 determinant:**

Under hypotheses similar to those in Theorem 9, we have the following formula for triple integrals:

29

### Example 4

Use Formula 13 to derive the formula for triple integration in spherical coordinates.

Solution:

Here the change of variables is given by

*x = *ρ sin φ cos θ *y = *ρ sin φ sin θ *z = *ρ cos φ

We compute the Jacobian as follows:

*Example 4 – Solution*

= cos φ (–ρ^{2} sin φ cos φ sin^{2 }θ – ρ^{2} sin φ cos φ cos^{2 }θ)
– ρ sin φ (ρ sin^{2} φ cos^{2 }θ *+ *ρ sin^{2} φ sin^{2 }θ)

= –ρ^{2} sin φ cos^{2} φ – ρ^{2} sin φ sin^{2} φ

= –ρ^{2} sin φ

cont’d

31

*Example 4 – Solution*

Therefore

and Formula 13 gives

*f(x, y, z) dV *

*= f(*ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ^{2} sin φ *d*ρ *d*θ *d*φ

cont’d