15.9

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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 dθ

where S is the region in the rθ-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 C¹ 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(u₁, v₁) = (x₁, y₁), then the point (x₁, y₁) is called the image of the point (u₁, v₁).

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 = u² – v² 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.

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Example 1 – Solution

The first side, S₁, 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 = u², y = 0, and so 0 ≤ x ≤ 1.

Thus S₁ is mapped into the line segment from (0, 0) to (1, 0) in the xy-plane.

The second side, S₂, is u = 1 (0 ≤ v ≤ 1) and, putting u = 1 in the given equations, we get

x = 1 – v² y = 2v

cont’d

11

Example 1 – Solution

Eliminating v, we obtain

x = 1 – 0 ≤ x ≤ 1

which is part of a parabola.

Similarly, S₃ 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, S₄ is given by u = 0 (0 ≤ v ≤ 1) whose image is x = –v², 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 (u₀, v₀) 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 (x₀, y₀) = T(u₀, v₀).

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 = v₀, whose image curve is given by the vector function r(u, v ).

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Change of Variables in Multiple Integrals

The tangent vector at (x₀, y₀) to this image curve is r_u = g_u(u₀, v₀) i + h_u(u₀, v₀) j

Similarly, the tangent vector at (x₀, y₀) to the image curve of the left side of S (namely, u = u₀) is

r_v = g_v(u₀, v₀) i + h_v(u₀, v₀) 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(u₀ + ∆u, v₀) – r(u₀, v₀) b = r(u₀, v₀ + ∆v) – r(u₀, v₀) shown in Figure 4.

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Change of Variables in Multiple Integrals

But

and so r(u₀ + ∆u, v₀) – r(u₀, v₀) ≈ ∆u r_u Similarly r(u₀, v₀ + ∆v) – r(u₀, v₀) ≈ ∆v r_v

This means that we can approximate R by a parallelogram determined by the vectors ∆u r_u and ∆v r_v. (See Figure 5.)

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_v| ∆u ∆v

Computing the cross product, we obtain

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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 (u₀, v₀).

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.)

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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

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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 rθ-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 rθ -plane to a polar rectangle in the xy-plane.

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Change of Variables in Multiple Integrals

The Jacobian of T is

Thus Theorem 9 gives

Triple Integrals

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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:

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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 (–ρ² sin φ cos φ sin² θ – ρ² sin φ cos φ cos² θ) – ρ sin φ (ρ sin² φ cos² θ + ρ sin² φ sin² θ)

= –ρ² sin φ cos² φ – ρ² sin φ sin² φ

= –ρ² sin φ

cont’d

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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) ρ² sin φ dρ dθ dφ

cont’d