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Chapter 15 – Multiple Integrals15.10 Change of Variables in Multiple Integrals

15.10 Change of Variables in Multiple Integrals

Objectives: How to change variables

for double and triple integrals

Carl Gustav Jacob Jacobi

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Change of Variable - Single 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 the Substitution Rule (Equation 6 in Section 5.5) as:

where x = g(u) and a = g(c), b = g(d).

( ) ( ( )) '( )b d

a cf x dx f g u g u du

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Change of Variables - DoubleA change of variables can also be useful in

double integrals.

◦ We have already seen one example of this: conversion to polar coordinates where the new variables r and θ are related to the old variables x and y by:

x = r cos θ

y = r sin θ◦

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Change of Variables - DoubleThe change of variables formula (Formula 2

in Section 15.4) can be written as:

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

( , ) ( cos , sin )R S

f x y dA f r r r dr d

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TransformationMore 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:

x = g(u, v) y = h(u, v)

◦ We sometimes write these as: x = x(u, v), y = y(u, v)

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C1 transformationWe usually assume that T is a C1

transformation.

◦ This means that g and h have continuous first-order partial derivatives.

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Image & One-to-one Transformation 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.

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Change of VariablesThe figure 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.

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Inverse Transform If T is a one-to-one transformation, it has an

inverse transformation T–1 from the xy–plane to the uv-plane.

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Double IntegralsNow, 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).

◦ Dimensions are ∆u and ∆v.

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Double IntegralsThe image of S is a region R in the xy-plane,

one of whose boundary points is: (x0, y0) = T(u0, v0)

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Double IntegralsWe can approximate R by a parallelogram

determined by the vectors ∆u ru and ∆v rv

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Double IntegralsThus, we can approximate the area of R by

the area of this parallelogram, which, from Section 12.4, is:

|(∆u ru) x (∆v rv)| = |ru x rv| ∆u ∆v

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Double IntegralsComputing the cross product, we obtain:

0

0

u v

x y x xx y u u u v

x y y yu ux y v u u vv u

i j k

r r k k

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JacobianThe determinant that arises in this

calculation is called the Jacobian of the transformation.◦ It is given a special notation.

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Definition - Jacobian of TThe Jacobian of the transformation T given

by x = g(u, v) and y = h(u, v) is:

( , )

( , )

x xx y x y x yu v

y yu v u v v u

u v

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Jacobian of TWith this notation, we can give an

approximation to the area ∆A of R:

where the Jacobian is evaluated at (u0, v0).

( , )

( , )

x yA u v

u v

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Math Fun FactThe Jacobian is named after the German

mathematician Carl Gustav Jacob Jacobi (1804–1851).

◦ The French mathematician Cauchy first used these special determinants involving partial derivatives.

◦ Jacobi, though, developed them into a method for evaluating multiple integrals.

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Example 1 – pg. 1020Find the Jacobian of the

transformation.

2. ,

4. ,s t s t

ux uv y

v

x e y e

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Change of Variables in a Double Integral – Theorem 9Suppose:

◦ T is a C1 transformation whose Jacobian is nonzero and that maps a region S in the uv-plane onto a region R in the xy-plane.

◦ f is continuous on R and that R and S are type I or type II plane regions.

◦ T is one-to-one, except perhaps on the boundary of S.

Then,( , )

( , ) ( ( , ), ( , ))( , )R S

x yf x y dA f x u v y u v du dv

u v

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Example 2 – pg. 1020 # 12Use the given transformation to

evaluate the integral.

4 8 , where is the parallelogram

with vertices (-1, 3), (1, -3), (3, -1), and (1, 5);

1 1, 3

4 4

R

x y dA R

x u v y v u

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Example 3 – pg. 1020 # 20Evaluate the integral by making

the appropriate change of variables.

2 2

, where is the rectangle enclosed

by the lines 0, 2, 0, and 3.

x y

R

x y e dA R

x y x y x y x y

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Triple IntegralsThere 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)

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Triple Integrals - Equation 12The Jacobian of T is this 3 x 3 determinant:

( , , )

( , , )

x x x

u v wx y z y y y

u v w u v wz z z

u v w

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Triple IntegralsUnder hypotheses similar to those in

Theorem 9, we have this formula for triple integrals:

( , , )

( , , )( ( , , ), ( , , ), ( , , ))

( , , )

R

S

f x y z dV

x y zf x u v w y u v w z u v w du dv dw

u v w

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Example 5 – pg. 1020 #5Find the Jacobian of the

transformation.

, ,u v w

x y zv w u