change of variables in integral on ℝ^n

change of variables in integral on

ℝ

n

Theorem 1

.

Let

g

:

X

→

Y

be a

diffeomorphism

between
open subsets

X

and

Y

of

R

n

.
Then for any

measurable function

f

:

Y

→

R

, and any

measurable
set

E

⊆

X

,

∫

E

f

⁢

(

g

⁢

(

x

)

)

⁢

|

det

⁡

D

⁡

g

⁢

(

x

)

|

⁢

𝑑

x

=

∫

g

⁢

(

E

)

f

⁢

(

y

)

⁢

𝑑

y

.

Also, if one of these

integrals

does not exist, then neither does the other.

This

theorem

is a

generalization

of the

substitution rule

for integrals from one-variable

calculus

.

To go from the left-hand side to the right-hand side or vice versa,
we can perform the formal substitutions:

y

=

g

⁢

(

x

)

,

d

⁢

y

=

g

⁢

(

d

⁢

x

)

=

|

det

⁡

D

⁡

g

⁢

(

x

)

|

⁢

d

⁢

x

.

The volume

scaling

factor

|

det

⁡

D

⁡

g

⁢

(

x

)

|

is sometimes called
the

Jacobian

or

Jacobian determinant

.

Theorem 1 is typically applied
when integrating over

ℝ

2

using

polar coordinates

,
or when integrating over

ℝ

3

using cylindrical or

spherical coordinates

.

Intuitively speaking, the image of a small cube centered at

x

,
under a

differentiable map

g

is approximately
the

parallelogram

resulting from the linear mapping

D

⁡

g

⁢

(

x

)

applied on that cube. If the volume of the original cube is

d

⁢

x

,
then the volume of the image parallelogram is

d

⁢

y

=

|

det

⁡

D

⁡

g

⁢

(

x

)

|

⁢

d

⁢

x

.
The integral

formula

in Theorem 1 follows for an
arbitrary set by approximating it by many numbers of small cubes,
and taking limits.

Figure 1:

Illustration of
linear approximation to

g

⁢

(

Q

)

by

x

+

D

⁡

g

⁢

(

x

)

⁢

(

Q

-

x

)

.

http://aux.

planetmath

.org/files/objects/7349/jacobian.py

Source program in Python for diagram

Proofs of Theorem 1 can be obtained
by making this procedure

rigorous

;
see

[

7

]

,

[

1

]

, or

[

3

]

.

A slightly stronger version of the theorem that does not require

g

to be a diffeomorphism
(i.e. that

g

is a

bijection

and has

non-singular

derivative

) is:

Theorem 2

.

Let

g

:

X

→

R

n

be continuously differentiable
on an open subset

X

of

R

n

.
Then for any measurable function

f

:

Y

→

R

, and
any measurable set

E

⊆

X

,

∫

E

f

⁢

(

g

⁢

(

x

)

)

⁢

|

det

⁡

D

⁡

g

⁢

(

x

)

|

⁢

𝑑

x

=

∫

g

⁢

(

E

)

f

⁢

(

y

)

⁢

#

⁢

g

|

E

-

1

⁢

(

y

)

⁢

d

⁢

y

,

where

#

⁢

g

|

E

-

1

⁢

(

y

)

∈

{

1

,

2

,

…

,

∞

}

counts the number of pre-images in

E

of

y

.

Observe that Theorem 2 (as well as its proof) includes
a special case of Sard’s Theorem.

The idea of Theorem 2 is that we may ignore those pieces of the set

E

that transform to zero volumes, and if the map

g

is not

one-to-one

,
then some pieces of the image

g

⁢

(

E

)

may be counted multiple times
in the left-hand integral.

These formulas can also be generalized for

Hausdorff measures

(

http://planetmath.org/AreaFormula

) on

ℝ

n

,
and non-differentiable, but

Lipschitz

,

functions

g

. See

[

4

]

or other geometric measure theory books for details.

References

1

T. M. Flett. “On

Transformations

in

ℝ

n

and a Theorem of Sard”.

American Mathematical Monthly

, Vol. 71, No. 6 (Jun–Jul 1964),
p. 623–629.

2

Gerald B. Folland.

Real Analysis

: Modern Techniques and Their Applications

,
second ed. Wiley-Interscience, 1999.

3

Miguel De Guzman. “

Change-of-Variables Formula

Without

Continuity

”.

American Mathematical Monthly

, Vol. 87, No. 9 (Nov 1980), p. 736–739.

4

Frank Morgan.

Geometric Measure Theory: A Beginner’s Guide

, second ed.
Academic Press, 1995.

5

James R. Munkres.

Analysis on Manifolds

. Westview Press, 1991.

6

Arthur Sard. “

http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07812-8/S0002-9904-1942-07812-8.pdf

The

Measure

of the

Critical Values

of Differentiable Maps”.

Bulletins of the

American Mathematical Society

, Vol. 48 (1942), No. 12, p. 883-890.

7

J. Schwartz. “The Formula for Change in Variables in a

Multiple Integral

”.

American Mathematical Monthly

, Vol. 61, No. 2 (Feb 1954), p. 81–95.

8

Michael Spivak.

Calculus on Manifolds

. Perseus Books, 1998.

Title

change of variables in integral on

ℝ

n

Canonical name

ChangeOfVariablesInIntegralOnmathbbRn

Date of creation

2013-03-22 15:29:32

Last modified on

2013-03-22 15:29:32

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

8

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 28A25

Classification

msc 26B15

Classification

msc 26B10

Synonym

integral substitution formula

Synonym

integral substitution rule

Synonym

change-of-variables formula

Related topic

JacobiDeterminant

Related topic

LebesgueMeasure

Related topic

AreaFormula

Related topic

PotentialOfHollowBall

Related topic

ExampleOfRiemannTripleIntegral

Related topic

ExampleOfRiemannDoubleIntegral

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