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The Change of Variables Theorem for Integrals

The case for double integrals will be considered first; i.e., those over two dimensional regions. However there is one preliminary that must be established before they are considered.

The Area of a Parallelogram
Defined by Two Vectors

Let ux and uy be the unit vectors in the x and y directions, respectively. Let two vectors be given by

A = axux + ayuy
and
B = bxux + byuy

The area P of the parallelogram for which the vectors A and B are two adjacent sides is given by

P = |A||B|sin(|θ|)

where |A| and |B| are the lengths of the two vectors and |θ| is the positive angle between them. Suppose A is the base of their parallelogram and B is the slanted side.

The area P can also be expressed as the value of the determinant created from the components of the vectors; i.e.,

| axbx |
P =det|          |
| ayby |

For more on the topic of determinants and area see determinants and area/volumes.


The Change of Variables
Theorem for Double Integrals

Let ∫∫Rf(x, y)dxdy be an integral over Euclidean 2-space. Let T be a transformation from (x, y) space to (u, v) space such that

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

and such that the inverse transformation

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

exists. This requirement limits the transformations to continuous ones. Let the inverse transformation be denoted as W.

Then

∫∫Rf(x, y)dxdy = ∫∫S|det(J)|f(x(u,v), y(u, v))dudv

where |det(J)| is the absolute value of the determinant of the derivative matrix J for the inverse transformation W. The derivative matrix for W is defined as

|(∂x/∂u)(∂x/∂v)|
J=|                         |
|(∂y/∂u)(∂y/∂v)|

This is known as the Jacobian matrix of the transformation. The Jacobian K for the transformation T is

|(∂u/∂x)(∂u/∂y)|
K=|                        |
|(∂v/∂x)(∂v/∂y)|

It is the inverse transformation that is relevant for the change of variables for integrals.

The relationship between J and K is that the composition of T and W or W and T is the identity transformation and hence the products KJ and JK are equal to the 2×2 identity matrix I. Thus

J = K-1
and
K = J-1

Furthermore, of course,

det(J) = 1/det(K)

The region S is the image of the transformation T of the region R.


Now consider the four corners of an area element ΔxΔy in (x, y) space

(x0, y0), (x0+Δx, y0),
(x0, y0+Δy), (x0+Δx, y0+Δy)

Under the transformation T and for small Δx and Δy the four corners are transformed to

(u0, v0),
(u0+(∂u/∂x)Δx, v0+(∂v/∂x)Δx),
(u0+(∂u/∂y)Δy, v0+(∂v/∂y)Δy)
(u0+(∂u/∂x)Δx + (∂u/∂y)Δy, v0+(∂v/∂x)Δx+(∂v/∂y)Δy)

The two vectors emanating from (u0, v0) are:

U = ((∂u/∂x)Δx, (∂v/∂x)Δx)
and
V = ((∂u/∂y)Δy, (∂v/∂y)Δy)

When a determinant is formed from the components of these vectors its value can be reduced to

|(∂u/∂x)(∂u/∂y)|
ΔuΔv =det|                         |ΔxΔy
|(∂v/∂x)(∂v/∂y)|

This is none other than

ΔuΔv = |det(K)|ΔxΔy

This means that

ΔxΔy = (1/|det(K)|)ΔuΔv = |det(J)|ΔuΔv

So, in changing the variables of integration in ∫∫Rf(x, y)dxdy from (x, y) to (u, v) through the transformation T what is needed is the Jacobian of the inverse transformation W.

Illustration

Let (x, y) be the Cartesian coordinates of the plane and (r, θ) the polar coordinates. The transformation of (x, y) to (r, θ) would be given by

r = (x² + y²)½
and
θ = tan-1(y/x)

The inverse transformation, that is to say the transformation of (r, θ) to (x, y), is

x = r·cos(θ)
and
y = r·sin(θ)

It is the Jacobian of this inverse transformation that is relevant in changing the variables in a double integral from (x, y) to (r, θ). That Jacobian is r so

∫∫f(x,y)dxdy → ∫∫F(r, θ)rdrdθ

The Case of Triple Integrals

Consider an integral of the form

∫∫∫Rf(x,y,z)dxdydz

If the variables of integration are to be changed from (x, y, z) to (u, v, w) then the integral in terms of (u, v, w) that is equal to the above triple integral is

∫∫∫S|det(J)|f(x(u,v,w),y(u,v,w),z(u,v,w))dudvdw

where J is the Jacobian derivative matrix of the transformation of (u,v,w) to (x,y,z); i.e.,

x = x(u,v,w)
y = y(u,v,w)
z = z(u,v,w)

That Jacobian derivative matrix is

| (∂x/∂u)(∂x/∂v)(∂x/∂w)|
J =|(∂y/∂u)(∂y/∂v)(∂y/∂w)|
|(∂z/∂u)(∂z/∂v)(∂z/∂w)|

The transformation from (x,y,z) to (u,v,w) is also relevant because the region of integration S in the (u,v,w) space is the image of R under that transformation.

Now consider the transformation of the volume element ΔuΔvΔw into the (x,y,z) space. There are eight corner points to the volume element, but it is only the following four that are relevant:

(u0, v0, w0), (u0+Δu, v0, w0)
(u0, v0+Δv, w0), (u0, v0, w0+Δw)

These can be viewed as the point (u0, v0, w0) with three vectors (Δu, 0, 0), (0, Δv, 0), (0, 0, Δw).

The point (u0, v0, w0) goes to the point (x0, y0, z0). The vector (Δu, 0, 0) induces the vector (((∂x/∂u)Δu, ((∂y/∂u)Δu, ((∂z/∂u)Δu) of deviations from (x0, y0, z0). Likewise the vector (0, Δv, 0) induces the vector ((∂x/∂v)Δv, ((∂y/∂v)Δv, ((∂z/∂v)Δv) of deviations. And finally the vector (0, 0, Δw) induces the vector (((∂x/∂w)Δw, ((∂y/∂w)Δw, ((∂z/∂u)Δw). These three vectors of deviations from the point (x0, y0, z0) determine a parallelpiped volume element in the (x, y, z) space. The volume of the parallelpiped is equal to the determinant of the matrix formed using the three vectors of deviations as columns. That matrix is

| (∂x/∂u)(∂x/∂v)(∂x/∂w)|
|(∂y/∂u)(∂y/∂v)(∂y/∂w)|
|(∂z/∂u)(∂z/∂v)(∂z/∂w)|

This matrix is simply the Jacobian of the transformation from (u,v,w) to (x,y,z), above denoted as J. Thus

∫∫∫Rf(x,y,z)dxdydz is equal to
∫∫∫S|det(J)|f(x(u,v,w),y(x(u,v,w),z(x(u,v,w))dudvdw

To rigorously establish the equality it is necessary to go back to the limiting process which defines the integrals and show that the partitioning of region R by the images of partitioning elements of S is valid.

The General Case

After going through the two and three dimensional cases it is clear how the general case of transforming the variables of integration from (x1, …, xn) to (u1, …, un) should be carried out; i.e.,

∫…∫Rf( (x1, …, xn)dx1…dxn
= ∫…∫Sf( (x1|det(J)|f( (x1(u1,…, un)), …, xn(u1,…, un))du1…dun

where J is the Jacobian matrix of the inverse transformation of (u1, …, un) to (x1, …, xn)

The inverse transformation is represented as

xj = xj(u1, …, un)
for j=1, 2, …, n.

When uk changes by an amount Δuk the x variables change by the amounts given by

((∂x1/∂uk)Δuk, …, (∂xn/∂uk)Δuk)
for j=1, 2, … n.

The transformation of the parallelotopes (parallelogram, parallelopiped, etc.) of an area/volume element Δu1*…*Δun to the (x1, …, xn) space creates an area/volume element whose generalized volume is

|det(M)|Δu1*…*Δun

where the j,k element of the matrix M is

(∂xj/∂uk)

But this means that the matrix M is none other than the Jacobian matrix for the transformation of the u variables to the x variables. Therefore the area/volume element of dxl*…*dxn is transformed to

|det(J)|dul*…*dun

(To be continued.)

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