Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

y

x

u=constant

v=constant

N

M

L

K

R

C

Figure 6.10 A region of integrationRoverlaid with a grid formed by the

family of curvesu=constantandv= constant. The parallelogramKLMN

defines the area elementdAuv.

express a multiple integral in terms of a new set of variables. We now consider

how to do this.

6.4.1 Change of variables in double integrals

Let us begin by examining the change of variables in a double integral. Suppose

that we require to change an integral

I=

∫∫

R

f(x, y)dx dy,

in terms of coordinatesxandy, into one expressed in new coordinatesuandv,

given in terms ofxandyby differentiable equationsu=u(x, y)andv=v(x, y)

with inversesx=x(u, v)andy=y(u, v). The regionRin thexy-plane and the

curveCthat bounds it will become a new regionR′and a new boundaryC′in

theuv-plane, and so we must change the limits of integration accordingly. Also,

the functionf(x, y) becomes a new functiong(u, v) of the new coordinates.

Now the part of the integral that requires most consideration is the area element.

In thexy-plane the element is the rectangular areadAxy=dx dygenerated by

constructing a grid of straight lines parallel to thex-andy- axes respectively.

Our task is to determine the corresponding area element in theuv-coordinates. In

general the corresponding elementdAuvwill not be the same shape asdAxy, but

this does not matter since all elements are infinitesimally small and the value of

the integrand is considered constant over them. Since the sides of the area element

are infinitesimal,dAuvwill in general have the shape of a parallelogram. We can

find the connection betweendAxyanddAuvby considering the grid formed by the

family of curvesu=constantandv= constant, as shown in figure 6.10. Sincev