Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

MULTIPLE INTEGRALS


z

x

y

C


R


T


S P Q


u=c 1

v=c 2

w=c 3

Figure 6.12 A three-dimensional region of integrationR, showing an el-
ement of volume inu, v, wcoordinates formed by the coordinate surfaces
u=constant,v=constant,w=constant.

evaluate the integrals over the inner and outer circles respectively, and we find


π

(
1 −e−a

2 )
<I^2 (a)<π

(
1 −e−^2 a

2 )
.

Taking the limita→∞, we findI^2 (a)→π. ThereforeI=



π, as we found previ-

ously. Substitutingx=



αyshows that the corresponding integral of exp(−αx^2 )

has the value



π/α. We use this result in the discussion of the normal distribution

in chapter 30.


6.4.3 Change of variables in triple integrals

A change of variable in a triple integral follows the same general lines as that for


a double integral. Suppose we wish to change variables fromx, y, ztou, v, w.


In thex, y, zcoordinates the element of volume is a cuboid of sidesdx, dy, dz


and volumedVxy z=dx dy dz. If, however, we divide up the total volume into


infinitesimal elements by constructing a grid formed from the coordinate surfaces


u=constant,v=constantandw= constant, then the element of volumedVuvw


in the new coordinates will have the shape of a parallelepiped whose faces are the


coordinate surfaces and whose edges are the curves formed by the intersections of


these surfaces (see figure 6.12). Along the line elementPQthe coordinatesvand

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