Page 231 - Schaum's Outline of Theory and Problems of Advanced Calculus

222 MULTIPLE INTEGRALS [CHAP. 9

The vector r from the origin O to point P is
r ¼ xi þ yj þ zk ¼ f ðu; v; wÞi þ gðu; v; wÞj þ hðu; v; wÞk
assuming that the transformation equations are x ¼ f ðu; v; wÞ; y ¼ gðu; v; wÞ, and z ¼ hðu; v; wÞ.
Tangent vectors to the coordinate curves corresponding to the intersection of pairs of coordinate
surfaces are given by @r=@u;@r=@v;@r=@w. Then the volume of the region r of Fig. 9-17 is given approxi-
mately by
@r @r

@r @ðx; y; zÞ
u v w

@u @v @w u v w ¼ @ðu; v; wÞ
The triple integral of Fðx; y; zÞ over the region is the limit of the sum

X
@ðx; y; zÞ
u v w

Ff f ðu; v; wÞ; gðu; v; wÞ; hðu; v; wÞg

@ðu; v; wÞ
An investigation reveals that this limit is

ðð ð

@ðx; y; zÞ
du dv dw
F f f ðu; v; wÞ; gðu; v; wÞ; hðu; v; wÞg

@ðu; v; wÞ
r 0
where r is the region in the uvw space into which the region r is mapped under the transformation.
0
Another method for justifying the above change of variables in triple integrals makes use of Stokes’
theorem (see Problem 10.84, Chapter 10).
9.14. What is the mass of a circular cylindrical body represented by the region
2
0 @ @ c; 0 @ @ 2 ; 0 @ z @ h, and with the density function ¼ z sin ?
h
ð ð 2 ð c
2
z sin d d dz ¼
M ¼
0 0 0
9.15. Use spherical coordinates to calculate the volume of a sphere of radius a.
a
ð ð =2 ð =2 4
2
V ¼ 8 a sin dr d d ¼ a 3
0 0 0 3
ðð ð
9.16. Express Fðx; y; zÞ dx dy dz in (a) cylindrical and (b) spherical coordinates.
r
(a) The transformation equations in cylindrical coordinates are x ¼ cos ; y ¼ sin ; z ¼ z.
As in Problem 6.39, Chapter 6, @ðx; y; zÞ=@ð ; ; zÞ¼ . Then by Problem 9.13 the triple integral
becomes
ðð ð
Gð ; ; zÞ d d dz
r 0
where r 0 is the region in the ; ; z space corresponding to r and where Gð ; ; z
Fð cos ; sin ; zÞ.
(b) The transformation equations in spherical coordinates are x ¼ r sin cos ; y ¼ r sin sin ; z ¼ r cos .
2
By Problem 6.101, Chapter 6, @ðx; y; zÞ=@ðr; ; Þ¼ r sin . Then by Problem 9.13 the triple
integral becomes
ððð
2
Hðr; ; Þr sin dr d d
r 0
where r is the region in the r; ; space corresponding to r, and where Hðr; ; Þ Fðr sin cos ,
0
r sin sin ; r cos Þ.