Page 221 - Schaum's Outline of Theory and Problems of Advanced Calculus

212 MULTIPLE INTEGRALS [CHAP. 9

Fig. 9-5

In the cylindrical case r ¼ cos i þ sin j þ zk and the set is
@r @r @r
¼ cos i þ sin j; ¼ sin i þ cos j; ¼ k
@ @ @z
@r @r @r
Therefore ¼ .
@ @ @z @r @r @r
That the geometric interpretation of d d dz is an infinitesimal rectangular parallele-
@ @ @z
piped suggests the differential element of volume in cylindrical coordinates is
dV ¼ d d dz

Thus, for an integrable but otherwise arbitrary function, Fð ; ; zÞ,of cylindrical coordinates, the
iterated triple integral takes the form
ð ð ð
z 2 g 2 ðzÞ f 2 ð ;zÞ
Fð ; ; zÞ d d dz
z 1 g 1 ðzÞ f 1 ð ;zÞ
The differential element of area for polar coordinates in the plane results by suppressing the z
coordinate. It is

@r
@r
d d
dA ¼
@ @
and the iterated form of the double integral is
ð ð
2 2 ð Þ
Fð ; Þ d d
1 1 ð Þ
The transformation equations relating spherical and rectangular Cartesian coordinates are
x ¼ r sin cos ; y ¼ r sin sin ; z ¼ r cos
In this case the coordinate surfaces are spheres, cones, and planes. (See Fig. 9-5.)