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 @x   @w   @y   @w   @z    @w
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is the Jacobian of x, y, and z with respect to u, v, and w. The results (9) and (11) correspond to change of variables for double and triple integrals. Generalizations to higher dimensions are easily made.
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THE DIFFERENTIAL ELEMENT OF AREA IN POLAR COORDINATES, DIFFERENTIAL ELEMENTS OF AREA IN CYLINDRAL AND SPHERICAL COORDINATES Of special interest is the di erential element of area, dA, for polar coordinates in the plane, and the di erential elements of volume, dV, for cylindrical and spherical coordinates in three space. With these in hand the double and triple integrals as expressed in these systems are seen to take the following forms. (See Fig. 9-5.) The transformation equations relating cylindrical coordinates to rectangular Cartesian ones appeared in 7, in particular, x  cos ; y  sin ; z z (See Fig. 9-5.) & ' @r @r @r At any point of the space (other than the origin), the set of vectors ; ; constitutes an @ @ @z orthogonal basis. The coordinate surfaces are circular cylinders, planes, and planes.
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MULTIPLE INTEGRALS
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[CHAP. 9
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Fig. 9-5
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In the cylindrical case r  cos i  sin j zk and the set is @r cos i sin j; @ Therefore @r  sin i  cos j; @ @r k @z
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@r @r @r . @ @ @z @r @r @r d d dz is an in nitesimal rectangular paralleleThat the geometric interpretation of @ @ @z piped suggests the di erential element of volume in cylindrical coordinates is dV  d d dz
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Thus, for an integrable but otherwise arbitrary function, F ; ; z , of cylindrical coordinates, the iterated triple integral takes the form z2 g2 z f2 ;z F ; ; z  d d dz
z1 g1 z f1 ;z
The di erential element of area for polar coordinates in the plane results by suppressing the z coordinate. It is    @r @r  dA   d d @ @ and the iterated form of the double integral is 2 2 
1 1 
F ;   d d
The transformation equations relating spherical and rectangular Cartesian coordinates are x r sin  cos ; y r sin  sin ; z r cos  (See Fig. 9-5.)
In this case the coordinate surfaces are spheres, cones, and planes.
CHAP. 9]
MULTIPLE INTEGRALS
Following the same pattern as with cylindrical coordinates we discover that dV r2 sin  dr d d and the iterated triple integral of F r; ;  has the spherical representation r2 2  2 r; F r; ;  r2 sin  dr d d
r1 1  1 r;
Of course, the order of these integrations may be adapted to the geometry. The coordinate surfaces in spherical coordinates are spheres, cones, and planes. constant, say, r a, then we obtain the di erential element of surface area dA a2 sin  d d The rst octant surface area of a sphere of radius a is =2 =2 =2 =2   2 a2 sin  d d a2 cos  0 d a2 d a2 2 0 0 0 0 Thus, the surface area of the sphere is 4a2 .
If r is held
Solved Problems
DOUBLE INTEGRALS 9.1. (a) (b) (c) Sketch the region r in the xy plane bounded by y x2 ; x 2; y 1. Give a physical interpreation to x2 y2 dx dy.
Evaluate the double integral in (b).
(a) The required region r is shown shaded in Fig. 9-6 below. (b) Since x2 y2 is the square of the distance from any point x; y to 0; 0 , we can consider the double integral as representing the polar moment of inertia (i.e., moment of inertia with respect to the origin) of the region r (assuming unit density).
Fig. 9-6
Fig. 9-7
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