Page 220 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 211

TRANSFORMATIONS OF MULTIPLE INTEGRALS
In evaluating a multiple integral over a region r,itis often convenient to use coordinates other than
rectangular, such as the curvilinear coordinates considered in Chapters 6 and 7.
If we let ðu; vÞ be curvilinear coordinates of points in a plane, there will be a set of transformation
equations x ¼ f ðu; vÞ; y ¼ gðu; vÞ mapping points ðx; yÞ of the xy plane into points ðu; vÞ of the uv plane.
In such case the region r of the xy plane is mapped into a region r of the uv plane. We then have
0
ðð ðð
@ðx; yÞ
du dv

Fðx; yÞ dx dy ¼ Gðu; vÞ ð9Þ

@ðu; vÞ
r r 0
where Gðu; vÞ Ff f ðu; vÞ; gðu; vÞg and

@x @x

@u
@ðx; yÞ
@v
@y
ð10Þ
@y
@ðu; vÞ

@u @v
is the Jacobian of x and y with respect to u and v (see Chapter 6).
Similarly if ðu; v; wÞ are curvilinear coordinates in three dimensions, there will be a set of transfor-
mation equations x ¼ f ðu; v; wÞ; y ¼ gðu; v; wÞ; z ¼ hðu; v; wÞ and we can write
ðð ð ðð ð
@ðx; y; zÞ
du dv dw

Fðx; y; zÞ dx dy dz ¼ Gðu; v; wÞ ð11Þ

@ðu; v; wÞ
r r 0
where Gðu; v; wÞ Fff ðu; v; wÞ; gðu; v; wÞ; hðu; v; wÞg and
@x @x

@x

@u @v @w

@y @y @y

@ðx; y; zÞ

ð12Þ
@u @v

@ðu; v; wÞ
@w
@z @z @z

@u @v @w

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.
THE DIFFERENTIAL ELEMENT OF AREA IN POLAR COORDINATES, DIFFERENTIAL
ELEMENTS OF AREA IN CYLINDRAL AND SPHERICAL COORDINATES
Of special interest is the differential element of area, dA, for polar coordinates in the plane, and the
differential 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 Chapter 7, in particular,
x ¼ cos ; y ¼ sin ; z ¼ z
The coordinate surfaces are circular cylinders, planes, and planes. (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.