James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 15.9 Change of Variables in Multiple Integrals 1059
The Jacobian of T is the following 3 3 3 determinant:
12
−x
−u
−sx, y, zd
−su, v, wd − −y
−u
−z
−u
−x
−v
−y
−v
−z
−v
−x
−w
−y
−w
−z
−w
Under hypotheses similar to those in Theorem 9, we have the following formula for triple
integrals:
13 y y f sx, y, zd dV −y y
R
S
y f sxsu, v, wd, ysu, v, wd, zsu, v, wdd Z
−sx, y, zd
−su, v, wd
Z du dv dw
ExamplE 4 Use Formula 13 to derive the formula for triple integration in spherical
coordinates.
SOLUtion Here the change of variables is given by
x − sin cos y − sin sin z − cos
We compute the Jacobian as follows:
Z
sin cos 2 sin sin
−sx, y, zd
−s, , d − sin sin sin cos
cos 0
2 sin sin
− cos Z
sin cos
Z
cos cos
cos sin
2 sin
cos cos
cos sin
Z 2 sin Z
sin cos
sin sin
2 sin sin
sin cos
Z
− cos s2 2 sin cos sin 2 2 2 sin cos cos 2 d
2 sin s sin 2 cos 2 1 sin 2 sin 2 d
− 2 2 sin cos 2 2 2 sin sin 2 − 2 2 sin
Since 0 < < , we have sin > 0. Therefore
Z
−sx, y, zd
−s, , d
Z − | 22 sin | − 2 sin
and Formula 13 gives
y y f sx, y, zd dV − y R
S
y f s sin cos , sin sin , cos d 2 sin d d d
which is equivalent to Formula 15.8.3.
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