Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

264 Multiple and Line Integrals [Ch. 7
3) the functional determinant (Jacobian)
of these functions
dx dx dx
dii ~dv dw
dy dy dy
D (u, v, w) da dv dw
dz dz dz
da dv dw
retains a constant sign in the region V, then we can make use of the formula
$ J \f(x,y,2)dxdydz =
(V)
\ \
\
f IT ( w L? w > )> ty( u y > * w )< du \
Fig. 98 Fig. 99
In particular,
1) for cylindrical coordinates r, cp, h (Fig. 98), where
we get / r\
X rcosrp, // rsinrp, z^-//,
dv dw.
2) for spherical coordinates r
cp, ap,
is the ( the latitude,
r the radius vector) (Fig. 99), where
x = r cos i|) cos 9, f/ = /-cosi|3 sin q), z / sin v|\
we have / r 2 cos i[).
Example 3. Passing to spherical coordinates, compute
JSJ
(V)
where V is a sphere of radius R.
Solution. For a sphere, the ranges of the spherical coordinates fp (longitude),
\|) (latitude), and r (radius vector) will be