Curvilinear Coordinate Systems 149
J(liiQ2,Q3) = hihzha. Try to derive this equality directly by expand-
ing the determinant in (3.1.48) and using the orthogonality conditions
(3.1.38). (b)c Verify that J(r,6,z) = r (cylindrical coordinates) and
J(r,6,<p) = r^2 simp (spherical coordinates).
Let G be a measurable domain in M^3 with cartesian coordinates 3£, and
let / = f{xi,X2,X3) be a continuous scalar field on G. A change of coor-
dinates % = (\k(xi,X2,xs), k = 1,2,3, maps every point (xi,X2,£3) to a
point (91,92,93), which is also in M^3. Under this map,
- The domain G transforms into another domain G (of course, there are
certain conditions on the functions c\k to ensure that the image of a domain
is again a domain);
- A small box with sides a,b,c around a point P = (xi,:r2,:E3) becomes a
small curvilinear box around the image of P; by (3.1.47), the volume of this
box is approximately abc\J(qi, 92,93)!, with approximation getting better
as max(a, 6, c) —> 0;
- The function / becomes / = /(9i,92,92) so that /(9i,92,93) =
/(xi(9i, 92,93), X2(9i, 92,93), X3(9i, 92,93))-
Using the general definition of the Riemann integral (3.1.13), we get the
following change of variables formula for the volume integral:
f(xux 2 ,x 3 )dV(X) = ^y"/(9i,92,93)| J(9i, 92,93) WQ)-
G G
(3.1.49)
Notice that the change of coordinates from X to Q is given by the functions
qfc, k — 1,2,3, whereas, to compute the Jacobian and to find the function
/, we need the functions Xk defining the change in the opposite direction.
A result similar to (3.1.49) holds for area integrals, if we write 9fc =
(\k(^i,x 2 ), k = 1,2, and 93 = £3.
EXERCISE 3.1.35? (a) Verify (3.1.49). (b) Write the analogs of (3.1.47) and
(3.1.49) in two dimensions. Hint: r[(0) xr^O) = J{qi,q2)(t- (c) Interpret
the vector parametric representation r = r(u, v), (u, v) G G, of a surface S
as a map from G to S and derive the familiar formula for the surface area
m(S) = // \\ru x rv\\dA(u,v).
G
Hint: show that a small rectangle with sides a,b in G becomes a curvilinear figure
on S with area approximately equal to ab\\ru x rv\\. Compare this with Exercise
3.1.15 on page 134.
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