Mathematics of Physics and Engineering

Curvilinear Coordinate Systems 145

the usual way; for example,

9f = nm /(gl+t,g2,g3)-/(gl,g2,g3)

dqi t-»o t

Unlike cartesian coordinates, the partial derivative df/dqk is not equal to

the directional derivative Dqkf- Instead, by (3.1.9), (3.1.35), and (3.1.36),

we conclude that

dqk

^- = V/ • r'fc(0) = hkVf-qk = hkD9hf. (3.1.41)

EXERCISE 3.1.28? (a) Convince yourself that, in general, df/dqk ^ Dqkf.

Hint: if P has Q coordinates (91,92,93), and Pt, the coordinates (91 +£,92,93),

then, in general, OPt ^= OP + tq 1 , but the equality does hold in cartesian coordi-

nates, (b) Verify (3.1.41).

Since the representation V/ = !Cfc=i(^/ ' 9fc)9fc holds in every or-

thonormal coordinate system, equality (3.1.41) implies the following for-

mula for computing the gradient in general orthogonal coordinates:

V/=l|^ 1 + l|^ 2 + l|^ 3 , (3.1-42)

h\ dq\ h 2 dq 2 h 3 dq 3

where the functions hi,h 2 ,h 3 are denned in (3.1.37).

EXERCISE 3.1.29.B (a) Verify that

Vf=

d

-Lr+

1

d

-L§+

d

-lz

dr r 86 dz

is the gradient in cylindrical coordinates, (b) Verify that

or r sin <p 06 r o<p

is the gradient in spherical coordinates, and verify that, for f(r, 6, ip) = g(r),

the result coincides with (3.1.33).

Next, we derive the expression for the DIVERGENCE in Q coordinates

using the definition (3.1.23). Recall that, to compute the divergence in the

cartesian coordinates, we consider a family of shrinking cubes with faces

parallel to the coordinate planes. The same approach works in any coordi-

nates by considering rectangular boxes whose sides are parallel to the local

basis vectors qi,q 2 ,q 3. Let P be a point with Q coordinates (01,02,03),

and, for sufficiently small a > 0, consider a rectangular box with vertices at