Mathematics of Physics and Engineering

140 Functions of Several Variables

then V^2 F = V^2 Fi z+V^2 F 2 j+V^2 F 3 k. (cf Verify that V^2 (fg) = gV^2 f+

Given a reference point O, there is a natural vector field, denoted by

r, which maps every point P to the corresponding position vector

OP. The corresponding scalar field r = \\r\\ is the distance from

OtoP.

The following exercise establishes a number of remarkable properties of

the fields r and r.

EXERCISE 3.1.22.C (a) Verify that if g

then

Vff(r) = g'(r) r

Hint: use cartesian coordinates so that r = x i+yj+z K, and r = \Jx^2 + y^2 + z^2.
(b) Verify that, in M", divr = n, n = 2,3. (c) Verify that, in R^2 , V^2 lnr =
0 for r ^ 0. (d) Verify that, in K^3 , V^2 r_1 = 0 for r ^ 0. Hint: for (c)
and (d), use (3.1.33) together with some of the identities in (3.1.30). (e) Let
h = h(t) be a differentiate function. Verify that curl(h(r) r) = 0. Hint:
the fastest way is to conclude from (3.1.33) that h(r)r = V/7(r) for a suitable
function H; then use (3.1.30). (f) Let b be a constant vector. Verify that

grad(6-r)=6. (3.1.34)

Hint: write the dot product in cartesian coordinates, (g) Let F = cr/r be a

central inverse-square force field in R^3 Prove that / F • dr is independent

of the path C joining any two points, as long as C does not pass through the

origin. Hint: show that F is conservative by finding f = f(r) so that F = V/.

EXERCISE 3.1.23.C Let F = cr/ra = (c/ra+l)r, where c and a are real

numbers. Verify that the flux of F across the sphere S with center at the

origin and radius R is equal to &-nR^2 ~ac. Hint: Use (3.1.22); Hs = f; on S,

r = R.

EXERCISE 3.1.24? Let BR be a ball with center at the origin and radius R.

Verify that limR^ 0 fff r~pdV = 0 for all p < 3.

BR

Our motivation of the definition of the curl suggests that curl is asso-

ciated with rotation; in fact, some books denote the curl of F by rotF.

Let us look closer at this connection with rotation. Recall that, for fixed

= g(t) is a differentiate function,

= ^-r. (3.1.33)