Problem 62P

The integral

is sometimes called the vector area of the surface S. If S happens to be flat, then |a| is the ordinary (scalar) area, obviously.

(a) Find the vector area of a hemispherical bowl of radius R.

(b) Show that a = 0 for any closed surface. [Hint: Use Prob. 1.61a.]

(c) Show that a is the same for all surfaces sharing the same boundary.

(d) Show that

Reference problem 1.61a

Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:

where the integral is around the boundary line. [Hint: One way to do it is to draw the cone subtended by the loop at the origin. Divide the conical surface up into infinitesimal triangular wedges, each with vertex at the origin and opposite side dl, and exploit the geometrical interpretation of the cross product (Fig. 1.8).]

(e) Show that

for any constant vector c. [Hint: Let T = c · r in Prob. 1.61e.]

figure 1.8

Reference problem 1.61a

Solution

Step 1 of 6

We need to Find the vector area of a hemispherical bowl of radius .

The vector area of the surface s is .

is the area element spanning from to and to on a spherical surface at constant radius .

The surface is considered to be in the northern hemisphere, the , , components are

The and components integrate to zero and component of is .

So the vector area of the surface s is .

.

The vector area of a hemispherical bowl of radius is found out to be .