The integral is sometimes called the vector area of the surface S. If S happens...

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