Advanced Calculus fi..

Chapter 5 Vector Integral Calculus . 275Figure 5.6closed path.(a) (b) .(a) Positive direction and (b) negative direction on a simplefh =Figure 5.7 Example 3.1If C is a closed curve, then there is no need to specify initial and terminal point,though the direction must be indicated. If C is a simple closed curve (traced justonce), then one need only specify which of the two possible directions is chosen.The notationsrefer to the two cases of Figs. 5.6(a) and 5.6(b). The counterclockwise arrow refersto what is roughly a counterclockwise direction on C; this will be termed the positivedirection (as for angular measure); the clockwise direction will be called the negativedirection. It should be noted that the direction can be specified by reference to the unittangent vector T in the direction of integration and the unit normal vector n that pointsto the outside of the region bounded by C; for the positive direction, n is 90" behindT, as in Fig. 5.6(a); for the negative direction, n is 90" ahead of T as in Fig. 5.6(b).EXAMPLE 3 To evaluate#y2dx +x2dy.Cwhere C is the triangle with vertices (1, O), (1, I), (0, 0), shown in Fig. 5.7, one hasto compute three integrals. The first is the integral from (0, 0) to (1, 0); along this