Advanced Calculus fi..

5. Evaluate by Green's theorem:a) $cay dx + bx dy on any path;Chapter 5 Vector Integral Calculus 287b) $ex sin y dx + ex cosy dy around the rectangle with vertices (0, O), (1, O), (1, in),(0,c) $(2x3 - y3) dx + (x3 + y3) dy around the circle x2 + y2 = 1;d) $cur ds, where u = grad ( x ~ and ~ ) C is the circle x2 + y2 = 1;e) $,u, ds, where v = (x2 + y2)i - 2xyj, and C is the circle x2 + y2 = 1, n being theouter normal;f) $c & [(x - 212 + y2] ds, where C is the circle x2 + Y2 = 1, n is the outer normal;9) §c C log [(x - 2)' + ),2,ds, where C and n are as in (f);h) $cf (x) dx + g(y) dy on any path.6. If r = xi + yj is the position vector of an arbitrary point (x, y), show thatr,, ds = area enclosed by C,Cn being the outer normal to C7. Check the answers to Problems 2(a), 3(a), (b), (c), and 4(a) following Section 5.3 byGreen's theorem.Let the functions P(x, y) and Q(x, y) be defined and continuous in a domain D.'Then the line integral j P dx + Q dy is said to be independent of path in D if, forevery pair of endpoints A and B in D, the value of the line integrallBPdx+ QdyIis the same for all paths C from A to B. The value of the integral will then in generaldepend on the choice of A and B, but not on the choice of the path joining them.Thus as in Fig. 5.13, the integrals on C,, C2, C3 have the same value.Figure 5.13Independence of path.