PROJECT. Independence of Representation. Dependence on

Chapter 10, Problem 10.1

(choose chapter or problem)

PROJECT. Independence of Representation. Dependence on Path. Consider the integral f F(r) dr, where F = [xy, _y2]. C (a) One path, several representations. Find the value of the integral when r = [cos t, sin t], 0 ~ t ~ 1712. Show that the value remains the same if you set t = - p or t = p2 or apply two other parametric transformations of your own choice. (b) Several paths. Evaluate the integral when C: y = x n , thus r = [t, t"l, 0 ~ t ~ 1, where n = 1,2,3, .... Note that these infinitely many paths have the same endpoints. (c) Limit. What is the limit in (b) as n --+ oo? Can you confirm your result by direct integration without referring to (b)? (d) Show path dependence with a simple example of your choice involving two paths.