Advanced Calculus fi..

288 Advanced Calculus. Fifth EditionFigure 5.14Construction of function F such that dF = P dx + Q dy.THEOREM I If the integral P dx + Q dy is independent of path in D, thenthere is a function F(x, y) defined in D such thatholds throughout D. Conversely, if a function F(x, y) can be found such that (5.46)holds in D, then j" P dx + Q dy is independent of path in D.Proof. Suppose first that the integral is independent of path in D. Then choose apoint (xo, yo) in D and let F(x, y) be defined as follows:dwhere the integral is taken on an arbitrary path in D joining (xO, yo) to (x, y).Since the integral is independent of path, the integral in (5.47) does indeed dependonly on the endpoint (x, y) and defines a function F(x, y). It remains to show thata~lax = P (X, y), a~lay = Q(X, y) in D.For a particular (x, y) in D, choose ( XI, y) so that xl # x and the line segmentfrom (xl, y) to (x, y) is in D, as shown in Fig. 5.14. Then, because of independenceof path, one hasHere xl and y are thought of as fixed, while (x, y) may vary along the line segment.Thus y has been restricted to a constant value, and F(x, y) is being considered asa function of x near a particular choice of x. The first integral on the right is thenindependent of x, while the second can be integrated along the line segment. Hencefor this fixed y,or, with the dummy variable x replaced by a t,F(x, y) = const + P(t, y)d't.1: