Identifying Potential Functions 1. Show F = (3x2 + 6xy, 3x2 + 6y) is conservative and ﬁnd the potential function f such that F = Vf . Answer: First, My = 6x = Nx . Since F is deﬁned for all (x, y), F is conservative. Method 1 (for ﬁnding f ): Use F · dr = f (P1 ) − f (P0 ) ⇒ f (P1 ) = f (P0 ) + C F · dr. C P1 = (x1 , y1 ) must be arbitrary. We can ﬁx P0 and C any way we want. For this problem take P0 = (0, 0) and C as the path shown. C1 : x = 0, y = y, ⇒ dx = 0, dy = dy C2 : x = x, y = y1 , ⇒ dx = dx, dy = 0 ⇒ f (x1 , y1 , z1 ) − f (0, 0, 0) = F · dr = C x1 = y1 C2 (x1 , y1 ) C1 C y1 M (x, 0) dx + 0 N (x1 , y) dy x 0 y1 = M dx + N dy y x1 6y dy + 0 3x2 + 6y1 dx = 3y12 + x13 + 3x21 y1 Path for problem 1 0 ⇒ f (x1 , y1 ) − f (0, 0) = 3y12 + x13 + 3x12 y1 = 3y12 + x13 + 3x12 y1 . ⇒ f (x, y) = 3y 2 + x3 + 3x2 y + C. Method 2: fx = 3x2 + 6xy ⇒ f = x3 + 3x2 y + g(y). ⇒ fy = 3x2 + g / (y) = 3x2 + 6y ⇒ g / (y) = 6y ⇒ g(y) = 3y 2 + C. ⇒ f (x, y) = x3 + 3x2 y + 3y 2 + C. In general method 1 is preferred because in 3 dimensions it will be easier. 2. Let F = (x + xy 2 ) i + (x2 y + 3y 2 ) j. Show F is a gradient ﬁeld and ﬁnd the potential function using both methods. Answer: We have M (x, y) = x + xy 2 and N (x, y) = x2 y + 3y 2 , so My = 2xy = Nx and F is deﬁned on all (x, y). Thusy, by Theorem 1, F is conservative. Method 1: Use the path shown. y f (P1 ) − f (0, 0) = F · dr = M dx + N dy. C (x1 , y1 ) C C1 : x = x, y = 0, ⇒ dx = dx, dy = 0 ⇒ M (x, 0) = x. C2 : x = x1 , y = y, ⇒ dx = 0, dy = dy ⇒ N (x1 , y) = x21 y + 3y 2 . x1 ⇒ F · dr = C ⇒ ⇔ y1 x dx + 0 f (x1 , y1 ) − f (0, 0) = x21 /2 + x21 y12 /2 f (x, y) = x2 /2 + x2 y 2 /2 + y 3 + C. Method 2: 0 x21 y + 3y 2 dy = x12 /2 + x12 y12 /2 + y13 . + C2 C1 x1 y13 x Path for problem 2 fx = x + y 2 ⇒ f = x2 /2 + x2 y 2 /2 + g(y) 1 ⇒ fy = x2 y + g / (y) = x2 y + 3y 2 ⇒ g / (y) = 3y 2 ⇒ g(y) = y 3 + C. ⇒ f (x, y) = x2 /2 + x2 y 2 /2 + y 3 + C. 2