V X C aunch to update E 2 / 5 - 99% Problem 2. If a vector field F = (P, Q)) is conservative in an open domain D CR2, meaning there exists a smooth scalar function f(x, y) such that F = Vf (in this case, the potential energy U is - f), then the potential function can be derived by a line integral. This is a fundamental theorem in multivariable calculus. a) Choose a fixed point (To, yo) ED and define f(x, y) for any arbitrary (x, y) ED as follows: f(x, y) = F . dr. J(20, 30) As we know, the integral is independent of the path of integration. Demonstrate the relations: fx (x, y) = P(x, y) and fy(x, y) = Q(x, y). Hint: Take the integration path y from (20, yo) to (x, y) as 71 + 72 where 71: (t, yo), te [ro, x], 72(t) = (x, t), te [yo, y] b) Utilize the theorem from part (a) and determine a potential for the conservative field F = (y2 -x2, 2xy). Calculate the work done by F to move a mass from (-1, 0) to (1, 0) along the path 7 (t ) = (t, +3 - t), te [-1, 1] 1 -1.C Q Search W P ENG 10:01 AM High winds soon US 2023-11-20 1