Alternative construction of potential functions in _2

Chapter 14, Problem 57

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Alternative construction of potential functions in _2 Assume that the vector field F is conservative in _2, so that the line integral 1C F # dr is independent of path. Use the following procedure to construct a potential function w for the vector field F = 8 f, g9 = 82x - y, -x + 2y9. a. Let A be 10, 02 and let B be an arbitrary point 1x, y2. Define w1x, y2 to be the work required to move an object from A to B, where w1A2 = 0. Let C1 be the path from A to 1x, 02 to B and let C2 be the path from A to 10, y2 to B. Draw a picture. b. Evaluate 1C1 F # dr = 1C1 f dx + g dy and conclude that w1x, y2 = x2 - xy + y2. c. Verify that the same potential function is obtained by evaluating the line integral over C2.

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