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Unformatted text preview: v F ? Which portions contribute negatively? b) Find the ux of v F through C by direct calculation (evaluating a line integral). Explain your answer using (a). c) Find the ux of v F through C using Greens theorem. Problem 2. (Thursday, 2 points) A harmonic function f ( x, y ) is one satisfying Laplaces equation: f xx + f yy = 0. Prove that if the eld v F is the gradient of a harmonic function, then the ux of v F across any simple closed curve is zero. Problem 3. (Thursday, 4 points) Let C be a dierentiable curve contained in the portion 0 x 1 of the rst quadrant, starting somewhere on the positive y-axis and ending somewhere on the line x = 1. Show that the ux of the vector eld v F = (2 xy-2 x 4 y ) + (4 x + 4 x 3 y 2-y 2 ) across C is always equal to-2. (Hint: apply Greens theorem to the region between C and the x-axis). 1...
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This note was uploaded on 04/28/2009 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
- Fall '08