Unformatted text preview: v F ? Which portions contribute negatively? b) Find the ²ux of v F through C by direct calculation (evaluating a line integral). Explain your answer using (a). c) Find the ²ux of v F through C using Green’s theorem. Problem 2. (Thursday, 2 points) A harmonic function f ( x, y ) is one satisfying Laplace’s equation: f xx + f yy = 0. Prove that if the ±eld v F is the gradient of a harmonic function, then the ²ux of v F across any simple closed curve is zero. Problem 3. (Thursday, 4 points) Let C be a di³erentiable curve contained in the portion 0 ≤ x ≤ 1 of the ±rst quadrant, starting somewhere on the positive yaxis and ending somewhere on the line x = 1. Show that the ²ux of the vector ±eld v F = (2 xy2 x 4 y )ˆ ı + (4 x + 4 x 3 y 2y 2 )ˆ across C is always equal to2. (Hint: apply Green’s theorem to the region between C and the xaxis). 1...
View
Full Document
 Fall '08
 Auroux
 Vector Calculus, Flux, Vector field, simple closed curve, harmonic function

Click to edit the document details