Using Green's Theorem, find the outward flux of F across the closed curve C.
-F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), (9, 0), and (0, 9)
A) 81
B) 0
C) 729
D) 162

Question

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. The outward flux of F across C is given by the line integral of F·n ds, where n is the outward normal. Using Green's Theorem, this is equivalent to the double integral over D of the divergence of F, ∇·F, dA. The divergence of F = (x - y)i + (x + y)j is ∂/∂x(x - y) + ∂/∂y(x + y) = 1 + 1 = 2. The area of the triangle is 1/2 * base * height = 1/2 * 9 * 9 = 40.5. The double integral of 2 over this area is 2 * 40.5 = 81.

Using Green's Theorem, Find the Outward Flux of F Across