Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), ( 3, 0), and (0, 10)
A) 30
B) 60
C) 300
D) 0

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. For a vector field $ \mathbf{F} = M\mathbf{i} + N\mathbf{j} $, the circulation form of Green's Theorem is given by:$$ \oint_C (M\,dx + N\,dy) = \iint_D \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA $$Given $ \mathbf{F} = (x - y)\mathbf{i} + (x + y)\mathbf{j} $, we have $ M = x - y $ and $ N = x + y $. Thus,$$ \frac{\partial N}{\partial x} = 1 \quad \text{and} \quad \frac{\partial M}{\partial y} = -1 $$Substituting these into the formula gives:$$ \iint_D (1 - (-1))\,dA = \iint_D 2\,dA $$The area D is a right triangle with base 3 and height 10, so its area is $ \frac{1}{2} \times 3 \times 10 = 15 $. Therefore, the double integral becomes:$$ \iint_D 2\,dA = 2 \times 15 = 30 $$Hence, the counterclockwise circulation of $ \mathbf{F} $ around the closed curve $ C $ is 30.

Using Green's Theorem, Compute the Counterclockwise Circulation of F Around