Calculate the flux of the field F across the closed plane curve C.
-F = xi + yj; the curve C is the closed counterclockwise path around the rectangle with vertices at (0, 0), (4, 0), (4, 3), and (0, 3)
A) 24
B) 7
C) 0
D) 25

Question

Quizplus · Accepted Answer

The flux of the field $ \mathbf{F} = x\mathbf{i} + y\mathbf{j} $ across the closed curve $ C $ can be calculated using the divergence theorem, which states that the flux through $ C $ is equal to the integral over the area enclosed by $ C $ of the divergence of $ \mathbf{F} $. The divergence of $ \mathbf{F} $ is $ \nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} = 1 + 1 = 2 $. The area of the rectangle is $ 4 \times 3 = 12 $. Therefore, the total flux is $ 2 \times 12 = 24 $.

Calculate the Flux of the Field F Across the Closed