Calculate the flux of the field F across the closed plane curve C.
-$\mathbf { F } = \mathbf { x } \mathbf { i } + \mathbf { y } \mathbf { j }$; the curve $\mathrm { C }$ is the circle $( \mathrm { x } + 5 ) ^ { 2 } + ( \mathrm { y } - 9 ) ^ { 2 } = 81$
A) 0
B) $162 \pi - 45$
C) $2 \pi$
D) $162 \pi$

Question

Quizplus · Accepted Answer

The divergence of the vector field $\mathbf{F} = \mathbf{x} \mathbf{i} + \mathbf{y} \mathbf{j}$ is 2. By the Divergence Theorem, the flux across the closed curve is the divergence times the area of the circle. The area of the circle is $\pi \times 81 = 81\pi$. Thus, the flux is $2 \times 81\pi = 162\pi$.

Calculate the Flux of the Field F Across the Closed F=xi+yj\mathbf { F } = \mathbf { x } \mathbf { i } + \mathbf { y } \mathbf { j }F=xi+yj

Calculate the Flux of the Field F Across the Closed $\mathbf { F } = \mathbf { x } \mathbf { i } + \mathbf { y } \mathbf { j }$