computing flux integral

Question

just given this question. compute the flux out of the unit circle, C. $$F(x,y)=\langle x+2y,3x+4y\rangle $$

i am not sure on how to solve this. Usually the flux would include Z function. please help!

Answer 1

It is a 2-dimensional flux. Evaluate $$\int_C{\bf F}\cdot{\bf n}\,ds=\int_0^{2\pi}<\cos(t)+2\sin(t),3\cos(t)+4\sin(t)>\cdot<\cos(t),\sin(t)>dt\\ =\int_0^{2\pi}(\cos^2(t)+10\sin(t)\cos(t)+ 4\sin^2(t))dt=\pi+0+4\pi=5\pi.$$ Note that by the planar divergence theorem, the same result can be obtained by evaluating $$\int_D (F_x+F_y)dxdy=\int_D (1+4)dxdy=5|D|=5\pi.$$ where $D$ is the unit disc.

Answer 2

The flux of $F(x,y)$ across $C$ is given by $\int_C F(x,y)\cdot n\,ds$, where $n$ is the outward normal vector to $C$. Using the planar divergence theorem, you could also calculate this integral as: $$\int_C F(x,y)\cdot n\,ds=\iint_D\nabla\cdot F(x,y)\,dA$$ Where $D$ is enclosed by $C$ (in this case $D$ is the unit disk).

Answer 3

Or using the great Gauss (divergence) theorem:

$$\nabla f=1+4=5\implies \iint_C\vec F\cdot\vec n\,d\vec s=\iint_R\nabla fdA=5\iint_RdA=5\cdot\pi\cdot1^2=5\pi$$