Find Flux Across Portion of Sphere in 3 Ways

[zcd]

Homework Statement

Find the flux of field $$\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$$ across the portion of the sphere $$x^{2}+y^{2}+z^{2}=4$$ in the first octant in the direction away from the origin in three ways:

a. using formula for flux when sphere is a level surface
b. using formula for flux when sphere is a parametric surface
c. using Stoke's theorem

Homework Equations

$$S=\iint \limits_{R}\frac{|\nabla f|}{|\nabla f\cdot\hat{p}|}dA$$

$$Flux=\iint \limits_{S} \mathbf{F}\cdot\mathbf{n} d\sigma=\iiint \limits_{V}\nabla\cdot\mathbf{F}dV$$

Stoke's theorem (I thought Stoke's theorem involved circulation and curl)?
$$\oint \limits_{C} \mathbf{F}\cdot d\mathbf{r}=\iint \limits_{S}\nabla\times\mathbf{F}\cdot\mathbf{n}d\sigma$$

Divergence theorem (we haven't learned this yet but I think it will apply better than stoke's theorem for part c)
$$\iint \limits_{S}\mathbf{F}\cdot\mathbf{n}d\sigma=\iiint \limits_{D} \nabla\cdot\mathbf{F}dV$$

The Attempt at a Solution

for part a:
$$Flux=\iint \limits_{S} \mathbf{F}\cdot\mathbf{n} d\sigma=\iiint \limits_{V}\nabla\cdot\mathbf{F}dV$$
the gradient vector is always perpendicular to a surface f(x,y,z)=c, so $$\mathbf{n}=\frac{\nabla f}{|\nabla f|}$$
this should lead to $$\iint \limits_{S} \frac{(\mathbf{F}\cdot\nabla f)|\nabla f|}{|\nabla f||\nabla f\cdot\hat{p}|}dA$$
after some simplifying and conversion to polar coordinates...
$$\int_{0}^{2\pi}\int_{0}^{2} r^{3}+r\sqrt{4-r^{2}}dr d\theta=...=\frac{40\pi}{3}$$

I temporarily skipped part b cause of the work involved for finding $$d\sigma=|\frac{\partial{\mathbf{r}}}{\partial{\phi}}\times\frac{\partial{\mathbf{r}}}{\partial{\theta}}|d\phi d\theta$$

for part c:
using divergence theorem to find flux-
$$\nabla\cdot\mathbf{F}=\frac{\partial{M}}{\partial{x}}+\frac{\partial{N}}{\partial{y}}+\frac{\partial{P}}{\partial{z}}=3$$
$$\iint \limits_{S}\mathbf{F}\cdot\mathbf{n}d\sigma=\iiint \limits_{D} \nabla\cdot\mathbf{F}dV$$
$$=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2}3\rho^{2}\sin\phi}d\rho d\phi d\theta=...=32\pi$$

and here's where I have the problem. The same flux from two different methods should not be different. Where in my work did I go wrong?

 

[LCKurtz]

You are given just a portion of a sphere which is in the first octant. There is no volume enclosed by it so the divergence theorem is irrelevant. The sphere is cut off by the 3 coordinate planes.

 

[zcd]

I overlooked the part about being in first octant, but how would I apply Stoke's theorem to find flux of the field across a surface? I thought Stoke's theorem was flux of the curl of the field?

Reworking part a:
$$\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{2}\frac{4r}{\sqrt{4-r^{2}}}dr=4\pi$$

 

[LCKurtz]

I agree Stokes' theorem doesn't seem appropriate since your $\vec F$ is not the curl of a field. Your answer or $4\pi$ is correct.

 

[zcd]

Thanks for the verification. Is there another theorem I can use to finish the problem? Or should I just solve for circulation and give 0 as an answer?