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zcd

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## Homework Statement

Find the flux of field [tex]\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}[/tex] across the portion of the sphere [tex]x^{2}+y^{2}+z^{2}=4[/tex] 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

[tex]S=\iint \limits_{R}\frac{|\nabla f|}{|\nabla f\cdot\hat{p}|}dA[/tex]

[tex]Flux=\iint \limits_{S} \mathbf{F}\cdot\mathbf{n} d\sigma=\iiint \limits_{V}\nabla\cdot\mathbf{F}dV[/tex]

Stoke's theorem (I thought Stoke's theorem involved circulation and curl)?

[tex]\oint \limits_{C} \mathbf{F}\cdot d\mathbf{r}=\iint \limits_{S}\nabla\times\mathbf{F}\cdot\mathbf{n}d\sigma[/tex]

Divergence theorem (we haven't learned this yet but I think it will apply better than stoke's theorem for part c)

[tex]\iint \limits_{S}\mathbf{F}\cdot\mathbf{n}d\sigma=\iiint \limits_{D} \nabla\cdot\mathbf{F}dV[/tex]

## The Attempt at a Solution

for part a:

[tex]Flux=\iint \limits_{S} \mathbf{F}\cdot\mathbf{n} d\sigma=\iiint \limits_{V}\nabla\cdot\mathbf{F}dV[/tex]

the gradient vector is always perpendicular to a surface f(x,y,z)=c, so [tex]\mathbf{n}=\frac{\nabla f}{|\nabla f|}[/tex]

this should lead to [tex]\iint \limits_{S} \frac{(\mathbf{F}\cdot\nabla f)|\nabla f|}{|\nabla f||\nabla f\cdot\hat{p}|}dA[/tex]

after some simplifying and conversion to polar coordinates...

[tex]\int_{0}^{2\pi}\int_{0}^{2} r^{3}+r\sqrt{4-r^{2}}dr d\theta=...=\frac{40\pi}{3}[/tex]

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

for part c:

using divergence theorem to find flux-

[tex]\nabla\cdot\mathbf{F}=\frac{\partial{M}}{\partial{x}}+\frac{\partial{N}}{\partial{y}}+\frac{\partial{P}}{\partial{z}}=3[/tex]

[tex]\iint \limits_{S}\mathbf{F}\cdot\mathbf{n}d\sigma=\iiint \limits_{D} \nabla\cdot\mathbf{F}dV[/tex]

[tex]=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2}3\rho^{2}\sin\phi}d\rho d\phi d\theta=...=32\pi[/tex]

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?

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