(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use the divergence theorem in three dimensions

[tex]\int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma[/tex]

to evaluate the flux of the vector field

V= (3x-2y)i + x^{4}zj + (1-2z)k

through the hemisphere bounded by the spherical surface x^{2}+y^{2}+z^{2}=a^{2}(for z>0) and the x-y plane

Hint: The direct evauation of the flux may not be the easiest way to proceed

2. Relevant equations

3. The attempt at a solution

I found it pretty simple which means I probably messed up (and I'm not sure what the hint is talking about

ok so the divergence is

[tex]\nabla \bullet V = 3-2=1 [/tex]

and the integral over the volume of the hemisphere (using spherical polar coordinates) is

[tex] \int_{0}^a r^2 d \tau \int_{0}^{2\pi} d\phi \int_{0}^{\pi/2} sin\theta d\theta = \frac{2\pi a^3}{3} [/tex]

So am I doing it completely wrong? I don't know the answer but if anyone could look through it and spot anything I would really appreciate it

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# Quick Divergence Theorem question

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