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Homework Statement
the problem is to calculate
[tex] \int (\nabla \cdot \vec{F}) d\tau [/tex]
over the region
[tex] x^2 + y^2 + x^2 \leq 25 [/tex]
where
[tex] \vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k}) [/tex]
in the simplest manner possible.
Homework Equations
divergence theorem!
The Attempt at a Solution
Write
[tex] \vec{F} = |\vec{r}|^2 \vec{r} = |\vec{r}|^3 \hat{r}, [/tex]
so
[tex] \vec{F} \cdot \hat{n} = \vec{F} \cdot \hat{r} = |\vec{r}|^3 \hat{r} \cdot \hat{r} = |\vec{r}|^3 = 125, [/tex]
since
[tex] \hat{n} = \hat{r} [/tex]
and
[tex] |\vec{r}| = 5 [/tex]
along the surface of the sphere.
Then, invoking the divergence theorem, we obtain:
[tex] \int (\nabla \cdot \vec{F}) d\tau = \oint_{\partial{\tau}} \vec{F} \cdot \hat{n} d\sigma = \oint_{\partial{\tau}} 125 d\sigma = 125 \cdot 4 \cdot \pi \cdot 5^2 [/tex]
the back of the book gives 100pi as the answer (and I've checked the errata for the book; no correction has been made). am i wrong? or is the book?
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