- #1

FaraDazed

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## Homework Statement

Using the fact that [itex] \nabla \cdot r^3 \vec{r} = 6 r^2[/itex] (where [itex]\vec{F(\vec{r})} = r^3 \vec{r}[/itex]) where S is the surface of a sphere of radius R centred at the origin.

## Homework Equations

[tex]

\int \int \int_V \nabla \cdot \vec{F} dV =\int \int_S \vec{F} \cdot d \vec{S}

[/tex]

That is meant to be a double surface integral but not sure how to do that in latex

## The Attempt at a Solution

Very new to this, so be kind :)

for the LHS

[tex]

\int \int \int_V \nabla \cdot \vec{F} dV = 6 \int \int \int_V r^2 dV \\

= 6 \int_0^R r^2 r^2 dr \int_0^{2 \pi} d\phi \int_0^{\pi} \sin{\theta} d\theta \\

= 6 [\frac{1}{5}r^5]_0^R [\phi]_0^{\pi} [\cos{\theta}]_0^{2 \pi} \\

= 6(\frac{1}{5} R^5) (2 \pi ) (2) = \frac{24}{5} \pi R^5

[/tex]

for the RHS

[tex]

\int \int_S \vec{F} \cdot d\vec{S} = \int \int_S r^3 \vec{r} \cdot d \vec{S} = \int \int_S r^3 \vec{r} \cdot \vec{\hat{n}} dS \\

[/tex]

Then I calc [itex]r^3 \vec{r} \cdot \vec{\hat{n}} [/itex]

[tex]

r^3 \vec{r} \cdot \vec{\hat{n}} = r^3 \vec{r} \cdot \frac{r^3 \vec{r}}{r^3 r} = \frac{r^2 r^3}{r} = r^4

[/tex]

so...

[tex]

\int \int_S r^3 \vec{r} \cdot \vec{\hat{n}} dS = \int \int_S r^4 dS \\

= \int_0^{2 \pi} d \phi \int_0^{\pi} R^4 R^2 \sin{\theta} d \theta \\

= R^6 [\phi]_0^{2 \pi} [\cos{\theta}]_0^{\pi} = 4 \pi R^6

[/tex]

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