# Flux: Surface integral of a sphere.

1. ### Kizaru

45
1. The problem statement, all variables and given/known data
Find the surface integral of $$\vec{r}$$ over a surface of a sphere of radius a and center at the origin. Also find the volume integral of $$\nabla$$ $$\bullet$$ $$\vec{r}$$.

2. Relevant equations
Divergence theorem.

3. The attempt at a solution
First I did the volume integral part of the divergence theorem. I obtained $$\nabla$$ $$\bullet$$ $$\vec{r}$$ = 1 + 1 + 1 = 3. So I figured, the answer must be 3*volume = 4$$\pi$$r$$^{3}$$ (I don't know why the pi looks like an exponent, but it's 4 pi r^3)

This answer seems like a correct one.

Now the surface integral I'm having trouble with. Knowing that the equation of the sphere is
x$$^{2}$$+y$$^{2}$$+z$$^{2}$$=a$$^{2}$$, I found $$\nabla$$ $$\bullet$$ (x$$^{2}$$+y$$^{2}$$+z$$^{2}$$) to obtain the normal. The $$\vec{r}$$ $$\bullet$$ $$\vec{n}$$ = 2x$$^{2}$$ + 2y$$^{2}$$ + 2z$$^{2}$$.

So I would integrate this over the surface in Cartesian coordinates, or convert to spherical and integrate? Is the normal suppose to be the normal unit vector? I appear to be obtaining the wrong answer no matter which way I am doing this. What exactly would the integral in cartesian coordinates contain for boundaries?

Thanks. Sorry if the latex syntax is not perfect.

2. ### LCKurtz

8,373
For the first integral your answer should be 4 pi a^3, not 4 pi r^3.

You probably aren't asked to find "the surface integral of $$\vec{r}$$ over a surface of a sphere". I'm guessing you are asked to find the flux integral for the vector field $$\vec{r}$$. In other words you are to calculate

$$\int\int_S \vec r \cdot \hat n\, dS$$

where $$\hat n$$ is the unit outward normal. In the case of your sphere your unit outward normal is:

$$\hat n = \frac {\vec r}{|\vec r |}$$

Now the natural way to do such an integral would be spherical coordinates. But when you evaluate $$\vec r \cdot \hat n$$ on the surface of the sphere you should see a shortcut.

3. ### Kizaru

45
Thanks!

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