# Flux through a sphere given a vector field

1. Dec 12, 2011

### takbq2

1. The problem statement, all variables and given/known data

Vector field F = (2siny-cosz, 2cosx+3sinz, cosy-2sinx)

Compute the flux of F through a sphere of radius one centered at the origin with respect to the outer unit normal.

2. Relevant equations

Divergence theorem/Gauss's Law

3. The attempt at a solution

The only way I know how to calculate flux is to look at F and let
2siny-cosz = M
2cosx+3sinz = N
cosy-2sinx = P

then do dM/dx dN/dy dP/dz

then use the divF integral to integrate dM/dx + dN/dy + dP/dz with respect to x,y,z

But I don't understand how else to do it when their derivatives are all 0.

Thanks a lot for any help

2. Dec 12, 2011

### LCKurtz

Do you know the saying "Don't look a gift horse in the mouth."? What do you get when you calculate $\iiint_V 0\, dv$?

3. Dec 12, 2011

### takbq2

I thought that you couldn't integrate 0

4. Dec 12, 2011

### LCKurtz

For example, in one variable$$\int_a^b 0\, dx = C|_a^b = C - C = 0$$

5. Dec 12, 2011

### takbq2

Well the triple integral yields zero then.
Does this mean that the vector field does not flow through the sphere? There is no flux?
Can I plot this?

Thanks, by the way.

6. Dec 12, 2011

### LCKurtz

It means that the total outward flow is zero. That doesn't mean no flux flows through the sphere, just that as much goes in as out in total. For example, a constant flux flowing right through the sphere would give a total flux of zero.

You're welcome.

7. Dec 12, 2011

### takbq2

So the flux is not zero, it is just out = in. From the information given is there anything else I can know about the flow?