Flux through a sphere given a vector field

[takbq2]

Homework Statement

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.

Homework Equations

Divergence theorem/Gauss's Law

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

 

[LCKurtz]

Homework Statement

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.

Homework Equations

Divergence theorem/Gauss's Law

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

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

 

[takbq2]

I thought that you couldn't integrate 0

 

[LCKurtz]

I thought that you couldn't integrate 0

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

 

[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.

 

[LCKurtz]

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.

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.

 

[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?