Flux through a sphere given a vector field

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The discussion focuses on calculating the flux of a vector field through a sphere using the divergence theorem. The vector field is defined, and the user attempts to compute the divergence but finds that the derivatives yield zero. This leads to confusion about whether the total flux through the sphere is zero, which is clarified by stating that a zero total flux indicates equal amounts of flow entering and exiting the sphere, not the absence of flux. The conversation concludes with a request for further insights into the flow characteristics of the vector field.
takbq2
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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
 
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takbq2 said:

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?
 
I thought that you couldn't integrate 0
 
takbq2 said:
I thought that you couldn't integrate 0

For example, in one variable\int_a^b 0\, dx = C|_a^b = C - C = 0
 
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.
 
takbq2 said:
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.
 
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?
 

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