Divergence theorem requires a conservative vector field?

[jesuslovesu]

Can anyone tell me whether or not the divergence theorem requires a conservative vector field? On a practice exam my professor gave a vector field that was nonconservative (I checked the curl) and proceeded to perform the divergence theorem to find the flux.

On one of my homework problems I forgot to check whether the field was conservative and it turns out that field was nonconservative but still got the correct answer. I don't know if that is a fluke or not.


In short: Do you need a conservative vector field in order to use the divergence theorem?

 

[Shooting Star]

In short: Do you need a conservative vector field in order to use the divergence theorem?

No. Only that the vector function have continuous derivatives.

 

[Moetasim]

Yes, the divergence theorem does require a conservative vector field. This is because the theorem is based on the fundamental theorem of calculus, which states that the integral of a conservative vector field over a closed surface is equal to the integral of the divergence of that field over the volume enclosed by the surface. If the vector field is not conservative, the two integrals will not be equal and the theorem cannot be applied. In your practice exam and homework problems, it is possible that the nonconservative vector fields happened to have a zero divergence, which would still result in the correct answer when using the divergence theorem. However, this is not always the case and it is important to always check for conservative behavior before applying the theorem.