When does the divergence theorem apply?

[EngageEngage]

As the thread title suggests, I'm having trouble realizing when the divergence theorem is applicable and when it is not. In some examples, I am instructed not to use it because it doesn't hold but on others I can use it. My first instinct was that it doesn't apply when the vector field isn't defined within the region of itnerest, but I realize that this can't be true because, for example, the electric field is undefined at the source point, but we can still use the theorem. So if someone could please help me out I would appreciate it greatly.

$$\int\int_{\partial V}\vec{F}\cdot\vec{n}dS = \int\int\int_{V}\nabla\cdot\vec{F} dV$$

 

[CrazyIvan]

The Divergence Theorem, in general, applies when you have some sort of vector field and a surface. Then, the triple integral of the divergence of the field is equal to the flux through the surface.

In terms of the electric field, the Divergence Theorem works because
$$\iiint_{V} \nabla \cdot \vec{F} dV = \frac{Q_{inside}}{\epsilon_{0}}$$

 

[EngageEngage]

Thank you for the reply. O yes, I am very familiar with the theorem and I have computed it many times. Its just that sometimes i run into problems when I cannot use the theorem (as instructed) but I haven't been able to figure out why I cannot use it in sometimes. Thus, I am really curious as to what the limitations of the theorem are and in what cases it doesn't work.

 

[HallsofIvy]

You are aware, I hope, that every theorem has "hypotheses". Any theorem applies when the hypotheses are true. What are the hypotheses of the divergence theorem?