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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 cant 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.

[tex]\int\int_{\partial V}\vec{F}\cdot\vec{n}dS = \int\int\int_{V}\nabla\cdot\vec{F} dV [/tex]

[tex]\int\int_{\partial V}\vec{F}\cdot\vec{n}dS = \int\int\int_{V}\nabla\cdot\vec{F} dV [/tex]

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