We need to generalize the scalar theorem to a fundamental theorem for
vector derivatives. However, we may end up having more than one! That
is because we can integrate over 1, 2 or all three dimensional domains
for scalar *and* vector functions defined in 3d Euclidean space.
Here is a non-exhaustive list of important integral types (some of which
you have encountered in introductory physics courses):

A line integral along some specified curvilinear path or around some specified loop :

You should recognize this type of integral from what you have learned about potential or potential energy or certain field integrals in Maxwell's Equations learned in introductory electricity and magnetism.

Next we have surface integrals (of the particular kind associated with the flux of a vector field):

for two common notations, the second one favored by e.g. Griffiths although I personally prefer the first one and it is more common in physics textbooks. In the first case, is an

[Aside: A closed line bounds an open surface. A closed surface bounds
an open volume. If you want to make your head hurt (in constructive
ways - we will need to think about things like this in relativity
theory) think about what a *closed volume* might bound...]

Finally, we have integration over a volume:

where is the (open) volume that might have been bounded by a closed , and I've indicated three different ways people write the volume element. Griffiths favors e.g. .

One doesn't have to integrate only scalar functions, and there are other line and surface integrals one can define or sensibly evaluate. For example all of:

might make sense in some context.