Vector Integration

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 $C$ :

$$\int_C \vV \cdot d\vell \textrm{\quad or \quad} \oint_C \vV \cdot d\vell$$

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):

$$\int_S \vV \cdot \hn dA = \int_S \vV \cdot d\va \textrm{\quad or \quad} \oint_S \vV \cdot d\va$$

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, $S$ is an open surface, which means it is a) (piecewise) bounded by a closed curve $C$ and the direction of the normal to the surface is arbitrary. In the second, $S$ is a closed surface - a surface that is topologically equivalent to soap bubble - in which case it encloses a volume. For example if we let $S$ be a square on the $xy$ -plane, we might chose to make $\hn dA = d\va = \hz dx dy$ , so you can see that in almost all cases you will have to at least mentally express $\hn$ explicitly in order to evaluate $d\va$ anyway.

[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:

$$\int_\mathcal{V} F dV = \int_\mathcal{V} F d^3r = \int_\mathcal{V} F d\tau$$

where $\mathcal{V}$ is the (open) volume that might have been bounded by a closed $S$ , and I've indicated three different ways people write the volume element. Griffiths favors e.g. $d\tau = dx\ dy\ dz$ .

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:

$$\int_C \vV d\ell \textrm{\quad or \quad} \int_S f da \textrm{\quad or \quad} \int_\mathcal{V} \vF d\tau$$

might make sense in some context.