Whole books are written on this!
Also, I don't know what you mean by "A.dV" dV is a scalar quantity, not a vector so you cannot take the dot product of a vector, A, with it.
If you are given a surface, S, you can always write it in terms of parametric equations, in terms of two parameters, say u and v: x(u,v), y(u,v), z(u,v). You can then write it as a vector equation in an obvious way: [itex]\vec{r}(u,v)= x(u,v)\vec{i}+ y(u,v)\vec{j}+ z(u,v)\vec{k}[/itex]. The "fundamental vector product" is the cross product of the two partial derivatives: [itex]\vec{r}_u\times\vec{r}_v[/itex] and the "vector differential of surface area" is [itex]\vec{r}_u\times\vec{r}_v dudv[/itex]. Of course, that points in opposite directions depending on the order of multiplication: that's because you need to determine an orientation of the surface.
For a path, which depends on one parameter, say t, [itex]\vec{r}= x(t)\vec{i}+ y(t)\vec{j}+ z(t)\vec{k}[/itex], we have [itex]d\vec{r}= x'\vec{i}dx+ y'\vec{j}dy+ z'\vec{k}dz[/itex]. A.dr is the dot product of A with that.
