In Section 2 we have seen how to calculate surface area and integrate scalar fields over surfaces. We now establish an extension similar to the step from scalar line integrals to vector line integrals.
We assume again that the surfaces we consider are smooth, that is, they are assumed to be images of parameterised surfaces for which:
- is a non-empty, compact and Jordan-measurable subset of ;
- the mapping is one-to-one;
- is continuously differentiable;
- the normal vector except possibly at a finite number of points;
(Notice, the condition that is compact can also be replaced by the condition that the surface is compact.)