These will be important in the next chapter. The idea is this. You have a surface S and a field of unit normal vectors n on S. That is, for each point of S there exists a unit normal. There is also a vector field F and you want to find ∫ SF ⋅ ndS. There is really nothing new here. You just need to compute the function F ⋅ n and then integrate it over the surface. Here is an example.
Example 16.2.1 Let F
First find the function
This follows because the normal is of the form
and the increment of surface area is then
The important thing to notice is that there is no new mathematics here. That which is new is the significance of a flux integral which will be discussed more in the next chapter. In short, this integral often has the interpretation of a measure of how fast something is crossing a surface.