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 ∫_{S}F ⋅ 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 26.2.1Let F

(x,y,z)

=

(x,x + z,y)

and let S be the hemispherex^{2} + y^{2} + z^{2} = 4,z ≥ 0. Let n be the unit normal to S which has nonnegative zcomponent. Find∫_{S}F ⋅ ndS.

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.