>>14301552are you talking about theorem 5? "outward flux" means the author is defining to be the unit vector that is normal to the curve C at any point, and oriented so that it points away from the region it encloses, i.e. it points outward. This means POSITIVE flux is fluid passing outward theough C, and NEGATIVE flux is fluid passing inward through C.
In english, theorem 5 says:
"The flux through C equals the the sum of all divergent bits over the region R"
An integral with vectors over a curve has been transformed into an integral over a region in the plane with no vectors. Handy!
