surface integral of f over the surface S:
If the surface is the graph of a function z=g(x,y) over a domain D, then
If the surface is given parametrically over the domain D, then
If it is possible to choose a unit normal vector n at every point on a surface (excluding boundary points) such that n varies continuously over S, then S is called an oriented surface, and the choice of n provides S with an orientation.
If S is a smooth orientable surface with parameterization 
,
then
For a closed surface (the boundary of a solid region E) the convention is that the positive orientation points outward from E.
If F is a continuous vector field defined on an oriented surface S with unit normal vector n, then the surface integral of F over S is
also known as the flux of F across S.
For a surface given as the graph of a function, this works out to
In the case of a parameterized surface,
Electric flux is the integral
where 
is an electric field. heat flow is an example of a
conservative vector field
where u(x,y,z) is the temperature function for a region, and K is the conductivity of the substance through which the heat is flowing. Then the rate of heat flow across the surface S in the body is then given by
The calculation of surface integrals is yet another example of the standard old trick of integration: the adding up of multitudinous small things weighted by some function to give a larger thing.
While it is enough to consider the general case of a parameterized surface, the special results are worked out in the case of a surface given as the graph of a function z=g(x,y).