Next: Volume Integrals
Up: Vectors
Previous: Line Integrals
Surface integrals often arise in Physics. For instance, the rate of
flow of a liquid of velocity through an infinitesimal
surface of vector area
is
(i.e., the product of the normal component of the velocity, , and the magnitude of the area, , where is the angle subtended between and ). The net rate of flow through a surface made up
of
very many infinitesimal surfaces is

(48) 
where is the angle subtended between a surface element and the local
flow velocity
. If the surface is closed, and the surface elements all point outward, then the integral is conventionally written

(49) 
In this case, the integral is often termed the flux of the velocity field
out of the closed surface .
Richard Fitzpatrick
20070714