Next: Volume integrals
Up: Vectors
Previous: Surface integrals
Surface integrals often occur during vector analysis. For instance, the rate of
flow of a liquid of velocity through an infinitesimal
surface of vector area
is
. The net rate of flow through a surface made up
of
lots of infinitesimal surfaces is

(86) 
where is the angle subtended between the normal to the surface and the
flow velocity.
Analogously to line integrals, most
surface integrals depend both on the surface and the rim.
But some (very important) integrals depend only on the rim, and not on the nature
of the surface which spans it. As an example of this, consider incompressible fluid
flow between two surfaces and which end on the same rim.
The volume between the surfaces is constant, so what goes in must come out, and

(87) 
It follows that

(88) 
depends only on the rim, and not on the form of surfaces and .
Next: Volume integrals
Up: Vectors
Previous: Surface integrals
Richard Fitzpatrick
20060202