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# Vector 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 very many infinitesimal surfaces is (A.95)

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 that end on the same rim. (See Figure A.23.) The volume between the surfaces is constant, so what goes in must come out, and (A.96)

It follows that (A.97)

depends only on the rim, and not on the form of surfaces and .   Next: Volume Integrals Up: Vectors and Vector Fields Previous: Surface Integrals
Richard Fitzpatrick 2016-03-31