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## Surface 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 2007-07-14