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Boundary Of a Solid Region

Gauss' Theorem shows a relationship between a triple integral over a region E in space and a surface integral over the boundary of E. It corresponds to Green's Theorem in the form

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Before stating Gauss' Theorem, we must explain what is meant by the surface integral over the boundary of a solid region E. In general, the boundary of E is made up of six surfaces corresponding to the six faces of a cube (Figure 13.6.7). Sometimes one or more faces will degenerate to a line or a point.

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Figure 13.6.7

The top and bottom faces of E are (x, y) surfaces, that is, they are given by equations z = c(x, y). However, the left and right faces of are (x, z) surfaces y = b(x, z), while the front and back faces of E are (y, z) surfaces of the form x = a(y, z). Surface integrals over oriented (x, z) and (y, z) surfaces are defined exactly as for (x, y) surfaces except that the variables are interchanged.

In the following discussion E is a solid region all of whose faces are smooth surfaces.

DEFINITION

The boundary of E, ∂E, is the union of the six faces of E oriented so that the outside surfaces are positive. The surface integral of a vector field F(x, y, z) over ∂E,

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is the sum of the surface integrals of F over the six faces of E. (See Figure 13.6.8.)

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Figure 13.6.8: Boundary of E

 


Last Update: 2006-11-22