Divergence Theorem (Gauss's Theorem) (continued)

In a Nut Shell: The Divergence Theorem extends the divergence form of Green’s

theorem from two to three dimensions.

In this case the line integral around a closed curve, C, is replaced by a surface integral

around a closed surface, S, and the area integral involving the divergence of the vector

field F is replaced by the volume integral of the divergence of the vector field, F .

So we go from ∫ F . n ds = ∫ ∫ div F dA C R	Green's Theorem
to ∫ ∫ F(x,y,z) . n dS = ∫ ∫ ∫ div F(x,y,z) dV S E	Divergence Theorem

Here F = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k

n = unit normal to the closed surface S

dS = element of area on surface S

div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

dV = element of volume for the solid region E

E = volume of solid region

Click here for alternate strategies in applying the Divergence Theorem.

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