Divergence Theorem (Gauss's Theorem) (continued)
Alternate Strategies in the application of the Divergence Theorem
∫ ∫ F . dS = ∫ ∫ F . n dS = ∫ ∫ ∫ div F dV
S S E
F(x,y,z) = vector field, dS = element of oriented surface, n = unit normal to S
dV = element of volume of region E
Method 1
Evaluate ∫ ∫ F . n dS directly over the surface S
S
Method 2
Transform the double integral on S ∫ ∫ F ● n dS
to a surface integral using n = (ru x rv) / | (ru x rv) |
which gives
∫ ∫ F ● (ru x rv) / | (ru x rv) |dS
and dS = | (ru x rv) | dA
to obtain the final result
∫ ∫ F ● n dS = ∫ ∫ F ● (ru x rv) dA
S R
where dA is the element of area on R, on the u-v plane
Method 3
Evaluate ∫ ∫ ∫ div F dV directly over the volume E
E
Side note: Some mathematicians refer to the Divergence Theorem as Gauss’ Theorem.
Click here for four examples illustrating the Divergence Theorem.
Copyright © 2017 Richard C. Coddington
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