Divergence_theorem_9-16

# Divergence_theorem_9-16 - THE DIVERGENCE THEOREM It is...

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1 THE DIVERGENCE THEOREM ± It is another generalization of Green’s theorem to 3D. ∫∫∫ ∫∫∫ ∫∫ ∫∫ = = = D D S S dV dV d n d F F S F S F r r r r r div ) ˆ ( ± NOTE: The theorem is sometimes referred to as Gauss’s Theorem or Gauss’s Divergence Theorem. ± Note that the boundary of D is a closed surface. We use the convention that the positive orientation is outward, that is, the unit normal vector n is direction outward from D . ± Let F be a vector field whose component functions have continuous partial derivatives on a open region that contains D . Then n : outward normal unit vector ± Let D be a simple solid region (For example, regions bounded by ellipsoids or rectangular boxes are simple solid regions). ± Let S be the boundary surface of D , given with positive (upward) orientation.

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2 Divergence Theorem (Cont.) dV F dS n F D S ) ( ) ˆ ( ∫∫∫ ∫∫ = r r ± The left side is the total flux out of S. The right side adds up small sources in the volume. ± If div
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Divergence_theorem_9-16 - THE DIVERGENCE THEOREM It is...

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