Math Link 1: Gauss' Divergence Theorem

Consider a closed surface S enclosing a convex region V. (V is convex when it has the property that if A, B are any two points belonging to V, all points of the line segment AB belong to V).

Suppose also that S is smooth, which means that it has a continuously turning tangent plane. S can be oriented (that is, given a direction) by drawing at each point a unit normal vector outward from V.

THEOREM: If is a vector-valued function whose component functions have continuous partial derivatives in V, then

(L22)
where divergence.

The assumptions about S and V can be generalized; for instance, to non-convex regions and/or to piecewise smooth surfaces (example: a finite circular cylinder with flat ends). But the simple assumptions made above are adequate for our needs in this module.


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