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Unformatted text preview: B Remark 176 In physics " U f (x)·$ (x) dHN #1 (x) represents the outward ﬂux
of a vector ﬁeld f across the boundary of a region U .
If E ) RN and f : E * RN is di"erentiable, then f is called a divergence-free
ﬁeld or solenoidal ﬁeld if
div f = 0.
Thus for a smooth solenoidal ﬁeld, the outward ﬂux across the boundary of a
regular set U is zero. 104 Corollary 177 The theorem continues to hold if U ) RN is open, bounded, and
its boundary consists of two sets E1 and E2 , where E1 is a closed set contained
in the ﬁnite union of compact surfaces of class C 1 and dimension less than
N ' 1, while for ever...
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- Spring '14