Appendix DProofs of Selected Results

D.1 Proof of Theorem 2.2

Suppose upper J is a closed trajectory in í’Ÿ. Then the inner product f left-parenthesis x right-parenthesis dot n left-parenthesis x right-parenthesis equals 0, where f left-parenthesis x right-parenthesis is the vector field defining (2.27) and n left-parenthesis x right-parenthesis is the outward normal to upper J. Consequently, the line integral around the closed orbit satisfies contour-integral Underscript upper J Endscripts f dot n d script l equals 0. Therefore, by Green's theorem we have

integral integral Underscript upper S Endscripts nabla dot upper F d upper A equals contour-integral Underscript upper J Endscripts f dot n d s equals 0 comma

where upper S is the region enclosed by upper J. Therefore, either nabla dot f is identically zero in , or it changes ...

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