Properties of Gradient Fields

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Presentation transcript:

Properties of Gradient Fields Theorem 5.8 (page 354 in Lovrić)

Properties of a Gradient Vector Field---Part I Let F be a C1 vector field defined on an open, connected set U  2 or (3) . The following statements are equivalent: F is a gradient vector field; F=f. For any oriented, simple closed curve c, F is path-independent: for any two oriented, simple curves c1 and c2 having the same initial an terminal points.

Properties of a Gradient Vector Field---Part II Let F be a C1 vector field defined on an open, connected set U  2 or (3) . F=f. Integral of F around any oriented, simple closed curve c is 0. F is path-independent. Each of (a), (b), and (c) imply Scalar curl of F is 0 (or curl F = 0 if U  3) . If, in addition, U is simply connected all are equivalent.