1 / 3

Properties of Gradient Fields

Properties of Gradient Fields. Theorem 5.8 (page 354 in Lovrić). Properties of a Gradient Vector Field---Part I. Let F be a C 1 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.

lok
Download Presentation

Properties of Gradient Fields

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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.

  3. 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 curlF = 0 if U  3) . • If, in addition, U is simply connected all are equivalent.

More Related