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5. Let D be an open, connected domain, and let F be a smooth vector field defined on D. Prove that the following statements are equivalent: (a) F is conservative in D (b) SF· dr for every piecewise smooth, closed curve C in D (c) Given any two points Po and P in D, ſ,F - dr has the same value for all piecewise smooth curves in D starting at Po and ending at P.

5. Let D be an open, connected domain, and let F be a smooth vector field defined on D. Prove that the following statements are equivalent: (a) F is conservative in D (b) SF· dr for every piecewise smooth, closed curve C in D (c) Given any two points Po and P in D, ſ,F - dr has the same value for all piecewise smooth curves in D starting at Po and ending at P.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Let D be an open, connected domain, and let F be a smooth vector field defined on D. Prove that the following statements are equivalent: (a) F is conservative in D (b) SoF· dr for every piecewise smooth, closed curve C in D (c) Given any two points Po and P in D, ſ,F· dr has the same value for all piecewise smooth curves in D starting at Po and ending at P.
Transcribed Image Text:5. Let D be an open, connected domain, and let F be a smooth vector field defined on D. Prove that the following statements are equivalent: (a) F is conservative in D (b) SoF· dr for every piecewise smooth, closed curve C in D (c) Given any two points Po and P in D, ſ,F· dr has the same value for all piecewise smooth curves in D starting at Po and ending at P.
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