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

Answered: Let D be an open, connected domain, and…

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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Question

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) fF- dr for every piecewise smooth, closed curve C in D
(c) Given any two points Po and P, in D, ſF· dr has the same value for
all piecewise smooth curves in D starting at Po and ending at P.

Expert Solution

Step 1

Given : $D$ is an open, connected domain, and let $F$ is a smooth vector field defined on $D$ .

To prove : following statements are equivalent

(a) $F$ is conservative in $D$

(b) $\int_{C} F \cdot d r = 0$ for every piecewise smooth, closed curve $C$ in $D$

(c) Given any two points $P_{0}$ and $P_{1}$ in $D, \int_{C} F \cdot d r$ has the same value for all piecewise smooth curves in $D$ starting at $P_{0}$ and ending at $P_{1}$ .

Pre-requisite : Fundamental Theorem on Line Integrals -

Suppose $C$ is a smooth curve given by $\vec{r} (t), a \leq t \leq b$ . Also suppose that $f$ is a function whose gradient vector, $\nabla f$ , is continuous on $C$ . Then