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Multivariable Calculus

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Checking a Conservative Vector Field

MVCALC-NFXGOY

For a vector field $\vec{F}$, which of the following statements are true?

I. If $\vec{F}$ is path independent, then the $\vec{F}$ is conservative.
II. If $\vec{F}$ has a potential function, then $\vec{F}$ is conservative.
III. If the line integral $\displaystyle \int_C \vec{F} ds = 0$ for some simple closed curve $C$, then $\vec{F}$ is conservative.
IV. If the curl of $\vec{F}$ is zero, then $\vec{F}$ is conservative.

A

I, II, III, and IV.

B

I, II, and IV only.

C

I and II only.

D

II only.

E

None of the above.