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I know that finding a potential is a sufficient condition to show that a vector field is conservative. My question is if the those statements are equivalent.

I've found a vector field which isn't conservative, does this imply that there is no potential to the vector field?

kind reg,

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If a implies b, then not-b implies not-a. You don't need a and b to be equivalent. – Chris Eagle Oct 6 '11 at 12:53

Having a potential function and being conservative are equivalent (under some mild assumptions).

Specifically, if a (continuous) vector field is conservative on an open connected region then it has a potential function.

And "Yes" if a vector field fails to be conservative, it cannot have a potential function.

Here are some notes I posted for one of my classes a few years ago...

A few notes:

1) I didn't list all assumptions everywhere (for example, I wasn't careful to say that I'm assuming things are continuous where needed).

2) In the notes a vector field which possesses a potential function is called a "gradient" vector field.

3) The relevant theorem is on page 5.

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