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,

  • 2
    $\begingroup$ If a implies b, then not-b implies not-a. You don't need a and b to be equivalent. $\endgroup$ – 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... http://mathsci2.appstate.edu/~cookwj/courses/math2130-fall2009/math2130-Line_Int_notes.pdf

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.