Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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,

share|improve this question
2  
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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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