Let$ F(x; y) = (g(t)x; g(t)y)$, where $t = x^{2} + y^{2}$ and g(t) is a function (of single variable) witch is continous for t > 0.

a) find the potential function for the vectorfield F or show that it is not conservative.

What I don't understand is , how to show that when I don't know how g look like ? it could be not defined on the origin , so that any line integral along a closed curve through (0,0) would not be 0 .

How in general can I show that a field is conservative ? should I find a potential and my field defined on simply connected region ?

For the field to be called conservative , does it the line integral have to be equal to zero for all closed curved ? can it be called conservative along the plane exept some places ?

Thanks in advance.


Since you do not know that $g$ is continuous at $(0,0)$, we're going to have to construct a potential function on $\Bbb R^2-\{(0,0)\}$. Since $g$ is, however, continuous on $t>0$, it has an antiderivative $G$ on $t>0$. Try looking at $$f(x,y) = \tfrac12 G(x^2+y^2).$$ What is its gradient?


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.