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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.

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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?

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