Potential functions

**rsq_a** · Jun8-09, 05:26 AM

I wasn't quite sure how to do the second part of this question:

Given $LaTeX Code: \\textbf{f}(x,y,z) = (y/(x^2+y^2), -x/(x^2+y^2), 0)$ where $LaTeX Code: (x,y) \\neq (0,0)$ , verify that $LaTeX Code: \\nabla \\times f = 0$ .

(A) Find a scalar field $LaTeX Code: \\phi$ such that $LaTeX Code: \\textbf{f} = \\nabla \\phi$ on $LaTeX Code: R_1 = \\{(x,y,z): y > 0\\}$ .

(B) Show that there does NOT exist $LaTeX Code: \\psi$ such that $LaTeX Code: \\textbf{f} = \\nabla\\psi$ on $LaTeX Code: R_2 = \\{(x,y,z): (x,y) \\neq (0,0)$

For (A), I found $LaTeX Code: \\phi =$ arctan(x) + arccot(x) - arctan(y/x).

I'm not sure how to do (B). In fact, I'm not even sure why it's true.

**rsq_a** · Jun8-09, 09:34 AM

Never mind. It's pretty easy to show just by considering a contour going around the pole (x,y) = (0, 0). Considering the path integral around r(t) = (cos t, sin t, 0), the integral gives $LaTeX Code: 2\\pi$ , whereas it should give 0 if a potential function did exist.