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Is this function symmetric?

  1. Mar 11, 2013 #1
    Today I've tried to investigate properties of a function [itex]f(x_1,x_2,x_3)[/itex] satisfying [itex]\nabla\times(f\mathbf{x})=0[/itex] in [itex]\mathbb{R}^3[/itex], where some degrees of the differentiability are assumed if needed.

    By some basic procedures, I've deduced that: there is a scalar function g such that [itex]f=\frac{1}{x_i}\frac{\partial{g}}{\partial{x_i}}[/itex] for i=1,2,3. Then I guess f is symmetric(i.e., f(x,y,z)=f(x,z,y)=f(y,x,z)=f(y,z,x)=f(z,x,y)=f(z,y,x)). Is it true? If not, is there a counterexample?
  2. jcsd
  3. Mar 11, 2013 #2


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    Staff: Mentor

    This is right, and you can even show that f can depend on ##\sqrt{x_1^2+x_2^2+x_3^2}## only.
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