F(x,y,z) Symmetry in Scalar Function g: Counterexample or Confirmation?

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SUMMARY

The discussion centers on the properties of a function f(x_1, x_2, x_3) that satisfies the condition ∇×(f𝑥) = 0 in ℝ³. It is established that there exists a scalar function g such that f = (1/x_i)(∂g/∂x_i) for i = 1, 2, 3. The symmetry of f is confirmed, as it holds true that f(x, y, z) = f(y, z, x) and similar permutations. Additionally, it is demonstrated that f can be expressed as a function of the radial distance √(x_1² + x_2² + x_3²).

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  • Understanding of vector calculus, specifically curl and divergence
  • Familiarity with scalar and vector fields in ℝ³
  • Knowledge of partial derivatives and their applications
  • Concept of symmetry in mathematical functions
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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?
 
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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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