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

In summary, the conversation discusses the properties of a function f(x_1,x_2,x_3) satisfying a specific equation in three-dimensional space. The speaker has deduced that the function can be represented by a scalar function g and that it is symmetric. They also ask if the function can depend on a specific variable only and if there are any counterexamples to this.
  • #1
hauho195
1
0
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?
 
Physics news on Phys.org
  • #2
This is right, and you can even show that f can depend on ##\sqrt{x_1^2+x_2^2+x_3^2}## only.
 

1. Is there an easy way to determine if a function is symmetric?

Yes, there are a few methods that can be used to determine if a function is symmetric. One method is to graph the function and see if it is symmetric about the y-axis. Another method is to check if the function is equal to its own inverse function.

2. Can a function be symmetric about more than one axis?

Yes, a function can be symmetric about more than one axis. For example, a function could be symmetric about both the x-axis and the y-axis, or it could be symmetric about the origin.

3. Are all symmetric functions also considered to be even functions?

Yes, all symmetric functions are also considered to be even functions. This means that the function is symmetric about the y-axis and has the property f(x) = f(-x).

4. What is the difference between a symmetric function and a periodic function?

A symmetric function is one that is symmetric about a specific axis or point, while a periodic function is one that repeats itself at regular intervals. A symmetric function can be periodic, but not all periodic functions are symmetric.

5. Can a function be both odd and symmetric?

No, a function cannot be both odd and symmetric. An odd function is one that is symmetric about the origin and has the property f(x) = -f(-x), while a symmetric function has the property f(x) = f(-x). These two properties cannot coexist in the same function.

Similar threads

I Lipschitz continuity of vector-valued function
    • Calculus
Replies
3
Views
1K
I Partial derivatives of the function f(x,y)
    • Calculus
Replies
6
Views
1K
MHB How to find F(x,y) for $f(x,y)= h(y)\hat{i} + g(x)\hat{j}?$
    • Calculus
Replies
11
Views
1K
I The Basic Area Problem (introduction to the topic of integrals)
    • Calculus
Replies
24
Views
2K
B Are continuous functions on sequentially compact sets u-continuous?
    • Calculus
Replies
18
Views
2K
MHB Implicit function theorem for f(x,y) = x^2+y^2-1
    • Calculus
Replies
1
Views
956
I How to prove that ##f## is integrable given that ##g## is integrable?
    • Calculus
Replies
30
Views
2K
MHB Equality with curl and gradient
    • Calculus
Replies
4
Views
2K
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
    • Calculus and Beyond Homework Help
Replies
2
Views
511
Z=x+iy; f(z)=u+iv; f'(z)=?; Cauchy-Riemann Conditions
    • Calculus
Replies
9
Views
2K

Hot Threads

Recent Insights

Back
Top