Symmetric function: Difference between revisions

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== Examples ==

* Consider the following real function:

::<math>f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3)</math>

:By definition, a symmetric function with n variables has the property that

<math>f(x_1,x_2,x_3)=(x-x_1)(x-x_2)(x-x_3)</math>

::<math>f(x_1,x_2,...,x_n)=f(x_2,x_1,...,x_n)=f(x_3,x_1,...,x_n,x_{n-1})</math> etc.

:In general, the function remains the same for every [[permutation]] of its variables. This means that, in this case:,

By definition, a symmetric function with n variables has the property that

::<math> (x-x_1)(x-x_2)(x-x_3)=(x-x_2)(x-x_1)(x-x_3)=(x-x_3)(x-x_1)(x-x_2)</math>, and so on, for all permutations of <math>x_1,x_2,x_3</math>

:and so on, for all permutations of <math>x_1,x_2,x_3</math>

* Consider the circle function:

<math>f(x_1,x_2,...,x_n)=f(x_2,x_1,...,x_n)=f(x_3,x_1,...,x_n,x_{n-1})</math> etc.

::<math>f(x,y)=x^2+y^2-r^2</math>

:If ''x'' and ''y'' are interchanged, the function becomes

In general, the function remains the same for every [[permutation]] of its variables. This means that, in this case:

::<math>f(y,x)=y^2+x^2-r^2</math>

:which yields gives exactly the same results as the original ''f''(''x'',''y'').

* Consider now the ellipse equation:function

<math> (x-x_1)(x-x_2)(x-x_3)=(x-x_2)(x-x_1)(x-x_3)=(x-x_3)(x-x_1)(x-x_2)</math>, and so on, for all permutations of <math>x_1,x_2,x_3</math>

::<math>f(x,y)=(\frac{x}{a})ax^2+(\frac{y}{b})by^2-r^2</math>

:If the ''x,y'' variablesand ''y'' are interchanged, the function becomes

* Consider the circle function:

::<math>f(y,x)=(\frac{y}{a})ay^2+(\frac{x}{b})bx^2-r^2.</math>

:This function is obviously not the same as the original if {{nowrap|1=''a'' ≠ ''b''}}, which makes it non-symmetric.

<math>f(x,y)=x^2+y^2-r^2</math>

If the x,y variables are interchanged the function becomes

<math>f(y,x)=y^2+x^2-r^2</math>

which yields gives exactly the same results as the original f(x,y). In this case, the symmetry of the function can be seen as a symmetry of rotation of the circle by 90 degrees.

* Consider now the ellipse equation:

<math>f(x,y)=(\frac{x}{a})^2+(\frac{y}{b})^2-r^2</math>

If x and y are interchanged, the function becomes

<math>f(y,x)=(\frac{y}{a})^2+(\frac{x}{b})^2-r^2</math>

where we effectively swapped the two semi axes. This function is obviously not the same as the original, which constitutes it non-symmetrical. Compared to the previous example, the 90 degree rotation symmetry is not maintained.

== Applications ==