Symmetric function: Difference between revisions

Revision as of 16:47, 9 May 2012

Revision as of 16:51, 9 May 2012

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== Examples ==

1)* Consider the following real function:

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

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<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>

2)* Consider the circle function:

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

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If the x,y variables are interchanged the function becomes

<math>f(x,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 around the axes x and y.

3)* Consider now the ellipse equation:

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

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If x and y are interchanged, the function becomes

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

,where we effectively swapped the two semi axes.