Editing Legendre transformation (section)

===Scaling properties===
The Legendre transformation has the following scaling properties: For {{math|''a'' > 0}},

<math display="block">f(x) = a \cdot g(x) \Rightarrow f^\star(p) = a \cdot g^\star\left(\frac{p}{a}\right) </math>
<math display="block">f(x) = g(a \cdot x) \Rightarrow f^\star(p) = g^\star\left(\frac{p}{a}\right).</math>

It follows that if a function is [[homogeneous function|homogeneous of degree {{mvar|r}}]] then its image under the Legendre transformation is a homogeneous function of degree {{mvar|s}}, where {{math|1=1/''r'' + 1/''s'' = 1}}. (Since {{math|1=''f''(''x'') = ''x<sup>r</sup>''/''r''}}, with {{math|''r'' > 1}}, implies {{math|1=''f''*(''p'') = ''p<sup>s</sup>''/''s''}}.) Thus, the only monomial whose degree is invariant under Legendre transform is the quadratic.