Symmetry of Second Derivatives - Non-symmetry

Non-symmetry

In the worst case symmetry fails. Given two variables near (0, 0) and two limiting processes on

corresponding to making h → 0 first, and to making k → 0 first. These processes need not commute (see interchange of limiting operations): it can matter, looking at the first-order terms, which is applied first. This leads to the construction of pathological examples in which second derivatives are non-symmetric. Given that the derivatives as Schwartz distributions are symmetric, this kind of example belongs in the 'fine' theory of real analysis.

The following example displays non-symmetry. Note that it does not violate Clairaut's theorem since the derivatives are not continuous at (0,0)

f(x,y) = \begin{cases} \frac{xy(x^2 - y^2)}{x^2+y^2} & \mbox{ for } (x, y) \ne (0, 0)\\ 0 & \mbox{ for } (x, y) = (0, 0). \end{cases}

The mixed partial derivatives of f exist and are continuous everywhere except at . Moreover

\frac {\partial}{\partial x} \left( \frac { \partial f }{ \partial y} \right) \ne \frac {\partial}{\partial y} \left( \frac { \partial f }{ \partial x} \right)

at .

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