# Today I Learned

## Symmetry of second derivatives

This wikipedia page shows an example of symmetry breaking of the second derivatives, that is $\partial_x\partial_y f \ne \partial_y\partial_x f$. The example is

\[f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & \mbox{ for } (x,y) \ne (0,0) \\ 0 & \mbox{ for } (x,y) = (0,0). \end{cases}\]A image of the function is shown on wikipedia.