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

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

An image of the function is provided by wikipedia.

Modified: by ;

L Ma (2015). 'Symmetry of second derivatives', Datumorphism, 02 April. Available at: https://datumorphism.leima.is/til/math/symmetry-of-second-derivatives/.

Current Ref:

• til/math/symmetry-of-second-derivatives.md