Distributional derivatives of euclidean norm

Question

does anybody know how to calculate the distributional derivatives of the function: $f: (x_1,x_2,...,x_n)\to \sqrt{x_1^2+...+x_n^2}$? I only know the way for $n=1$, but I don't see how to do this in higher dimensions. Thanks in advance.

Answer 1

For functions like these, the distributional derivative can be found by just normally differentiating (kids these days, skipping calculus to learn about Sobolev spaces :)).

The partial of this function with respect to $x_i$ is just found by the chain rule as $x_i \cdot (x_1^2+\cdots + x_n^2)^{-1/2}$. This function has a singularity at the origin, yes, but the origin is a set of measure 0, and so it doesn't matter that our expression isn't defined there if we just want a distributional derivative.

For higher-order derivatives, proceed the same way if $n > 1,$ since the full space minus the origin is still connected, and so even though you don't have continuity there, you can still differentiate normally (this is the difference in one-dimension, and why Dirac masses show up as you say you can calculate--because the function is discontinuous in a way that really can't be resolved).