I have the following question. Suppose I have a function from $\mathbb{R}^2\to\mathbb{R}$ which only depends on the first coordinate. I know that the function viewed as a function from $\mathbb{R}\to\mathbb{R}$ is weakly differentiable, is in then weakly differentiable on $\mathbb{R}^2$, with weak derivative in the non-constant direction the one dimensional one and the other direction 0? Can you generalize that, in a way that if a function only depends on the first $k$ coordinates and is weakly differentiable in thos coordinates, then it is differentiable in general? Thanks a lot :)
2026-03-27 23:30:42.1774654242
Constant weak differentaible functions?!
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Okay, silly question, here my own answer. Just do a case distinction in which direction you are taking the derivative. If you take it in the non constant direction, have the outer integral over the constant direction, and use your assumptions on the inside. If you integrate over the constant direction, do it the other way around and you will see that youll get zero as an answer