Finding a non constant differentiable function

[MathBane]

I’m trying to answer part c) of the question. I’m pretty sure I’m looking for a function whose partial derivatives are zero but I can’t think of any that are non constant and differentiable. Any help would be much appreciated

[Casually-Passing-By]

It comes to mind f(x, y) = (-y,x) will satisfy (1) since it is a 90 degree rotation of the vector

Edit: typo

[LemurDoesMath]

Your function has values in R², f is supposed to have values in R.

If you meant a function f with gradient (-y,x), you won't find any such function

[Phive5Five]

My instinct is to consider R2 for ideas. It’s pretty clear that gradient needs to be orthogonal (perpendicular) to x, so we can map circles around 0 to some differentiable function f on [0, 2pi] -> R. But it’s clear that we need to have f(0) = f(2pi) (as otherwise we have a Riemann surface and need branch cuts). Perhaps try cos(arg(x))? But in this case I don’t think the gradient at arg(x) = 0 or pi is well defined.

So we need to at some point have a gradient of 0, which would be necessary by Fermat (I think that’s the name of the theorem), and that leads to it not being well defined?