Let $f, g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be $C^\infty$ such that $f'_x = g'_x, f'_y=g'_y$, then $f=g+c$ for some $c\in \mathbb{R}$
#Show that if f and g share partial derivatives, then they are a constant apart
cunning marsh
hoary widgetBOT
Let $f, g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be $C^\infty$ such that $f'_x = ...
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novel reefBOT
Casiel368
cunning marsh
Real analysis tag is probably wrong, but I wasn't sure if #1015578016606343218 would be adequate for proof-based exercises.
late thunder
Real analysis tag is probably wrong, but I wasn't sure if <#1015578016606343218>...
what have you tried?
also I assume the primes are typo
cunning marsh
Writing it down, just like I do every time I try to solve non-general examples, but it gets harsh quickly
late thunder
ok, but what have you tried that's relevant to trying to solve the question?
cunning marsh
also I assume the primes are typo
That's the notation I use. Just to make it more explicit that those are derivatives
cunning marsh
I was told otherwise, that it was a valid notation too, but anyway
late thunder
but anyway, what have you tried other than.. writing the question?
cunning marsh
I meant writing the integrals
novel reefBOT
Omegabet_
cunning marsh
And then I get stuck there. I would usually perform the integration but I can't do that here.
cunning marsh
But anyway, hint: set $h:=f-g$
Oh that seems way nicer
late thunder
so it amounts to showing, under the hypotheses, that h is a constant function