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How to prove: if $\partial _x f(x,y)=\partial _y f(x,y)$, then $f(x,y)$can be expressed as $f(x,y)=g(x+y)$.

This is from a paper:

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DanielB
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    This is abuse of notation. I suppose it means that $f$ depends on $x$ and $y$ only via the sum $x+y$, but that is more clearly expressed as $f(x,y)=g(x+y)$. – RobPratt Nov 15 '22 at 03:26
  • Thanks RobPratt you are right, I correct it to prevent misunderstanding. – DanielB Nov 15 '22 at 03:34

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Parameterise the problem: $\frac{d}{dt}f(x,y) = \frac{dx}{dt}f_{x}(x,y) + \frac{dy}{dt}f_{y}(x,y) = 0$

Hence $\frac{dx}{dt} = 1, \ \frac{dy}{dt} = -1, \ \frac{df}{dt} = 0$ $$x = t + c_1, \ y = -t + c_2$$

$f(x,y)$ is a constant with respect to t hence the solution must be of the form $g(x+y)$.

tom
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