Assume $f(x,y) \in C^{(1)}(\Bbb{R}^2)$,If $$\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(x,y)}{\partial y}$$ for all $(x,y) \in \Bbb{R}^2$. Show that there exists a function $g(t)$,such that $f(x,y)=g(x+y)$.
2 Answers
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Your given condition implies that the directional derivative in the $(1,-1)$ direction is always zero. So $f(x,y)$ is constant on each line $x + y = c$...
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$\begingroup$ @ Zarrax : I think , g looks as g(t) = f(x,t-x) for all $ x \in \mathbb R$. Am i right. $\endgroup$ May 29, 2014 at 7:12
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$\begingroup$ Right. So define $y = t - x$ here, or equivalently $t = x + y$. $\endgroup$– ZarraxMay 29, 2014 at 19:03
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Hint: Show that the level curves of $f$ are orthogonal to $(1,1)$ (assuming $\nabla f\ne\mathbf 0$).