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I would like to find the solution (if there is one) of the following system of PDE: $$ f(x,y)\xi(x,y)=\partial_x f(x,y),\text{ and }f(x,y)\eta(x,y)=\partial_y f(x,y),$$ where $(x,y)\in U\subset\mathbb{R}^2$ and $f\in\mathcal{C}^2(U,\mathbb{R})$, as well as $\xi$ and $\eta$. Could somebody provide a method to solve this system or directs me towards some textbook on this topic?

Thank you!

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    Are $\xi$ and $\eta$ supposed to be given? Otherwise you could take $f$ to be any nonzero $C^2$ function and use these equations as definitions of $\xi$ and $\eta$. – 2017-02-22
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    Assuming $\xi, \eta$ are prescribed data and $f$ is positive, you can write this as $\nabla \log f = (\xi, \eta)$, reducing this to the problem of integrating a field to get a potential. – 2017-02-22
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    ... and of course in order for this to work, you need $$\frac{\partial \xi}{\partial y} = \frac{\partial \eta}{\partial x}$$ – 2017-02-22
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    $\xi$ and $\eta$ are prescribed data and I really like the idea to reduce this PDE to the problem of integrating a field to get a potential. @AnthonyCarapetis: Could you make it into an answer please? – 2017-02-22

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