at the moment I seem to struggle with a problem, which I assume is more trivial than I think. The problem I'm facing is integrating partial derivatives in order to reconstruct a root function. For this, I found very helpful posts like
- [1]: How to integrate a total derivative?
- [2]: Is it possible to reverse a gradient ($\vec{\nabla}$) operation?
I especially liked the Ansatz from one user of [2]. However, I cannot reconstruct the original function with the suggested methods, when one or more integrands appear in the denominator of one or more partial derivatives. One example:
$$ \begin{aligned} A(x,y) &= \frac{x^2 + y^2}{x^2 - y^2} \\ \\ \Leftrightarrow \frac{\partial A}{\partial x} &= - \frac{4 x y^{2}}{\left(x^{2} - y^{2}\right)^{2}} \\ \frac{\partial A}{\partial y} &= \frac{4 x^{2} y}{\left(x^{2} - y^{2}\right)^{2}} \\ \\ \Leftrightarrow \int \frac{\partial A}{\partial x} dx &= \frac{2 y^{2}}{x^{2} - y^{2}} + h(y) \\ \int \frac{\partial A}{\partial y} dy &= \frac{2 x^{2}}{x^{2} - y^{2}} + g(x) \end{aligned} $$
The methods suggests to integrate one of the partial derivatives with respect to their derivative variable, assume an integration constant depending on the remaining derivative variable and look for missing parts. However, with the fraction structure, the terms always containt both variables. I'm clearly missing something out but I cannot see my mistake. Probably partial fraction decomposition might come in handy (though I don't see how).
Originally, I wanted to recalculate a derivation from Griffiths "Exact Space-Times in Einstein's General Relativity" (2012) (p. 249), where two partial derivatives $$ \begin{aligned} \psi_{,y} &= \frac{y \left(x - 1\right) \left(x + 1\right)}{\left(x - y\right) \left(x + y\right) \left(p^{2} x^{2} + q^{2} y^{2} - 1\right)} \\ \psi_{,x} &= - \frac{x \left(y - 1\right) \left(y + 1\right)}{\left(x - y\right) \left(x + y\right) \left(p^{2} x^{2} + q^{2} y^{2} - 1\right)}\end{aligned} $$
are defined ($p$ and $q$ fullfil $p^2 + q^2 = 1$, $\log m$ is the integration constant) with a short hint to their solution:
$$ e^{2 \psi} = m^2 \frac{p^2 x^2 + q^2 y^2 - 1}{x^2 - y^2} $$
I hope I explained my problem as thoroughly as possible and I'm looking forward to your answers!
Best regards, fwillo