I have a problem where I am to prove Lagrange's minimal surfaces equation: $$(1+q²)r-2pqs+(1+p^2)t=0$$
Using Monge's notations: $$p:=\frac{\partial f}{\partial x};q:=\frac{\partial f}{\partial y};r:=\frac{\partial^2 f}{\partial x^2};s:=\frac{\partial^2 f}{\partial x\partial y};t:=\frac{\partial^2 f}{\partial y^2};$$
Where $f\in \mathcal{C}^2(\Delta \subset \mathbb{R}^2, \mathbb{R})$ is the minimal surface (any other function with the same values on the border of $\Delta$ has a bigger surface over it).
In the previous step, I have proven that for all $h\in \mathcal{C}^2$:
$$ \int\int_\Delta\frac{p\frac{\partial h}{\partial x}+q\frac{\partial h}{\partial y}}{\sqrt{1+p^2+q^2}}dxdy=0$$
I now have to prove that:
$$ \int\int_\Delta h \frac{(1+q^2)r-2pqs+(1+p^2)t}{\sqrt{1+p^2+q^2}}dxdy=0$$
I've understood that an integration by parts is to be done, however when I do apply it I run into a couple of quirks: I have a minus sign on the top of the fraction inside the integral (which is okay since the integral is linear) but I also have $(1+p^2+q^2)^{3/2}$ underneath. I am pretty sure my integration by parts is correct, so what am I doing wrong? Is there an extra step required, or am I using the wrong method?