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Trying to understand the derivation of Charpit's equations for a non-linear first order PDE given in these lecture notes https://courses.maths.ox.ac.uk/node/view_material/309 in page 31.

I'm not overly certain why in (119a) and (119b) the derivative of F with respect to x and y respectively is negative on the RHS. I'm fairly certain this is just a basic misunderstanding of using the multivariable chain rule, but I can't really figure out why, so any help would be appreciated.

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$F(p,q,u,x,y)=0$ and $\frac{\partial u}{\partial x}=p$ and $\frac{\partial u}{\partial y}=q$

$$ \frac{\partial F}{\partial p} \frac{\partial p}{\partial x} +\frac{\partial F}{\partial q} \frac{\partial q}{\partial x} +\frac{\partial F}{\partial u} \frac{\partial u}{\partial x} +\frac{\partial F}{\partial x}=0 $$ Which implies $$\frac{\partial F}{\partial p} \frac{\partial p}{\partial x} +\frac{\partial F}{\partial q} \frac{\partial q}{\partial x} =-p\frac{\partial F}{\partial u}-\frac{\partial F}{\partial x} $$

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