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This question concerns physic problem but since it is a strict math problem I thought it would be better to ask here in the math forum. How can one prove how one can rewrite the equation dz=Mdx+Ndy with the fractions $M=\frac{\delta z}{\delta x}_y$ and $N=\frac{\delta z}{\delta y}_x$. I need a rigorous proof not just an assumption.

torgny
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By the multivariable chain rule, if $z = z(x,y)$, we have $$dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy$$

Now, $\{ dx, dy \}$ constitutes a basis for the differential forms on the $xy$-space, so if also $dz = M dx + N dy$ then we must have $$M = \frac{\partial z}{\partial x}$$ $$N = \frac{\partial z}{\partial y}$$

md2perpe
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  • What do you mean by basis? I believe that is why I am lost as that may act as a constraint for the validity of the substitution? – torgny Jun 03 '17 at 16:36
  • Do you know any linear algebra? All differentials on the $xy$-space can be written as $M dx + N dy$ for some functions $M$, $N$. You don't need more than two differentials (here $dx$ and $dy$) and taking fewer won't be enough to represent all differentials. – md2perpe Jun 03 '17 at 16:41