Partial Derivative Identity

Question

I have a question relating to why we can equate two partial derivative formulas.

I have an implicit equation $F(p,V,T)=0$ and do not understand why the following relation is true in this case; $$\left(\frac{ \partial p}{\partial V}\right)_T = -\frac{\left(\frac{ \partial F}{\partial V}\right)_{p,T}}{\left(\frac{ \partial F}{\partial p}\right)_{V,T}}$$

Would this relation only make sense if we could express $p$ as an explicit function in terms of $V,T$, ie. that $p=p(V,T)$.

Part of the implicit function theorem is that you can express $p$ as a unique function of $V,T$ if the elements on the right hand side are finite. — Fabian, Apr 29 '17 at 12:45
Just look at the conditions of the implicit function theorem. — Fabian, Apr 29 '17 at 12:48

score 1 · Answer 1 · answered Apr 29 '17 at 12:55

1

after the formula $$F(x,y)=0$$ and differentiate this with respect to $x$ we get $$F_x+F_y\cdot y'=0$$ we get $$(F_V)_{p,T}+(F_p)_{V,T}(p_V)_T=0$$

answered Apr 29 '17 at 12:55

Dr. Sonnhard Graubner

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Partial Derivative Identity

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