= Reciprocity rule for constrained partial derivatives
{title2=$(\partial y/\partial x)_z(\partial x/\partial y)_z=1$}
On a regular <level set> $F(x,y,z)=0$, holding $z$ fixed gives $F_x\,dx+F_y\,dy=0$. If $F$ is <continuously differentiable> and both $F_x$ and $F_y$ are nonzero, the <implicit function theorem> gives $(\partial y/\partial x)_z=-F_x/F_y$ and $(\partial x/\partial y)_z=-F_y/F_x$. Multiplication proves the identity. The same held-fixed variable is essential, and a zero denominator invalidates the corresponding quotient rather than creating an infinite factor that may be cancelled formally.
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