# Unambiguous derivative notation in Spivak's "Calculus on Manifolds"

I don't understand Spivak's comment at the end that $$f$$ means something different on the two sides of the equation. Don't they both refer to the same function?

Also, the expression $$D_1(f \circ (g, h))$$ isn't clear about which variable should be first. The first var of $$f$$ is $$u$$, but the first variable of $$g, h$$ is $$x$$. So I'm wondering what that statement means since this notation purports to remove ambiguities.

I'm self-studying to prepare for grad school after a long gap, so I don't have a professor to consult. Thank you for any advice.

• In $D_1(f\circ (g,h))$ you are differentiating the first variable of the function $f\circ(g,h)$ defined by $(f\circ(g,h))(x,y)=f(g(x,y),h(x,y))$. Hopefully this clears up what the first variable should be (it's the $x$ in what I wrote above) Apr 22 at 18:47
• The left hand side refers to the partial derivative of $(x,y)\mapsto f(u(x,y),v(x,y))$ while the partial derivatives of $f$ on right hand side refer to the function $(u,v)\mapsto f(u,v).$ These derivatives are evaluted at $(u(x,y),v(x,y)).$ Apr 22 at 19:21

No, they don't refer to the same function from a strict mathematical point of view: $$f_{\text{left}}(x,y) = f_{\text{right}}(u(x,y), v(x,y)).$$