So Lee's smooth manifolds defines the chain rule on functions between manifolds as:
$$d(F\circ G)=dF\circ dG$$
Where $F:M\rightarrow N$ and $G:N\rightarrow L$ where $M,N$ and $L$ are smooth manifolds. However, I do not see how this reduces to the "classic" chain rule on $\mathbb{R}$ that we all know and love. If we let $M=N=L=\mathbb{R}$ then how does the above line reduce to:
$$\frac{d}{dx}(F\circ G)=(\frac{d}{dx}F\circ G)\frac{d}{dx}G$$
Or since I'm not sure notationally thats correct:
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$
I cannot see how these two are equivalent, though I'm honestly assuming I have gotten lost in the notation and don't really understand what $dF$ is.
Best Answer
The chain rule is
$$ d(f\circ g)_p=df_{g(p)}\circ dg_p $$
This is exactly the same if $M=\mathbb{R}^m$ and $N=\mathbb{R}^n$, but in this case we generally use a different notation as
$$ \partial (f\circ g)(p)=(\partial f\circ g)(p)\circ \partial g(p) $$
As the derivatives at a point are linear we generally just write the composition of linear maps by yuxtaposition, given
$$ \partial (f\circ g)(p)=(\partial f\circ g)(p)\partial g(p) $$
or
$$ d(f\circ g)_p=df_{g(p)} dg_p $$
Above $\partial f$ represent the Fréchet derivative of $f$.