From Lee's Intro to Smooth Manifolds
I don't see how this is a straightforward application the chain rule. The idea of having different coordinates on the domain and codomain is throwing me off. I would have naively written
$$\left.\frac{\partial}{\partial x^i} \right|_p (f \circ F)=\frac{\partial f}{\partial F^j}(F(p))\frac{\partial F^j}{\partial x^i}(p),$$
since if I write things more explicitly, i.e. since $x=(x^1,..,x^n) \in \mathbb{R^n}$ and $F(x)=(F^1(x),..,F^m(x)) \in \mathbb{R^m},$ we can write $f\circ F$ as
$$f \circ F(x)=f(F^1(x^1,..,x^n),…,F^m(x^1,..,x^n)).$$
Then, disregarding evaluation of the original partial at the point $p$, we have by the multivariable chain rule that
$$\frac{\partial}{\partial x^i} (f \circ F(x))=\frac{\partial}{\partial x^i} (f(F^1(x^1,..,x^n),…,F^m(x^1,..,x^n)))=\frac{\partial f}{\partial F^j}\frac{\partial F^j}{\partial x^i}(x).$$
So either what did I do wrong in my application of the chain rule or where do the coordinates $(y^i)$ come into play?
Best Answer
To sum it up: Both are right, but Lee is talking about something else.
For a start, you should not be thrown off by the fact that the coordinates in $V$ have names $y^j$. The function $f$ is defined on $V$. It is in the first place a function ${\bf y}\mapsto f({\bf y})\in{\mathbb R}$, hence a function of the $y^j$. It does not make sense to write ${\partial f\over\partial F^j}$, since you can only partially differentiate with respect to coordinate variables, not with respect to functions. Therefore I'd write your chain rule as $$\frac{\partial}{\partial x^i} (f \circ F(x))=\frac{\partial}{\partial x^i} (f(F^1(x^1,..,x^n),...,F^m(x^1,..,x^n)))=\frac{\partial f}{\partial y^j}\frac{\partial F^j}{\partial x^i}(x).$$
The linked passage in Lee's book is not about partial derivatives and the chain rule per se. Instead it talks about the effect of the derivative $dF_p$ on the tangent vectors at $p$. The "special" tangent vectors ${\partial\over\partial x_i}\biggr|_p$ (introduced before, I hope) form a basis of $T_p{\mathbb R}^n$. Therefore Lee is out to compute their images $$dF_p\left({\partial\over\partial x_i}\biggr|_p\right)\in T_q{\mathbb R}^m\qquad\qquad(q:=F(p))$$ in terms of the "special" tangent vectors ${\partial\over\partial y^j}\biggr|_q$ forming a basis of $T_q{\mathbb R}^m$. In order to find the resulting matrix he looks at the effect $$dF_p\left({\partial\over\partial x_i}\biggr|_p\right).f$$ of $dF_p\left({\partial\over\partial x_i}\biggr|_p\right)$ on an arbitrary $f$ defined in the neighborhood of $q$. In the resulting computation he then needs the chain rule as quoted by you.