I am currently reading through "Riemannian Geometry" by M. P. do Carmo and have some trouble with the notation just before Definition 2.6 where it says that
Let $\alpha:(-\epsilon,\epsilon) \to \mathbb{R}^n$ be a differentiable curve in $\mathbb{R}^n$, with $\alpha(0) = p$. Write
$$
\alpha(t) = (x_1(t),\dots,x_n(t)), \quad t \in (-\epsilon,\epsilon), \quad (x_1,\dots,x_n) \in \mathbb{R}^n.
$$
Then $\alpha^\prime(0) = (x_1^\prime(0),\dots,x^\prime_n(0)) = v \in \mathbb{R}^n$. Now let $f$ be a differentiable function defined in a neighborhood of $p$. We can restrict $f$ to the curve $\alpha$ and express the directional derivative with resprect to the vector $v \in \mathbb{R}^n$ as
$$
\frac{d(f\circ \alpha)}{dt}\Big\vert_{t = 0} = \sum^{n}_{i=1}\frac{\partial f}{\partial x_i}\Big\vert_{t=0} \frac{dx_i}{dt}\Big\vert_{t=0} = \left(\sum^n_{i=1} x^\prime_i(0)\frac{\partial}{\partial x_i}\right)f.
$$
What I don't understand is the last equality. If I add the dependence on $t$ it reads
$$
\frac{d(f\circ \alpha)(t)}{dt}\Big\vert_{t = 0} = \sum^{n}_{i=1}\frac{\partial f(x_i(t))}{\partial x_i}\Big\vert_{t=0} \frac{dx_i(t)}{dt}\Big\vert_{t=0}
$$
so how is $\frac{\partial f(x_i(t))}{\partial x_i}\Big\vert_{t=0}$ evaluated or why is it not evaluated at $t = 0$?
Best Answer
do Carmo is imprecise.
The phrase "restrict $f$ to the curve $α$" is a bit unusual. Usually a function is restricted to a subset of ist domain, not to a map into its domain. Actually do Carmo considers the composition $f \circ \alpha$ whose domain is $\alpha^{-1}(U)$ (where $U$ is the domain of $f$). Note that $\alpha^{-1}(U)$ is an open neigbhorhood of $0$.
The derivative $$\frac{d(f\circ \alpha)}{dt}\Big\vert_{t = 0} = (f\circ \alpha)'(0)$$ can be computed by the chain rule of multivariable calculus. Using the Jacobians it reads as
$$J(f\circ \alpha)(0) = Jf(\alpha(0)) \cdot J\alpha(0) = Jf(p) \cdot J\alpha(0)$$
$J(f\circ \alpha)(0)$ is a $(1 \times 1)$-matrix with entry $(f\circ \alpha)'(0)$. Moreover $Jf(p) = (\frac{\partial f}{\partial x_1}(p) \ldots\frac{\partial f}{\partial x_n}(p))^T $ and $J\alpha(0) = (x'_1(0) \ldots x'_n(0))$, thus the matrix product gives
$$(f\circ \alpha)'(0) = \sum_{i=1}^n\frac{\partial f}{\partial x_i}(p)x'_i(0) .$$ You can write the RHS as $$\sum_{i=1}^n\frac{\partial f}{\partial x_i}\Big\vert_p \frac{dx_i}{dt}\Big\vert_0. $$ Writing $\frac{\partial f}{\partial x_i}\Big\vert_0$ does not make any sense. It is mistake in the book, or at least a typo.