Morse lemma question Consider the statement of the Morse lemma:
Let $b$ be a non-degenerate critical point of $f:M \to \mathbb R$. Then there exists a chart $(x_1, ..., x_n)$ in a neighborhood $U$ of $b$ such that $\color{red}{x_i (b) = 0}$ and 
$$ f(x) = f(b) -x_1^2 - ... - x_\alpha^2 + x_{\alpha + 1^2} + ... + x_n^2$$ 
I am not clear why it is $x_i (b) = 0$ in red. Should it not be $(x_1 (0), ..., x_n(0)) = b$ instead? Assuming $f(x)$ means $f \circ (x_1,...,x_n)$.
 A: Here is the explanation in all the details: 
Recall that a smooth structure on a topological $n$-manifold $M$ is defined via a collection of homeomorphisms $\phi_\alpha: U_\alpha\to V_\alpha\subset M$, where $U_\alpha\subset R^n$; the collection of maps $\phi_\alpha$ satisfies smoothness condition on the transition maps. The maps $\phi_\alpha$ are called charts for the smooth structure. Sometimes, one uses the name "chart" for the map $\psi_\alpha=\phi_\alpha^{-1}$ as well. Sometimes, one also uses the name "local coordinates" for the components of the vector-functions $\psi_\alpha: V_\alpha\to R^n$; these local coordinates are sometimes denoted $x_i$, but, I think, it is best to avoid doing when you are learning basics. 
In the setting of Morse lemma, one has: For every critical point $b\in M$ of a Morse function $f: M\to R$, there exists a chart $\phi_\alpha$ with $b\in V_\alpha, b=\phi_\alpha(0)$, so that the map $F: U_\alpha\to R$, $F_\alpha=f\circ \phi_\alpha$, has the form
$$
F_\alpha(x_1,...,x_n)= f(b)+ \sum_{i=1}^n \epsilon_i x_i^2,
$$
where $\epsilon_i\in \{-1, 1\}$ for every $i\in \{1,...,n\}$. 
