Let $I \subseteq \mathbb{R}$ be an open interval and $\vec{f}: \mathbb{R}^2 \to \mathbb{R}^2, \vec{y}: I \to \mathbb{R}^2$ be functions such that the system $\vec{y}' = \vec{f} \circ \vec{y}$ has an isolated critical point at $\vec{0}$. Then we say the above system is locally linear at $\vec{0}$ if there is a linear map $A: \mathbb{R}^2 \to \mathbb{R}^2$ and a function $\vec{g}: \mathbb{R}^2 \to \mathbb{R}^2$ such that $$\vec{y}' = A\vec{y} + \vec{g} \circ \vec{y}$$ and $$\lim_{\vec{\epsilon} \to \vec{0}} \frac{||\vec{g}(\vec{\epsilon})||}{||\vec{\epsilon}||} = 0$$
Now in the lecture notes, it is stated that if $\vec{f}$ is a twice continuously differentiable vector function, then the system is locally linear at $\vec{0}$. While in the textbook, it is stated that suppose we write $\vec{f} = \begin{pmatrix} f_1 \\ f_2 \end{pmatrix} $, then the system is locally linear at $\vec{0}$ if both $f_1$ and $f_2$ has continuous second order partial derivatives.
Can anyone explain what does it mean for a vector function to be twice differentiable and what is the relationship between the condition in the lecture notes and the condition in the textbook?
I have asked the professor who wrote the lecture notes. It turns out that a vector function $\vec{f} = \begin{pmatrix} f_1 \\ \vdots \\ f_n \end{pmatrix}$ is defined to be $m^{\text{th}}$ times continuously differentiable if all partial derivatives (including the mixed ones) up to order $m$ of all $f_j$ are continuous. So the two conditions are (trivially) the same by definition.