Velocity is a vector in Newtonian mechanics

[BookWei]

I studied the vector analysis in Arfken and Weber's textbook : Mathematical Methods for Physicists 5th edition.
In this book they give the definition of vectors in N dimensions as the following:
The set of $N$ quantities $V_{j}$ is said to be the components of an N-dimensional vector $V$
if and only if their values relative to the rotated coordinate axes are given by
$$V_{i}^{'}=\sum_{j=1}^N a_{ij}V_{j},\;i=1,2,...,N$$
From the definition of $a_{ij}$ as the cosine of the angle between the positive $x_{i}^{'}$ direction
and the positive $x_{j}$ direction we may write (Cartesian coordinates)
$$a_{ij}=\frac {\partial x_{i}^{'}} {\partial x_{j}}$$

Can we use the same mathematical method (or tensor analysis) to prove the velocity is a vector in
Newtonian mechanics?
Many thanks!

 

[kuruman]

Can we use the same mathematical method (or tensor analysis) to prove the velocity is a vector in
Newtonian mechanics?

Why not? If you have a position vector $\vec X$ that transforms according to $X_{i}^{'}=\sum^3_{j=1} a_{ij}X_{j}$, then when you take the time derivative, $\dot{X}_{i}^{'}=\sum_{j=1}^3 a_{ij}\dot{X}_{j}$. This says that $\dot{\vec X}$ transforms as a vector.