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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!