Velocity is a vector in Newtonian mechanics

  • #1
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Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
kuruman
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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.
 

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