Vector Calculus (7) - Definitions and Operations of a Scalar and a Vector

17. Definitions of a Scalar and a Vector  in Terms of Transformation Properties

 

$ x_{i}'=\sum_{j}^{}\lambda _{ij}x_{j} $의 좌표변환($ \sum_{j}^{}\lambda _{ij}\lambda _{kj}=\delta _{ik} $)

 

$ \phi $ is unaffected → scalar

$ A_{i}'=\sum_{j}^{}\lambda _{ij}A_{j} $ is transformed from the $ x_{i} $ system to the $ x_{i}' $ system → vector

 

 

18. Properties of Scalar and Vector

 

$ \vec{A}, \vec{B} $: Vector, $ \phi ,\psi ,\xi $: Scalar

 

1) Commutative Law

$ A_{i}+B_{i}=B_{i}+A_{i} $

$ \phi +\psi =\psi+\phi $

 

2) Associative Law

$ A_{i}+(B_{i}+C_{i})=(A_{i}+B_{i})++C_{i} $

$ \phi +(\psi+\xi ) =(\phi+\psi)+\xi $

 

3) Multiplication

$ \xi \vec{A}=\vec{B} $: Vector

$ \xi \phi =\psi $: Scalar

 

$ B_{i}'=\sum_{j}^{}\lambda _{ij}B_{j}=\sum_{j}^{}\lambda _{ij}\xi A_{j}=\xi\sum_{j}^{}\lambda _{ij}A_{j}=\xi A_{i}' $