Title:

Author:

Mechanics

Copyright:

T. Tsujioka

Advanced Seminar

Mechanics

0. Mathematical Preliminaries

1. Vectors and coordinate systems

__We describe briefly about vectors and related concepts necessary to obtain notions and carry out calculations in Mechanics.

Notations vary; by boldface: * a*, by arrow: , or by bra-ket: . We freely use these notations at will. We may also use

sqrt( ) for by typographical reasons.

__Vectors are defined algebraically by *n*-tuple of numbers:

____* a* = (

____

or, geometrically by an arrow directed from O to A: .

__Vectors follow a set of rules called Axiom of Vector Space^{1*} with addition and scalar multiplication.
Products among vectors are defined as two types: scalar product and vector product, depending on the results: scalars or vectors.

____Scalar Product__ Def. (Algebraically) = *a*_{1}*b*_{1} + *a*_{2}*b*_{2} + *a*_{3}*b*_{3}

_____________Def. (Geometrically) ___________

where denotes the angle formed by * a* and

The magnitude is defined by . From these definitions,

note that if

A unit vector may be constructed by any vector as .

__

______________Def. (Geometrically) the direction is determined by ‘the right-hand screw rule’

__________________and the magnitude is equal to the area spanned by * a* and

__When observers set the coordinate system, each component of a vector may be regarded as the projection onto each coordinate axis with which each basis vector is associated: , which is nothing but the inner product of a vector * x* with the basis vector:

__The coordinate system is not restricted to the rectangular system. Many convenient coordinate systems are chosen to describe the physical situation, depending on the symmetric property of a problem. Typical

i) Rectangular system

__Fig. 1-3__

ii) Cylindrical system

iii) Spherical system