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T. Tsujioka
Advanced Seminar
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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 = (a1, a2, ..., an), etc.
____b = (b1, b2, b3) in 3-D space,
or, geometrically by an arrow directed from O to A: .

__Vectors follow a set of rules called Axiom of Vector Space1* 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) = a1b1 + a2b2 + a3b3
_____________Def. (Geometrically) ___________
where denotes the angle formed by a and b.
The magnitude is defined by . From these definitions,
note that if a = b, i.e., the angle between two vectors = 90, then .
A unit vector may be constructed by any vector as .
__Vector Product Def. (Algebraically) ab = ex(a2b3a3b2) + two ciclic cofactor terms:

______________Def. (Geometrically) the direction is determined by ‘the right-hand screw rule’
__________________and the magnitude is equal to the area spanned by a and b, i.e., .


__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: eix. The basis vectors move together with the axes of the coordinate system, and so, ‘co’-varies, whereas, the component of a vector ‘contra’-varies to the basis vectors.
__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 orthogonal coordinate systems are


i) Rectangular system

Fig. 1-3

ii) Cylindrical system
iii) Spherical system


Fig. 1-1


Fig. 1-2



Remark 1*













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