ABSTRACT
A vectorwill be understood as an element of a vector spaceV . InV , an operation of addition of two elements and multiplication of an element by a scalar is defined in a usual way, that is, for any two vectors u and v
u + v ∈ V (2.1.1)
and
λu ∈ V (2.1.2)
where λ is a scalar. The inner product of u and v will be denoted by u · v. If Cartesian coordinates are
introduced in such a way that the set of vectors {ei} = {e1, e2, e3}with an origin 0 stands for an orthonormal basis, and if u is a vector and x is a point of a three-dimensional Euclidean space E3, then Cartesian coordinates of u and x are given by
ui = u · ei, xi = x · ei (2.1.3)
Apart from the direct vector notation, we will use indicial notation in which subscripts range is from 1 to 3 and a summation convention over repeated subscripts is observed. For example,
u · v ≡ 3∑ i=1
uivi = uivi (2.1.4)
From the definition of an orthonormal basis {ei}, it follows that
ei · ej = δij (i, j = 1, 2, 3) (2.1.5) 19
ij by
δij = { 1 if i = j 0 if i = j (2.1.6)
We introduce the permutation symbol ijk, also called the alternating symbol or the alternator, defined by
ijk =
⎧⎪⎨ ⎪⎩
1 if (ijk) is an even permutation of (123),
−1 if (ijk) is an odd permutation of (123), 0 otherwise, that is, if two subscripts are repeated.
