52
1 Preliminaries
base vectors with unit length is called the rectangular Cartesian base (basis), it is
called the Cartesian base (basis) for short.
In three-dimensional rectangular coordinate system, the base vectors e 1 , e 2 and
e 3 are perpendicular to each other and the modulus of each base vector is unity, that
is
e 1 · e 1 = 1, e 2 · e 2 = 1, e 3 · e 3 = 1, e 1 · e 2 = 0, e 2 · e 3 = 0, e 3 · e 1 = 0
These six relations can be concisely expressed by the relation
e i · e j = δ i j
(1.6.3)
where
δ i j =
1 i = j
0 i = j
(1.6.4)
The symbol δ i j is called the Kronecker delta or Kronecker symbol. This definition shows that δ i j has symmetry, it has nothing to do with the index sequence,
namely
δ i j = δ ji
(1.6.5)
The components of the Kronecker symbol are the elements of unit matrix, in
three-dimensional space which can be expressed as
δ 11 δ 12 δ 13
δ 21 δ 22 δ 23
δ 31 δ 32 δ 33
=
1 0 0
0 1 0
0 0 1
(1.6.6)
The operation rule of the Kronecker symbol can be mastered by the following
examples. For instance
δ ii = δ 11 + δ 22 + δ 33 = 3
(1.6.7)
⎧
⎨
⎩
δ 1m a m = δ 11 a 1 + δ 12 a 2 + δ 13 a 3 = a 1
δ 2m a m = δ 21 a 1 + δ 22 a 2 + δ 23 a 3 = a 2
δ 3m a m = δ 31 a 1 + δ 32 a 2 + δ 33 a 3 = a 3
(1.6.8)
The general expression of Eq. (1.6.8) is
δ im a m = a i
(1.6.9)
1 Preliminaries
base vectors with unit length is called the rectangular Cartesian base (basis), it is
called the Cartesian base (basis) for short.
In three-dimensional rectangular coordinate system, the base vectors e 1 , e 2 and
e 3 are perpendicular to each other and the modulus of each base vector is unity, that
is
e 1 · e 1 = 1, e 2 · e 2 = 1, e 3 · e 3 = 1, e 1 · e 2 = 0, e 2 · e 3 = 0, e 3 · e 1 = 0
These six relations can be concisely expressed by the relation
e i · e j = δ i j
(1.6.3)
where
δ i j =
1 i = j
0 i = j
(1.6.4)
The symbol δ i j is called the Kronecker delta or Kronecker symbol. This definition shows that δ i j has symmetry, it has nothing to do with the index sequence,
namely
δ i j = δ ji
(1.6.5)
The components of the Kronecker symbol are the elements of unit matrix, in
three-dimensional space which can be expressed as
δ 11 δ 12 δ 13
δ 21 δ 22 δ 23
δ 31 δ 32 δ 33
=
1 0 0
0 1 0
0 0 1
(1.6.6)
The operation rule of the Kronecker symbol can be mastered by the following
examples. For instance
δ ii = δ 11 + δ 22 + δ 33 = 3
(1.6.7)
⎧
⎨
⎩
δ 1m a m = δ 11 a 1 + δ 12 a 2 + δ 13 a 3 = a 1
δ 2m a m = δ 21 a 1 + δ 22 a 2 + δ 23 a 3 = a 2
δ 3m a m = δ 31 a 1 + δ 32 a 2 + δ 33 a 3 = a 3
(1.6.8)
The general expression of Eq. (1.6.8) is
δ im a m = a i
(1.6.9)