54
1 Preliminaries
div(r
n r) = nr
n−1 x i x i
r
+ r
n
δ ii = (n + 3)r
n
Quod erat demonstrandum.
In three-dimensional rectangular coordinate system, the base vectors e i satisfy
the following cross-product relations
e 1 × e 2 = e 3 ; e 2 × e 3 = e 1 ; e 3 × e 1 = e 2 ; e 2 × e 1 = −e 3 ; e 3 × e 2 = −e 1 ; e 1 × e 3 = −e 2
and when i = j, there is e i × e j = 0.
The above relationships can be uniformly represented as
e i × e j =
⎧
⎨
⎩
e k if i = j = k and i, j, k permute in cyclic order
−e k if i = j = k, and i, j, k permute not in a cyclic order
0 if any two of indices are the same
(1.6.14)
The above expression can be easily expressed as
e i × e j = ε i jk e k (i, j, k = 1, 2, 3)
(1.6.15)
where, ε i jk is called the Ricci symbol, permutation symbol or alternating tensor.
It can be expressed as
ε i jk =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
1 if (i, j, k) permute in cyclic order of (1, 2, 3), (2, 3, 1) or (3, 1, 2)
−1 if (i, j, k) permute in cyclic order of (3, 2, 1), (1, 3, 2) or (2, 1, 3)
0 if (i, j, k) permute in cyclic order of (3, 2, 1), (1, 3, 2) or (2, 1, 3)
0 if in indices of two any (i, j, k) are the same
(1.6.16)
or
ε i jk =
1
2
(i − j)( j − k)(k − 1) (i, j, k = 1, 2, 3)
(1.6.17)
Dot-multiplying both sides of Eq. (1.6.15) by e k , we obtain
(e i × e j ) · e k = ε i jk (i, j, k = 1, 2, 3)
(1.6.18)
The left side of Eq. (1.6.18) is the mixed product of trivector. It follows that the
physical meaning of the permutation symbol ε i jk is the volume of a cube taking the
base vectors e i , e j and e k as the three edges.
The cross product of two vectors can use permutation symbol to represent
1 Preliminaries
div(r
n r) = nr
n−1 x i x i
r
+ r
n
δ ii = (n + 3)r
n
Quod erat demonstrandum.
In three-dimensional rectangular coordinate system, the base vectors e i satisfy
the following cross-product relations
e 1 × e 2 = e 3 ; e 2 × e 3 = e 1 ; e 3 × e 1 = e 2 ; e 2 × e 1 = −e 3 ; e 3 × e 2 = −e 1 ; e 1 × e 3 = −e 2
and when i = j, there is e i × e j = 0.
The above relationships can be uniformly represented as
e i × e j =
⎧
⎨
⎩
e k if i = j = k and i, j, k permute in cyclic order
−e k if i = j = k, and i, j, k permute not in a cyclic order
0 if any two of indices are the same
(1.6.14)
The above expression can be easily expressed as
e i × e j = ε i jk e k (i, j, k = 1, 2, 3)
(1.6.15)
where, ε i jk is called the Ricci symbol, permutation symbol or alternating tensor.
It can be expressed as
ε i jk =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
1 if (i, j, k) permute in cyclic order of (1, 2, 3), (2, 3, 1) or (3, 1, 2)
−1 if (i, j, k) permute in cyclic order of (3, 2, 1), (1, 3, 2) or (2, 1, 3)
0 if (i, j, k) permute in cyclic order of (3, 2, 1), (1, 3, 2) or (2, 1, 3)
0 if in indices of two any (i, j, k) are the same
(1.6.16)
or
ε i jk =
1
2
(i − j)( j − k)(k − 1) (i, j, k = 1, 2, 3)
(1.6.17)
Dot-multiplying both sides of Eq. (1.6.15) by e k , we obtain
(e i × e j ) · e k = ε i jk (i, j, k = 1, 2, 3)
(1.6.18)
The left side of Eq. (1.6.18) is the mixed product of trivector. It follows that the
physical meaning of the permutation symbol ε i jk is the volume of a cube taking the
base vectors e i , e j and e k as the three edges.
The cross product of two vectors can use permutation symbol to represent