346
Remarks to the Notation
The scalar product of a vector and a tensor (matrix) is a vector,
a · ε =
3
r =1
a r ε r 1 ,
3
r =1
a r ε r 2 ,
3
r =1
a r ε r 3
,
ε · b =
3
r =1
ε 1r b r ,
3
r =1
ε 2r b r ,
3
r =1
ε 3r b r
.
The scalar product of two tensors is a tensor,
ε · σ =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
3
r =1
ε 1r σ r 1
3
r =1
ε 1r σ r 2
3
r =1
ε 1r σ r 3
3
r =1
ε 2r σ r 1
3
r =1
ε 2r σ r 2
3
r =1
ε 2r σ r 3
3
r =1
ε 3r σ r 1
3
r =1
ε 3r σ r 2
3
r =1
ε 3r σ r 3
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
Should the scalar product sign ‘ · ’ be set twice then summation must be done twice
correspondingly,
ε · · σ =
3
r =1
3
s=1
ε rs σ sr .
The Kronecker symbol δ ik is defined according to
δ ik =
⎧
⎨
⎩
1
0
for
i = k .
i = k .
The Levi- Civita symbol i jk is defined according to
123 = 231 = 312 = +1 ,
213 = 132 = 321 = −1 ,
i jk = 0 otherweise .
A cross × respresents the vector product,
a × b
i
=
3
r, s=1
irs a r b s ,
a × ε
ik
=
3
r, s=1
irs a r ε sk ,
σ × b
ik
=
3
r, s=1
krs σ ir b s .