Remarks to the Notation
All calculations refer to Cartesian coordinates.
The indices 1, 2, 3 stand for x, y, z , so a 1 = a x , a 2 = a y , a 3 = a z .
Multicomponent quantities are characterised by bold face typing,
a = (a 1 , a 2 , a 3 ) , F = (F 1 , F 2 , F 3 ) ,
ε =
⎛
⎜
⎝
ε 11 ε 12 ε 13
ε 21 ε 22 ε 23
ε 31 ε 32 ε 33
⎞
⎟
⎠ , σ =
⎛
⎜
⎝
σ 11 σ 12 σ 13
σ 21 σ 22 σ 23
σ 31 σ 32 σ 33
⎞
⎟
⎠ .
Bold face typing is not used for vectors and tensors if these are used for one dimensional problems where one component quantities are concerned, see Chap. 22.
A dot ‘ · ’ between two quantities describes the scalar product.
As usual ‘ := ’ denotes equal by definition.
The scalar product of two vectors is a number,
a · b =
3
r =1
a r b r .
The point can be dropped for products with numbers,
λ · ν = λ ν .
The quantity v v · dA created from vector v and vector dA is therefore a vector
according to
v v · A =
v 1
3
r =1
v r d A r , v 2
3
r =1
v r d A r , v 3
3
r =1
v r d A r
.
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2
345
All calculations refer to Cartesian coordinates.
The indices 1, 2, 3 stand for x, y, z , so a 1 = a x , a 2 = a y , a 3 = a z .
Multicomponent quantities are characterised by bold face typing,
a = (a 1 , a 2 , a 3 ) , F = (F 1 , F 2 , F 3 ) ,
ε =
⎛
⎜
⎝
ε 11 ε 12 ε 13
ε 21 ε 22 ε 23
ε 31 ε 32 ε 33
⎞
⎟
⎠ , σ =
⎛
⎜
⎝
σ 11 σ 12 σ 13
σ 21 σ 22 σ 23
σ 31 σ 32 σ 33
⎞
⎟
⎠ .
Bold face typing is not used for vectors and tensors if these are used for one dimensional problems where one component quantities are concerned, see Chap. 22.
A dot ‘ · ’ between two quantities describes the scalar product.
As usual ‘ := ’ denotes equal by definition.
The scalar product of two vectors is a number,
a · b =
3
r =1
a r b r .
The point can be dropped for products with numbers,
λ · ν = λ ν .
The quantity v v · dA created from vector v and vector dA is therefore a vector
according to
v v · A =
v 1
3
r =1
v r d A r , v 2
3
r =1
v r d A r , v 3
3
r =1
v r d A r
.
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2
345