Tensors for Physics

2.5 Definitions of Vectors and Tensors in Physics 25 with the quantityC′linked withCby C′μν=UμλUνκCλκ. (2.50) The relation (2.50 ...

26 2Basics Fig. 2.6 Parity operation: r→−r The parity operatorP, when applied on any functionf(r), yieldsf(−r): Pf(r)=f(−r). (2. ...

2.6 Parity 27 2.6.3 Consequences for Linear Relations The electromagnetic interaction underlying all relevant interactions encou ...

28 2Basics Pμ=ε 0 (χμν(^1 )Eν+χμνλ(^2 )EνEλ+χμνλκ(^3 ) EνEλEκ+...). (2.59) Hereχμν(^1 )≡χμνis the linear susceptibility tensor. ...

2.7 Differentiation of Vectors and Tensors with Respect to a Parameter 29 The parity operation is also timeless. Thus it commute ...

30 2Basics 1-direction. Forw>0, the motion runs counterclockwise, i.e. in the mathemat- ically positive sense. AssumingR=cons ...

2.8 Time Reversal 31 does physics allow the backward motion just as well as the forward motion. If the answer to this question i ...

32 2Basics Table 2.1The parity and time-reversal behavior of some vectors Physical quantity r v p a F w L T E B Parity − − − − − ...

Chapter 3 Symmetry of Second Rank Tensors, Cross Product Abstract This chapter deals with the symmetry of second rank tensors an ...

34 3 Symmetry of Second Rank Tensors, Cross Product 5.1 Isotropic and Symmetric Traceless Parts The symmetric part of a second r ...

3.1 Symmetry 35 Again, notice that summation indices can have different names, as long as no index appears more than twice. The ...

36 3 Symmetry of Second Rank Tensors, Cross Product 3.1.5 Fourth Rank Projections Tensors The decomposition (3.4) of a second ra ...

3.1 Symmetry 37 3.1.6 Preliminary Remarks on “Antisymmetric Part and Vector” The three independent components of the antisymmetr ...

38 3 Symmetry of Second Rank Tensors, Cross Product ab:= ⎛ ⎝ a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 ...

3.2 Dyadics 39 The further contractionμ=νleads to ab:cd=aμbλcλdμ = 1 2 (a·cb·d+a·db·c)− 1 3 a·bc·d. (3.25) For the special casea ...

40 3 Symmetry of Second Rank Tensors, Cross Product 3.3 Antisymmetric Part, Vector Product 3.3.1 Dual Relation. As already menti ...

3.3 Antisymmetric Part, Vector Product 41 Use of (3.31)fora′ 1 and of the corresponding expressions fora′ 2 ,a′ 3 (obtained by t ...

42 3 Symmetry of Second Rank Tensors, Cross Product To verify this expression, e.g. useμ=1 and note thatδ 11 =1,δ 21 =δ 31 =0. T ...

3.4 Applications of the Vector Product 43 Fig. 3.1 Vector product 3.4 Applications of the Vector Product 3.4.1 Orbital Angular M ...

44 3 Symmetry of Second Rank Tensors, Cross Product Fig. 3.2 Motion on a straight line The time change of the orbital angular mo ...