Tensors for Physics

3.4 Applications of the Vector Product 45 wherewis an axial vector which is perpendicular to the plane of motion. The magnitude ...

46 3 Symmetry of Second Rank Tensors, Cross Product Hereeandu, withe·u=0, are orthogonal unit vectors. The parameterρis the radi ...

Chapter 4 Epsilon-Tensor Abstract Thethirdrankepsilon-tensorisusedtoformulatethedualrelationbetween an antisymmetric second rank ...

48 4 Epsilon-Tensor The epsilon-tensor is totally antisymmetric, i.e. it changes sign, when two indices are interchanged. It is ...

4.1 Definition, Properties 49 εμνλεμ′ν′λ′= ∣ ∣ ∣ ∣ ∣ ∣ δμμ′δμν′δμλ′ δνμ′δνν′δνλ′ δλμ′δλν′δλλ′ ∣ ∣ ∣ ∣ ∣ ∣ . (4.8) The rows and c ...

50 4 Epsilon-Tensor 4.1.3 Antisymmetric Tensor Linked with a Vector With the help of the epsilon-tensor and its properties, the ...

4.2 Multiple Vector Products 51 a×(b×c)=a·cb−a·bc. (4.17) Noticethat thepositionof theparenthesis(.. .)is essential inthis case. ...

52 4 Epsilon-Tensor When the center of the circle is chosen as the originr=0, the position vectorr is perpendicular tow, thusr·w ...

4.3 Applications 53 Here ( r⊥(i) ) 2 = ( r(i) ) 2 − ( ̂wνrν(i) ) 2 is the square of the shortest distance of mass pointifrom the ...

54 4 Epsilon-Tensor This 2Dε-tensor can also be expressed in matrix notation: εij= ( 01 − 10 ) . (4.28) In 3D, a corresponding n ...

Chapter 5 Symmetric Second Rank Tensors Abstract This chapter deals with properties and applications of symmetric second rank te ...

56 5 Symmetric Second Rank Tensors In solid state mechanics, the symmetric traceless part is commonly referred to deviatoricpart ...

5.2 Principal Values 57 the tensorSisisotropic, i.e. it is proportional to the unit tensor: Sμν=Sδμν. (5.6) This follows from (5 ...

58 5 Symmetric Second Rank Tensors 5.2.4 Biaxial Tensors. The general symmetric second rank tensor with three different principa ...

5.2 Principal Values 59 In matrix notation, (5.12) is equivalent to Sμν=S ̄ ⎛ ⎝ 100 010 001 ⎞ ⎠+^2 3 s ⎛ ⎝ −^1200 0 − 210 001 ⎞ ...

60 5 Symmetric Second Rank Tensors Now the case is considered, wherevis not parallel tou. By symmetry, one of the principal dire ...

5.3 Applications 61 with the moment of inertia Θ=m 1 d 12 +m 2 d 22 =m 12 d^2. (5.19) Herem 12 =m 1 m 2 /(m 1 +m 2 )is the reduc ...

62 5 Symmetric Second Rank Tensors A molecule with three different moments of inertia is referred to asasymmetric topmolecule. O ...

5.3 Applications 63 5.3.3 Molecular Polarizability Tensor An electric fieldEcauses a slight average shift of the electrons in an ...

64 5 Symmetric Second Rank Tensors The index of refractionν(i)for linearly polarized light with the electric field vector parall ...