Tensors for Physics
8.5 Further Applications in Electrodynamics 147 however, and also in an anisotropic linear medium with ενμ =0, the Maxwell stre ...
148 8 Integration of Fields Notice, no specific form of an interrelation betweenEandDor betweenBandH has been used. So the energ ...
8.5 Further Applications in Electrodynamics 149 one has Eλ=ε− 01 ε−λκ^1 Dκ, Hλ=μ− 01 μ−λκ^1 Bκ, and consequently uel= 1 2 ε 0 −^ ...
150 8 Integration of Fields Putting terms together, one arrives at ρEμ+εμνλjνBλ+εμνλ ∂ ∂t (DνBλ) =Eμ∇νDν−εμνλDνελκτ∇κEτ+εμνλενκτ ...
8.5 Further Applications in Electrodynamics 151 of the Maxwell stress tensor. The total stress tensor is Tνμ=Tνμel+T mag νμ. (8. ...
152 8 Integration of Fields Notice that εκνμTνμ=εκνμDνEμ+εκνμBνHμ=(D×E)κ+(B×H)κ. (8.128) Thus integration of the local balance e ...
Part II Advanced Topics ...
Chapter 9 Irreducible Tensors Abstract At the begin of the more advanced part of the book, irreducible, i.e. symmetric traceless ...
156 9 Irreducible Tensors seeSects.3.1.2andChap. 6 .Thenumberofindependentcomponentsofanirreducible tensor is of rankis 2 + 1. ...
9.1 Definition and Examples 157 In many cases, the explicit form of the symmetric traceless higher rank tensors are not needed. ...
158 9 Irreducible Tensors with N= ! ( 2 − 1 )!! = 1 · 2 · 3 ···(− 1 )· 1 · 3 · 5 ···( 2 − 3 )·( 2 − 1 ) . (9.11) For a pr ...
9.4 Cartesian and Spherical Tensors 159 with thespherical components a(m)=a·(e(m))∗=(− 1 )ma·e(−m). (9.19) Explicitly, the relat ...
160 9 Irreducible Tensors The isomorphic Cartesian tensor has the same number of independent components, though it is not obviou ...
9.4 Cartesian and Spherical Tensors 161 Y 2 (±^2 )=c( 2 ) 1 2 √ 2 r−^2 (x±iy)^2 =c( 2 ) 1 2 √ 2 sin^2 θexp(± 2 iφ). (9.28) For a ...
162 9 Irreducible Tensors H 4 ≡Hμνλκ(^4 ) r̂μr̂νr̂λr̂κ =x^4 +y^4 +z^4 − 3 5 . (9.32) Herex,y,zstand for the components of the un ...
Chapter 10 Multipole Potentials Abstract In this chapter descending and ascending multipole potentials are introduced, their pro ...
164 10 Multipole Potentials The same applies for a-fold spatial differentiation ofr−^1. In this spirit, Cartesian tensors of ra ...
10.1 Descending Multipoles 165 ∇μ∇νr−^1 =Xμν(r)− 4 π 3 δμνδ(r), (10.7) whereXμν(r)is given by (10.5). To verify (10.7), integrat ...
166 10 Multipole Potentials Multiplication ofXμ 1 μ 2 ···μby the components ofryields a tensor of rank+1. The contraction with ...
10.2 Ascending Multipoles 167 are also solutions of the Laplace equation. These are theascending multipole potentials. They are ...
«
4
5
6
7
8
9
10
11
12
13
»
Free download pdf