Tensors for Physics

168 10 Multipole Potentials cf. (8.81). This force can be written as the negative gradient ofqφ,viz.Fμ=−∇μqφ, where φ= Q 4 πε 0 ...

10.3 Multipole Expansion and Multipole Moments in Electrostatics 169 |r−r′|−^1 = ∑∞ = 0 1 ! (− 1 )∂r−^1 ∂rμ 1 ∂rμ 2 ···∂rμ ...

170 10 Multipole Potentials and theoctupole moment Qμνλ= ∫ ρ(r) 15 rμrνrλd^3 r. (10.30) Up to=3, the expansion (10.27) of the e ...

10.3 Multipole Expansion and Multipole Moments in Electrostatics 171 and are proportional tor−^2 ,r−^3 andr−^4 , respectively. A ...

172 10 Multipole Potentials The hexadecapole moment, corresponding to=4, and the higher even moments are also non-zero. However ...

10.4 Further Applications in Electrodynamics 173 The center of the sphere is put atr=0. The electric field, far from the sphere, ...

174 10 Multipole Potentials cf. Sect.7.5. Thus the electric polarizationP=D−ε 0 E,cf.(7.58)ofdielectric material is determined b ...

10.4 Further Applications in Electrodynamics 175 An expansion of the potential in powers ofr′yields φ(r+r′)=φ(r)+rμ′∇μφ(r)+ 1 2 ...

176 10 Multipole Potentials As expected, this force is also obtained as a spatial derivative of the energyW,as given by (10.45), ...

10.4 Further Applications in Electrodynamics 177 The interaction between two magnetic dipoles has the same functional form, the ...

178 10 Multipole Potentials In thecreeping flow approximation, applicable for slow velocities, the nonlinear terms involvingvν∇ν ...

10.5 Applications in Hydrodynamics 179 of the functionspandvare ∇μp=−AXμνVν, and ∇μvν=a′r̂μVν+b′r̂μXνλVλ−bXμνλ. The divergence o ...

180 10 Multipole Potentials According to (8.91), the force acting on a solid body is given by the surface integral − ∮ ∂Vnνpνμd ...

10.5 Applications in Hydrodynamics 181 With the help of the multipoles in D dimensions, expressions analogous to the Stokes forc ...

Chapter 11 Isotropic Tensors Abstract This chapter deals with isotropic Cartesian tensors. Firstly, the isotropic Delta-tensors ...

184 11 Isotropic Tensors 11.2Δ-Tensors 11.2.1 Definition and Examples Of special interest among the tensors of rank 2are those, ...

11.2 Δ-Tensors 185 Δ()μ 1 μ 2 ···μ− 1 λ,μ′ 1 μ′ 2 ···μ′− 1 λ = 2 + 1 2 − 1 Δ(μ−^1 ) 1 μ 2 ···μ− 1 ,μ′ 1 μ′ 2 ···μ′− 1 . ...

186 11 Isotropic Tensors 11.2.3 Δ-Tensors as Derivatives of Multipole Potentials The-fold spatial derivative of the-th rank ir ...

11.3 Generalized Cross Product,-Tensors 187 Letwbe a vector andAa tensor of rank. The generalized cross product is given by (w ...

188 11 Isotropic Tensors 11.3.2 Properties ofh-Tensors. The-tensors are antisymmetric against the exchange of the fore and hind ...