Tensors for Physics

352 17 Tensor Dynamics defines time-correlation functions. It is assumed that the distribution underlying the average is station ...

17.1 Time-Correlation Functions and Spectral Functions 353 The pertainingspectral functions Sij are the Fourier-Laplace transfor ...

354 17 Tensor Dynamics Fig. 17.1Depolarized Rayleigh scattering, VH- and HH-geometries. Thedouble arrowsindicate the directions ...

17.1 Time-Correlation Functions and Spectral Functions 355 SLor(ω)=π−^1 τ 1 +ω^2 τ^2 =π−^1 ν ω^2 +ν^2 ,ν=τ−^1. (17.10) Theline w ...

356 17 Tensor Dynamics The unit vectorhis parallel to the magnetic field. Consider the HH-geometry and puthperpendicular to both ...

17.1 Time-Correlation Functions and Spectral Functions 357 In a spatial Fourier transform of this equation, the LaplacianΔis rep ...

358 17 Tensor Dynamics The basis tensorsTiare defined by Tμν^0 = √ 3 2 ezμeνz, Tμν^1 = 1 2 √ 2 ( eμxeνx−eyμeyν ) , Tμν^2 = √ 2 e ...

17.2 Nonlinear Relaxation, Component Notation 359 The scalar constructed from the triple product of these tensors is determined ...

360 17 Tensor Dynamics The quantityQi=− √ 6 Tμνiaνλaλμis explicitly given by Q 0 =− 3 a 02 + 3 (a^21 +a^22 )− 3 2 (a^23 +a^24 ), ...

17.2 Nonlinear Relaxation, Component Notation 361 In the absence of a flow, one hasaμνst =aeqTμν^0 , when thez-direction is put ...

362 17 Tensor Dynamics 17.3 Alignment Tensor Subjected to a Shear Flow 17.3.1 Dynamic Equations for the Components In the presen ...

17.3 Alignment Tensor Subjected to a Shear Flow 363 for the flow alignment angleχ. For nematics composed of rod-like particles t ...

364 17 Tensor Dynamics 0.50.25 0 0.250.50. 75 1 a 2 0 0.2 0.4 0.6 0.8 1 1.2 a 1 0.250.20.150.10.05 0 0.050.1 a 3 0.1 0.05 0 0.05 ...

17.3 Alignment Tensor Subjected to a Shear Flow 365 17.3.3 Flow Properties The type of orientational behavior strongly affects t ...

366 17 Tensor Dynamics 17.4.1 Formulation of the Model Here, the stress tensor rather than the pressure tensor is used. The symm ...

17.4 Nonlinear Maxwell Model 367 scalar stressπisπ=B/( 2 C)± √ B^2 /( 4 C^2 )−A/C, provided thatA<B∗/( 4 C), otherwise one ha ...

368 17 Tensor Dynamics flow responsible for the friction occurs in a layer which is approximately constant, a constantvelocityco ...

Chapter 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations Abstract This chapter provides an outlook onto Special Relat ...

370 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations is invariant, for two coordinate systems moving with a constant ...

18.1 Lorentz Transformation 371 xixi=−(r 12 +r^22 +r 32 )+c^2 t^2 =−r^2 +c^2 t^2. (18.4) The condition (18.1) for the Lorentz in ...