Christoffel

The Christoffel symbols (Spiegel, 1974), appearing in the second term on the right-hand side of Equation (5.57), are defined as... 

Volume 42 Laboratory Studies of Heterogeneous Catalytic Processes by E.G. Christoffel, revised and edited by Z. Paal Volume 43 Catalytic Processes under Unsteady-State Conditions by Yu. Sh. Metros... 

Wuttke W, Jarry H, Christoffel V, Spengler B, Seidlova-Wuttke D. Chaste tree (Vitex agnuS Castus)—pharmacology and clinical indications Phytomedicine 2003 10 348-357. 

Jarry H, Thelen P, Christoffel V, Spengler B, Wuttke W. Cimicifuga racemosa extract BNO 1055 inhibits proliferation of the human prostate cancer cell line LNCaP. Phytomedicine 2005 12 178-182. 

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

R. C. Wester, J. Christoffel, T. Hartway, N. Poblete, H. I. Maibach, and J. Forsell. Human cadaver skin viability for in vitro percutaneous absorption Storage and detrimental effects of heat-separation and freezing. Pharm. Res. 15 82-84 (1998). 

Lavigne, J.V., Gibbons, R.D., Christoffel, K.K., Arend, R., Rosenbaum, D., Binns, H., et al. (1996) Prevalence rates and correlates of psychiatric disorders among preschool children. J Am Acad Child Adolesc Psychiatry 35 204-214. 

Butterweck V, Christoffel V, Nahrstedt A, Petereit F, Spengler B, Winterhoff H. Step by step removal of hyperforin and hypericin activity profile of different Hypericum preparations in behavioral models. Life Sci 2003 73 627-639. 

Schrader E, Schwankl W, Sieder C, Christoffel V. Comparison of the bioavailability of beta-aescin after single oral administration of two different drug formulations containing an extract of horse-chestnut seeds. Pharmazie 1995 50 623-627. 

All crystals are anisotropic many other structures also have elastic anisotropy. The propagation of elastic waves in anisotropic media is described by the Christoffel equation. This still depends on Newton s law and Hooke s law, but it is expressed in tensor form so that elastic anisotropy may be included. The tensor description of elastic stiffness was summarized in 6.2, especially eqns (6.23)—(6.29). The Christoffel equation is... 

T is the Christoffel matrix. The elements of for arbitrary (i.e. triclinic) symmetry are given in Table 11.1(a). The direction cosines /, are defined by... 

The solutions to the Christoffel equation can be found by rewriting it in the form... 

Like the isotropic wave equation the Christoffel equation has three solutions, although in general there is no degeneracy except along symmetry directions. The motions of the particles are orthogonal for the three solutions, but not necessarily exactly parallel or perpendicular to the propagation direction, and so the waves are described as quasi-longitudinal or quasi-shear. 

The solution to the Christoffel equation for a given material can be plotted in k-space as surfaces of k/w. These are known as slowness surfaces, because they represent the reciprocal of phase velocity. Figure 11.1 is a representation of slowness surfaces of nickel in three-dimensional k-space. Because longitudinal waves have the greatest velocity and therefore... 

In order to calculate the reflectance function for waves from a fluid incident on an anisotropic surface, the Christoffel equation must be solved with the... 

If the z direction is taken as the normal to the surface, and the x direction is taken as also lying in the plane of incidence, then in calculating the Christoffel matrix T for arbitrary crystallographic orientation the stiffness matrix must be transformed to these new coordinates. In the notation of (11.2), and for an incident wavevector (k x, k y, k z) in the fluid, the requirements of Snell s law become... 

The Christoffel equation yields a sextic in lz. For each of these solutions the normalized particle velocity vector v can be found. Hence the traction is... 

Introducing the definition (28) of the Christoffel symbol, it is possible to write... 

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