Metric tensor components

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

We shall denote the space time coordinates by a (which as a four-vector is denoted by a light face x) with x° — t, x1 = x, af = y, xz = z x — ai0,x. We shall use a metric tensor grMV = gliV with components... 

The most important second order tensor is the metric tensor g, whose components in a Riemann space are defined by the relations... 

It is obvious that gab z) is independent both of the choice of inertial frame at z, with its corresponding natural coordinate system (v), and the choice of curve x(X). The elements of g are known as the components of the metric tensor in this coordinate system. Expression (39) is the required generalization that allows evaluation of 4> at all points in terms of gab %) and the curve x(A). 

It should be noted that the metric factors represent diagonal elements of a transformation matrix. It is therefore prudent to check the off-diagonal components to ensure that the new coordinate system is indeed orthogonal. In general, the elements of the metric tensor are given as [257]... 

The measuring vectors associated with the /th body-frame component of the total angular momentum can be obtained as described in the following sections. As seen from Eq. (110), the contravariant metric tensor, where the rotational part has been expressed in the components of the internal angular momentum, is inverse to the covariant metric tensor, where the rotational part has been given in terms of the components of the rotational velocity. Thus, the vectors are reciprocal to the vectors e ) that is, they obey Eq. (120) ... 

Here, and // denote respectively the local mole fraction and local electrochemical potential of the charged lipid species in that particular leaflet, g is the metric tensor defined on the leaflet surface, and Di p represents the diffusion coefficient of charged lipids. Note that Diip should not affect the equilibrium state. The local electrochemical potentials, in turn, are derived from the free energy functional that itself depends on local lipid component densities membrane curvature. This property results in a self-consistent formulation, which remains as the main computational task. 

This quaternion differential is a generalization of the Riemannian metric. The 4-vector quaternion fields qn (x) then replace the second-rank, symmetric tensor fields gM v(x) as the fundamental metric of the spacetime. The metric field q (x) is a 4-vector, whose four components are each quaternion-valued. This is then a 16-component field, rather than the 10-component metric tensor field g v of the standard Riemannian form. 

The solutions of the latter equations are the 10 components of the symmetric second-rank metric tensor g iv. The solutions of the factorized equations (46a) [or (46b) are the 16 components of the quaternion metrical field qp (or q ). We will now see that this 16-component metrical quaternion field, indeed, incorporates the gravitational and the electromagnetic fields in terms of their earlier tensor representations. Gravitation entails 10 of the components in the symmetric second-rank tensor g iv. Electromagnetism entails 6 of the components (the 3 components of the electric field and the three components of the magnetic field), as incorporated in the second-rank antisymmetric tensor Fpv. 

The flux tensor components Q Q, and are the products of the coordinate transformation metrics by the corresponding Cartesian quantities + uyQ y + USXM,... 

Magnitude of local acceleration of gravity vector g (295) determinant of fundamental tensor (318)-(319) Function of C (25)-(26) Covariant component of metric tensor (318)... 

The components of the reciprocal metric tensor are the same in each equation whereas the indices hkl are integers characteristic of individual lines. Theoreti-... 

The tensor g with components ga or g°" is known as the metric tensor or the fundamental tensor. In terms of g the length of any vector is defined by any of the following,... 

In Seet. 4.2, we need veetor spaee with abasis whieh is formed by A linear independent vectors gp p =, ..., k) which are not generally perpendicular or of unit length [12, 18, 19]. Sueh nonorthogonal basis, we eall a contravariant one. Covariant components of the so called metric tensor are defined by... 

A metric tensor with matrix 9pq is obviously symmetrical and regular (this last assertion is necessary and sufficient for the linear independence of gp in the basis of k orthonormal vectors in this space, we obtain det g , as a product of two determinants first of them having the rows and second one having the columns formed from Cartesian components of gp and gq. Because of the linear independence of these k vectors, every determinant and therefore also det g , is nonzero and conversely). Contravariant components gP of the metric tensor are defined by inversion... 

It is common practice to obtain the seemingly covariant components of the 7 matrices from the seemingly contravariant ones by multiplication with the metric tensor of Eq. (3.8) as... 

The quantities /in, /112, /122 form the covariant components of the surface metric tensor which has determinant... 

We note that the components of the contravariant metric tensor axe obtained from the components of the covariant metric tensor [gijj by the equation... 

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