Vector contravariant component

The quantities /" are the contravariant components of a vector in the coordinate system X. They give an actual vector only when multiplied by the unit vector e = hvev. If the unit vectors along the coordinate lines have a scale inverse to the coordinate, e = ev/hv so that... 

The are referred to as the contravariant components of f along these basic unit vectors. It is found that... 

The stress tensor of the fluid particle (X,t) at time t, x(X, can be expanded either in terms of unit vectors at the present time t and Cartesian components Tij or in terms of the convected base vectors at position X and time t and contravariant components as shown in the following ... 

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... 

In Eqs. (1-5), the vector x and the scalar t denote spatial coordinates (x ) and time, respectively. The vector u signifies the displacement vector with contravariant components u, the components s denote the contavariant components of the stress tensor, and the vectors , Uo, and vo are prescribed functions. The vectors u(x, f) and t(x, t) represent time dependent prescribed boundary conditions on the parts T and of the boundary F, respectively, and p denotes mass density. Finally, n, signify the components of the outward unit normal to F(. It should be noted that this set of equations is supplemented by the equilibrium of angular momentum (a generalized symmetry condition on the stress tensor), the material law, and the kinematic relationships between strains/rigid rotations and the spatial derivatives of displacements. 

In section 3.1.2 we introduced the contravariant space-time 4-vector whose components are denoted by superscript indices and are given by Eq. (3.7). It is essential to realize that the metric g may be employed in order to lower (or raise) indices of any vector (within one inertial system IS) according to... 

The 4-gradient has been written as a row vector above solely for our convenience it still is to be interpreted mathematically as a column vector, of course. Being defined as the derivative with respect to the contravariant components x, the 4-gradient dpi is naturally a covariant vector since its transformation property under Lorentz transformations is given by... 

The components of the 4-potential are given by AT (, A). Note that the vector potential A = A, A, A ) contains the contravariant components of the 4-potential. According to what follows Eq. (5.54), we need to add a Lorentz scalar to the (scalar) Dirac Hamiltonian in order to preserve Lorentz covariance. This Lorentz scalar shall depend on the 4-potential. The simplest choice is a linear dependence on the 4-potential and by multiplication with 7H we obtain the desired Lorentz scalar. Minimal coupling thus means the following substitution for the 4-momentum operator... 

In (2.1) apices i, j, k, I etc. indicate (contravariant) components of the three vectors under consideration, i.e. P, and E. They are, respectively, the dipole moment per volume unit of the perturbed material, the (permanent) dipole moment per volume unit of the unperturbed material, and the perturbing external electric field. Of course, apices run fi om 1 to 3 and we can assume that 1 stands for the X component, 2 for the y component and 3 for the z component of each vector with respect to a common reference system, R = [0,(x,y,z). Partial derivatives in (2.1) depends on two or more indices. These derivatives are the components, in P = 0, (x,y,z), of the various susceptibility tensors. In particular, first order derivatives, which depend on two indices, are 3 = 9 in total and are the components of the linear or 1st order susceptibility tensor, second order derivatives... 

In (2.16), which is the microscopic analogue of (2.1), apices i, j, k, I etc. indicate (contravariant) components of the three vectors jx ed E, which are, respectively, the ground state electric dipole moment of the unperturbed molecule. 

Similar to vectors, based on the transfomiation properties of the second tensors the following three types of covariant, contravariant and mixed components are defined... 

Although it is customary to refer to covariant and contravariant vectors, this may be misleading. Any vector can be described in terms of its contravariant or its covariant components with equal validity. There is no reason other than numerical simplicity for the preference of one set of components over the other. 

The products of the components of two covariant vectors, taken in pairs, form the components of a covariant tensor. If the vectors are contravariant,... 

The fractional coordinates in crystallography, referred to direct and reciprocal cells respectively, are examples of covariant and contravariant vector components. 

It has been already noted that the rate of a steady-state reaction can be regarded as a vector in the P-dimensional space specified by its components, which are the rates along the basic routes. In terms of linear algebra, the above result means that when the basis of routes is transformed the reaction rate vector along these routes is transformed contravariantly. 

The signs of spatial components are reversed in the corresponding contravariant 4-vectors, indicated by x etc. Dirac matrices are represented in a form appropriate to a 2-component fermion theory [38], in which helicity y5 is diagonal for zero-mass fermions,... 

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) ... 

Geometric algebra approach offers some advantages over other methods presented in the literature. First of all, atomic position vectors themselves are manipulated instead of their components, and hence all expressions are simple at each stage of derivation. This is not the case when Cartesian components and back substitutions are used to obtain contravariant measuring vectors [57]. As a... 

Consider a 3-D domain that can be adequately described by the generalized curvilinear coordinate system (u, v, w) and that its mappings are adequately smooth to allow consistent definitions. Then, any vector F can be decomposed into three components with respect to the contravariant a , a , a or the covariant a , a, a,a, linearly independent basis system as... 

An even more compact formulation is obtained in covariant notation. We will follow the advice of Sakurai [38, p.6] and not introduce the Minkowski space metric g p, since the distinction between covariant and contravariant 4-vectors is not needed at the level of special relativity. We shall, however, employ the Einstein summation convention in which repeated Greek indices implies summation over the components a = 1,2,3,4 of a 4-vector. From the 4-gradient... 

Generalized contravariant velocity component in configuration space (336) Generalized vector gradient in configuration space (337) Superficial velocity vector in a porous medium (171) heat-flux vector (307)-(308)... 

By this means the theory of the underlying space may be reduced to the simultaneous affine geometry of this set of affine spaces. The tensors provide a suitable aid for treatment of this simultaneous-affine theory. As first example we take contravariant vectors or contravariant tensors of first rank. That is a geometrical object that contains four components in each coordinate system... 

It is because of this difference in behaviour between the null component and the others in the case of a covariant and a contravariant vector, that the projective tensor calculus is non-trivial. Otherwise we could think that these tensors simply arise from the combination of affine tensors. In fact, a decomposition into affine tensors happens instead, but it happens differently for covariant and contravariant tensors. 

The behave under gauge transformation like the components of a projective contravariant vector. 

Because our homogeneous coordinates behave like the components of a contravariant vector under coordinate transformation we may interpret the equation = A as characreristic of one and only one point in each tangent space. The index can here be arbitrary but different from zero, i.e. the function A is of the form... 

We know that the Z are components of a contravariant vector of index -1. We next consider the projective derivative... 

Take as given a point x, a specific representation and five arbitrary numbers X°, X, ... X. We now combine the totality of all contravariant projective vectors of fixed index whose components assume the values X°, X, ... X in the given presentation, into a single geometrical objecf. The vectors of a... 

The differentials of the coordinates x, ..., x and the differential of the gauge variable transform exactly like the components of a contravariant vector ... 

Specifically, these are contravariant four-vectors, with their component labels written as superscripts. The spacetime interval (9.83) can be represented as a scalar product if we define associated covariant four-vectors as the row matrices... 

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... 

After the preliminaries presented above we can now precisely define vectors and tensors in Minkowski space by their transformation properties under Lorentz transformations. Each four-component quantity A, which features the same transformation property as the contravariant space-time vector as given by Eq. (3.12),... 

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