Contravariant component

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

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... 

We are often interested in the rms thermal displacements in A. They correspond to the contravariant components UJk along covariant axes of unit length, rather than along the non-unit length a, b, c axes. The rms displacements are obtained from... 

Here subscripts correspond to the covariant components of the spinor, which are related to the contravariant components (superscripts) through the metric spinor... 

Couette flow has a velocity field whose contravariant components are (2,22,29)... 

Conversely, the divergence of F uses the contravariant components and becomes... 

To select the correct basis for (3.65), covariant (contravariant) components should be expressed as a function of their contravariant (covariant) counterparts, via the pertinent system metrics. Unfortunately, the differentiation of these metrics produces the demanding Cristoffel symbols that obstruct the solution of (3.65). This difficulty can be circumvented by classical... 

To ensure the consistent termination of infinite space, again observe the boundary-adjacent cell of Figure 3.7. The contravariant component is evaluated by linear interpolation over the covariant ones at the white-dotted points, while the gradient is calculated through a linear extrapolation. A significant feature of (3.83)—(3.85) is their fully conservative nature that has a serious impact on the profile of the FDTD analysis. If the prior aspects are inserted into the complete form of Maxwell s equations (see Chapter 2), one acquires... 

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. 

The trivial constant space-time shift a does not affect the determination of the inverse transformation and is therefore neglected in the following. Anticipating the discussion in section 3.1.4, it is further convenient to define the fully contravariant components of the metric to be identical to the fully covariant components g v as already introduced in Eq. (3.8), g = gfiv, and to note that the metric is both symmetric and its own inverse,... 

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... 

Although the metric g has been defined as constant by Eq. (3.8), we have just shown that it perfectly fits into the definition of a tensor and features the correct transformation property under Lorentz transformations. Another very useful fourth-rank Lorentz tensor is the totally antisymmetric Levi-Civit (pseudo-)tensor whose contravariant components are defined 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... 

As a generalization of the three-dimensional Levi-Civiti symbol defined in chapter 2 by Eq. (2.9) we have introduced the four-dimensional totally antisymmetric (pseudo-)tensor whose contravariant components have been defined by... 

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... 

We remember that if the reference system is orthonormal (as often it is) there is no difference between contravariant components (indicated by apices) and covariant components (pedices) so the components can be all indicated with pedices, as it is usually done. Also, it should remembered that some authors include the numeric coefficients of the Taylor series in the definition of tensor components [1]. 

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. 

Similarly, for contravariant components A "(, t ) anda9 (x, t) of a tensor of second order in convected and fixed coordinate systems, respectively, we have... 

The contravariant components g i of the metric tensor are obtained by (Hawkins... 

KINEMATICS AND STRESSES OF DEFORMABLE BODIES 43 and for contravariant components we have... 

Comparison of Eq. (3.17) with (3.6) shows that the use of the Jaumann derivative of a in the classical Maxwell model gives rise to the material functions that are quite different in form as compared to when the contravariant components of the convected derivative of a are used in the classical Maxwell model. It is of great interest to... 

Note that when using the Jaumann derivative one obtains the identical expressions for the material functions, regardless of whether the covariant or contravariant components of tensors a and d are employed. 

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