Contravariant derivative

As observed in (2.159), 8 v/8t, 8cv/8t denotes a part of v excluding the change of basis, which corresponds to the change of v when the observer is moving along the same coordinate system of the deformed body. 8 v/8t is known as the contravariant derivative or upper convected rate, and 8cv/8t is known as the covariant derivative or lower convected rate. ... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

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... 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

It will be assumed throughout that H and are symmetric positive-definite tensors. We write the mobility as a contravariant tensor, with raised bead indices, to reflect its function the mobility H may be contracted with a covariant force vector Fv (e.g., the derivative of a potential energy) to produce a resulting contravariant velocity Fy. 

There are several major implications of the Jacobi identity (40), so it is helpful to give some background for its derivation. On the U(l) level, consider the following field tensors in c = 1 units and contravariant covariant notation in Minkowski spacetime ... 

In these expressions differentiation and one-index transformations refer to the g integrals only of the Fock matrix [Eqs. (235) and (236)], treating the t elements as densities. The Fock matrix density elements in (Dj 1 and to the contravariant representation. If first derivative integrals in the MO basis is reduced to two occupied and two unoccupied indices (Handy et al., 1986). Note that 7] + T2 [Eqs. (257) and (258)] has the same structure as the <2) part of the MRCI Hessian (129). 

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... 

In what follows, we describe the derivation of Eq. (166). The gradients of the Euler angles can be related to the contravariant rotational measuring vectors via the mobile velocity to(, y (a) of the body frame. The mobile velocity can be obtained from [32,34]... 

We note in passing that the first three terms in the last equation are closely related to the convective derivative of a contravariant tensor [see, for example, Oldroyd (59), Lodge ((46), Eq. 12.58), Fredrickson (30)],... 

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

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 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 contravariant metric tensor a in the full space can be subdivided in a similar manner. Elowever, a subset of a in the reduced space is not simply a single submatrix, since the partial derivatives that define each term... 

The expression (18) features in the calculation of surface gradients. (An alternative derivation for the normal is also available through n = ti x t2.) It is useful to introduce the covariant basis vectors and in terms of the contravariant ones as follows ... 

A further derivation of Grad E to any gives a two-dimensional field of combinations of d Ejdx dx with i and k. It is usually arranged in a matrix, named the Hessian matrix (after Otto Hesse, 1811-1874). In general, co- or contravariant characteristic cannot be assigned to the only partial derivatives of this matrix under a coordinate transformation, because there are mixed terms coming out of the chain rule. The new terms are connected with the coordinate system and can be compressed in special symbols. The matrix... 

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... 

The contravariant base vectors can be derived from equations (4) and (5) as ... 

Rearranging Equations (16), (17), substituting the covariant tangent base vectors in the deformed state in Equations (14) (15) and the contravariant tangent base vectors in the initial configuration in Equations (5), the rotated deformation tensor F can be derived as ... 

In Chapter 3, we show that the contravariant and covariant components, respectively, of the convected derivative of the stress tensor give rise to different expressions for the material functions in steady-state simple shear flow. When compared with experimental data, it turns out that the material functions predicted from the contfavariant components of the convected derivative of the stress tensor give rise to a correct trend, while the material functions predicted from the covariant components of the convected derivative of the stress tensor do not. 

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