Tensor transposing

Jaumann s derivative ordinary total derivative trace of the tensor, trx = Xu, vorticity tensor transpose of the tensor Vv... 

To obtain the transpose of a tensor T the indices of its components (originally given a.s Tpg) are transposed such that... 

The stretch tensor is not indifferent but invariant under a rotation of frame. Taking the material derivative and the transpose of the first of these, and using the results in (A.23)... 

For orientation measurements, this tensor also needs to be expressed in the coordinate system OXYZ, axrz, using the matrix transformation u.xyz = Oaxyz / where O is a matrix whose elements are the direction cosines of the coordinate axes and is its transposed matrix [44]. 

The tensor mean-square displacements of atom j, is the time average , where u is the 3 x 1 column matrix of the displacements of atom j along the Cartesian axes, and T indicates the transpose. Since the normal modes are independent of each other, cross terms between modes disappear in the averaging. The result is... 

More generally, contracting the transposed projection tensor with a force to its right (or the projection tensor with a force to its left) produces a constrained force given by the sum of the original force and the constraint force induced by it. Such constrained forces may have nonzero hard components, but, on contraction with induce velocities that do not. 

In this expression, I is the identity tensor, W and (VV)T are, respectively, the velocity-gradient tensor and its transpose (Appendix B.2). 

In this book, vector quantities such as x and y above are normally column vectors. When necessary, row vectors are indicated by use of the transpose (e.g., r). If the components of x and y refer to coordinate axes [e.g., orthogonal coordinate axes ( i, 2, 3) aligned with a particular choice of right, forward, and up in a laboratory], the square matrix M is a rank-two tensor.9 In this book we denote tensors of rank two and higher using boldface symbols (i.e., M). If x is an applied force and y is the material response to the force (such as a flux), M is a rank-two material-property tensor. For example, the full anisotropic form of Ohm s law gives a charge flux Jq in terms of an applied electric field E as... 

This pattern—a rank-one tensor is transformed by a single matrix multiplication and a rank-two tensor is transformed by two matrix multiplications—holds for tensors of any rank. If A is an orthogonal transformation, such as a rigid rotation or a rigid rotation combined with a reflection, its inverse is its transpose. For example, if R is a rotation, RijRji = 8, where 5 is the Kronecker delta, defined as... 

Example 15.4-3 Both the piezoelectric effect and the Pockels effect involve coupling between a vector and a symmetric 7(2). The structure of K is therefore similar in the two cases, the only difference being that the 6 x 3 matrix [rqi is the transpose of the 3x6 matrix [diq where i 1, 2, 3 denote the vector components and q= l,. .., 6 denote the components of the symmetric 7(2) in the usual (Voigt) notation. Determine the structure of the piezoelectric tensor for a crystal of C3v symmetry. 

Here, pa,- is the bead momentum vector and u(rm. f) = iyrV is the linear streaming velocity profile, where y = dux/dy is the shear strain rate. Doll s method has now been replaced by the SLLOD algorithm (Evans and Morriss, 1984), where the Cartesian components that couple to the strain rate tensor are transposed (Equation (11)). 

Furthermore, traditional notation for scalars, vectors and variables will be adopted. A scalar of fixed value, e.g., the number of factors in a model, is represented by an italicized capitol, A. An italicized lowercase letter, e.g., the nth factor, represents a scalar of arbitrary value. All vectors are column vectors designated by lowercase bold, e.g., x. Matrices are given by uppercase bold, e.g., X, and cubes (third-order tensors by uppercase open-face letters, e.g., R. Transposes of matrices and vectors, defined by switching the row and column indices, is designated with a superscript T, e.g., xT. The transpose of a cube need not be defined for this chapter. Subscripts designate a specific element of a higher-order tensor, where the initial order is inferred by the number of subscripts associated with the scalar. 

Equation 10.1 is a second-rank tensor with transpose symmetry. The normal components of stress are the diagonal elements and the shear components of stress are the nondiagonal elements. Although Eq. 10.1 has the appearance of a [3 x 3] matrix, it is a physical quantity that, for one set of axes, is specified by nine components, whereas a transformation matrix is an array of coefficients relating two sets of axes. The tensor coefficients determine how the three components of the force vector, /, transmitted across a small surface element, vary as different values are given to the components of a unit vector / perpendicular to the face (representing the face orientation) ... 

In equations (l)-(7), vector notation is employed Vis the gradient operator, U is the unit tensor, two dots ( ) imply that the tensors are to be contracted twice, and the superscript T denotes the transpose of the tensor. The symbols appearing here are summarized in Table 1.1. 

Adjoints of tensor operators require some discussion. The adjoint of a linear operator is the transposed complex conjugate. 

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