Vector-tensor identities

By subtracting the mechanical-energy contributions from the total energy equation, a thermal energy equation can be derived. It is this equation that proves to be most useful in the solution of chemically reacting flow problems. By a vector-tensor identity for symmetric tensors, the work-rate term in the previous sections can be expanded as... 

Next, a vector-tensor identity is employed to expand the convective momentum flux term in (8-23)... 

This identity is verified by employing summation notation for v and the del operator V in rectangular coordinates, because unit vectors Sy, and are not functions of position. This strategy applies to all vector-tensor identities because all unit vectors in rectangular coordinates can be moved to the left of... 

Even though this vector-tensor identity was verified using summation notation in rectangular coordinates, it is valid in any coordinate system. It is extremely tedious to verify vector-tensor identities that involve the del operator in curvilinear coordinate systems because the unit vectors exhibit spatial dependence. Now it is possible to combine terms in the equation of motion due to the accumulation rate process and convective momentum flux. Equations (8-24) and (8-25) yield ... 

This represents a balance between forces due to convective momentum flux, fluid pressure, and gravity. The vector-tensor identity presented in Problem 8-7 is used to re-express forces due to convective momentum flux ... 

It is necessary to introduce the following vector-tensor identity for convective momentum flux (i.e., pv Vv) in the equation of motion (see Problem 8-7) ... 

The inertial and geometrical projection tensors, and associated reciprocal vectors, are identical for models with equal masses for all beads, in which the mass tensor is proportional to the identity. 

Problem 2-1. Some Vector and Tensor Identities. Prove the following identities, where e/> is a scalar, u and v are vectors, T is a second-order tensor, and e is the third-order alternating tensor ... 

Cartesian component notation offers an extremely convenient shorthand representation for vectors, tensors, and vector calculus operations. In this formalism, we represent vectors or tensors in terms of their typical components. For example, we can represent a vector A in terms of its typical component Ait where the index i has possible values 1, 2, or 3. Hence we represent the position vector x as x, and the (vector) gradient operator V as Note that there is nothing special about the letter that is chosen to represent the index. We could equally well write x . x/ . or x , as long as we remember that, whatever letter we choose, its possible values are 1, 2, or 3. The second-order identity tensor I is represented by its components %, defined to be equal to 1 when i = j and to be equal to 0 if z j. The third-order alternating tensor e is represented by its components ,< , defined as... 

One particularly useful feature of the Cartesian index notation is that it provides a very convenient framework for working out vector and tensor identities. Two simple examples follow ... 

We use bold-face italic symbols for vectors and bold-face Greek symbols for second-order tensors dot-product operations enclosed in () are scalars, those in [ ] are vectors, and those in are second-order tensors. The vector-tensor notation and conventions are identical with those used by Bird, Stewart, and Lightfoot (8) unless otherwise indicated. 

The vector-tensor algebra in (8-19) is analogous to multiplying a 1 x 3 row matrix for n by a 3 x 3 identity matrix for the unit tensor, defined by If... 

In each of the right-hand terms the tensor is operating on two dx vectors. Because these vectors are identical, we can reverse the operations, operating on the other one first However, changing die operation order requires the tensor to be symmetric. D is, but W is not (see eqn 2.2.17). Thus the last term is 0 and we have... 

By use of vector and tensor identities, this expression may be rearranged to ... 

The identical transformation, equation (6), of the electromagnetic vector potentials was found before to leave the fields unaffected or gauge invariant. The fields Atl are not gauge invariant, but the fields described by the tensor, equation (33)... 

The product of any 3N vector with this geometric projection tensor isolates the soft component of that vector. The geometrical projection tensor is a symmetric tensor, like the Euclidean identity and unlike the dynamical projection tensor. To reflect this fact, its bead indices are written directly above and below one another, with no offset to indicate whether the implicit Cartesian index associated with each bead index acts to the right or left. 

The first term on the right-hand side of the identity above involves the divergence of the stress tensor, which also appears in the vector form of the momentum (Navier-Stokes) equations, Eq. 3.53. The momentum equation can be easily rearranged as... 

But the Minkowski spacetime R4 has trivial cohomology. This means that the Maxwell equation implies that. is a closed 2-form, so it is also an exact form and we can write. = d d, where ( is another potential 1-form in the Minkowski space. Now the dynamical equation becomes another Bianchi identity. This simple idea is a consequence of the electromagnetic duality, which is an exact symmetry in vacuum. In tensor components, with sJ = A dx and ((i = C(1dxt we have b iV = c, /tv — and b iV = SMCV - SvC or, in vector components... 

In order to obtain Green s identities for the flow field (u,p), a vector z is defined as the dot product of the stress tensor a(u, p) and a second solenoidal vector field v (divergence-free). The divergence or Gauss Theorem (10.1.1) is applied to the vector z... 

This expression is identical in form to equation (5.20). In the case of Raman scattering, however, it is necessary to compute the average Raman tensor, (. For a transversely isotropic system, the segment is free to spin about the r. axis, and the vector ni is averaged over the unit circle normal to r . In addition to (n() = 0 and equation (5.22), we require the result,... 

Differentiation of a tensor with respect to a scalar does not change its rank. The spatial differentiation of a tensor raises its rank by unity, and identical to multiplication by the vector V, called del or Hamiltonian operator or the nabla... 

It is important to make the distinction, and state the exact relation, between the measured observable and the actual hyperpolarizability tensor component(s), even if, for specific experimental conditions and molecular symmetries, their values turn out to be identical. What is also important is the fact that only one experimental condition is favorable for EFISHG, namely parallel polarizations for all optical and static fields. This leads to only one observable, resulting in only a single value to be deduced. It is not possible with EFISHG to determine more than one tensor component hence, one often contends either with the approximation that P. was determined, or with the statement that / , was obtained, in any of the above-mentioned relations to the individual tensor components. Even then, the assumption that the dipole moment vector and the vector part of the third-rank tensor along the molec-... 

The traveling-wave excitation described by Eq. (21) affects the dielectric tensor, as described by Eq. (15). The effects can be detected by a variably delayed probe pulse that is phase matched for coherent scattering, that is, collinear (in practice, nearly collinear) with the excitation pulse and the vibrational wave vector. Since the probe pulse follows the excitation pulse through the sample at the same speed c/n (neglecting dispersion), it surfs along a crest or null of the vibrational wave. The probe pulse therefore encounters each region of the sample with identical coherent vibrational distortion. 

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