Dyadic product

The velocity-gradient tensor is VV and the operator indicates the dyadic product of two tensors, which produces a scalar. Work is a scalar quantity. 

Equation (9.15) defines the scalar product (or inner product, dot product vjv7), and (9.16) defines the corresponding dyadic product (or outer product y7yl) of vectors v -and Vj, which are different kinds of mathematical object. 

This equation shows that the stress contribution tensor is essentially a dyadic product of the end-to-end vector r and the statistical force /, which is exerted by the chain on the considered end-point. The angular brackets indicate the averaging with the aid of the mentioned distribution function. Eq. (2.25) can be explained as follows Factor rt in the brackets gives the probability that the mentioned statistical force actually contributes to the stress. This factor gives the projection of the end-to-end vector of the chain on the normal of the considered sectional plane. If a unit area plane is considered, as is usual in stress-analysis, the said projection gives that part of the unit of volume, from which molecules possessing just this projection, actually contribute to the stress on the sectional plane. 

A few words of explanation are not useless in order to understand this formalism. As a consequence of mixing, the medium is assumed to have a lamellar structure and n is a unit vector which remains normal to the material slices undergoing deformations in the velocity field, n n denotes a dyadic product (the dyadic product of vectors a and b is the tensor a.jbj) and 13 n n denotes the scalar product of the two tensors (the scalar product of tensors i = Tij and W = is the scalar quantity T W = E Z T j wji)- Assume that we start with two miscible fluids A J and B (having for instance different colors). Upon mixing, we obtain a lamellar marbled structure characterized by a striation thickness 6 and a specific "interfacial" area av. If the fluid is incompressible, avS = 1. Then,by application of (7-1)... 

By taking the scalar dyadic product with the external electric field, Eqn. (73) yields the electric field induced at nucleus I by the perturbed electron via a feedback effect. The total effective field experienced by the nucleus is hence... 

We define a tensor G as the external or dyadic product of two vectors a and b from a Euclidean space by the formula... 

The last term is somewhat complicated containing a dyadic product, vv. To be able to reformulate this expression into Cartesian coordinates we have to apply rigorous mathematical algorithms. 

The product is a second-order tensor, or a dyadic product. 

If they are independent of space, they are constant and the magnetic field varies linearly with space. Because the magnetic field B is a vector with components Bx, By, and B, the magnetic-field gradient is a second-rank tensor with nine components. It can be written as the dyadic product of the gradient operator V and the magnetic field. 

A second key notational convention that will be invoked repeatedly is the definition of the dyadic product given by (a 0 b)v = a(b v). Another key convention that will be used repeatedly is the notation that a, refers to the set ai, a2,..., a v. When carrying out integrations, volumes will typically be denoted by Q while the boundary of such a volume will be denoted as 9. ... 

Note that the term(5/(5x-(Up,2A ) is a scalar product of the spatial gradient vector and the second-order tensor produced by the dyadic product of velocity vectors. 

The quantity in square brackets is a second-order tensor, formed as a sum of dyadic products of the vectors t(e,) and e, for i = 1,2, and 3. This second-order tensor is known... 

According to the summation convention, we must sum over any repeated index over all possible values of that index. So the scalar product produces a scalar that is equal to A i Si -(- A2B2 + A3B3, whereas the vector product is a vector, the /th component of which is SijkAjBit (so, for example, the component in the 1 direction is A2B2 — A2B2), and the dyadic product is a second-order tensor with a typical component A, Bj (if we consider all possible combinations of i and j, there are clearly nine independent components). 

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation = HAq and its analogue H Ag - = Aq - must hold (where Ag - = g - g and similarly for Aq - ). These equations, which have only n components, are obviously insufficient to determine the n(n + l)/2 independent components of the Hessian or its inverse. Therefore, the updating is arbitrary to a certain extent. It is desirable to have an updating scheme that converges to the exact Hessian for a quadratic function, preserves the quasi-Newton conditions obtained in previous steps, and—for minimization—keeps the Hessian positive definite. Updating can be performed on either F or its inverse, the approximate Hessian. In the former case repeated matrix inversion can be avoided. All updates use dyadic products, usually built... 

The formation and the disappearance of the various species with a certain degree of polymerization can be balanced and clarified in the following scheme. The scheme looks like a dyadic product... 

The components of a tensor can be formed from the components of two vectors, two as a pair for one component. Take two vectors v and w. The tensor formed from v and w is called the dyadic product of v and w. The symbol to represent it is simply vw ... 

The tensor product of two vectors, also called the dyadic product, is denoted by a <8) b or just ab, and is an order two tensor with components... 

A nucleus with nonzero nuclear quadrupole moment is subject to electric quadrupole interaction if it is experiencing an inhomogeneous electric field. The latter is characterized by the electric field gradient (EFG) tensor defined by = —V°VVj where indicates dyadic product (the matrix product of a column vector and a row vector resulting in a square matrix). The Vij element of the EFG tensor is given by... 

Although the distribution function provides a general description of the orientation state in the suspension, the numerical solution of the Fokker-Planck equation is computationally expensive. One needs a more compact and efficient description of fiber orientation for use in modeling of process. A proper approach is to use orientation tensors (Advani and Tucker 1987). Orientation tensors are defined in term of the ensemble average of the dyadic products of the unit vector p, i.e.. 

Given two second-order tensors A (Ay) and B (Bij), we have different types of products the dyadic product, the single dot product and the double dot product. The dyadic product is a fourth-order tensor, written as... 

Now, rearranging the vector-dyadic product term in the last equation as... 

The components of the matrix Nj given by eq 138 define a tensor of second-rank N which is the dyadic product n n... 

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