Vector dyadic product

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

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)... 

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

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. 

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). 

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.. 

P is the projection matrix projecting any vector onto a direction orthogonal to the gradient. In contrast, P projects onto the gradient direction itself. P is the dyadic product of the normed gradient vector with itself, and, therefore, it is of rank 1. Then, P is of rank (n-1). It is (proof by calculation)... 

For all vectors z perpendicular to v, the equality M z = M z holds. Notice, the correction matrix (second right-hand term) in Eq.(15) is generated by the dyadic product of the vectors v and (q-M p). The... 

The static component of the stress tensor [8, 9] is defined as the dyadic product of the force f acting at contact c with the corresponding branch vector, where every contact contributes with ist force and its branch vector, if the particle lies in the averaging volume... 

As the name implies, this is a scalar quantity. The scalar product is sometimes called the dot product. The dyadic product of these two vectors is defined as... 

The dyadic product, sometimes called the tensor product [256, 273], is denoted by the symbol (S) and can be defined as having the property that for any given vectors a and b... 

Use of Dirac notation allows us to recognize at a glance that v) is a column vector, (v is the adjoint row vector, (v v) is the scalar product of these two vectors, and v)(v is a corresponding matrix dyadic, all referring to underlying object v. Further examples of Dirac notation are shown in Sidebar 9.2. 

This result requires some additional explanation. The expression (r2l — rr) will occur frequently in the following analysis, summed for both electrons and nuclei. The symbol 1 is the unit dyadic and is represented by a unit matrix. The product rr, which is not to be confused with either a scalar product or a vector product, is also a dyadic. Explicitly the above expression is evaluated in the following manner ... 

This principle as originally stated by Curie in 1908, is quantities whose tensorial characters differ by an odd number of ranks cannot interact (couple) in an isotropic medium. Consider a flow J, with tensorial rank m. The value of m is zero for a scalar, it is unity for a vector, and it is two for a dyadic. If a conjugate force A) also has a tensorial rank m, than the coefficient Ltj is a scalar, and is consistent with the isotropic character of the system. The coefficients Lij are determined by the isotropic medium they need not vanish, and hence the flow J, and the force A) can interact or couple. If a force A) has a tensorial rank different from m by an even integer k, then Ltj has a tensor at rank k. In this case, Lfj Xj is a tensor product. Since a tensor coefficient Lt] of even rank is also consistent with the isotropic character of the... 

A dyadic behaves like a two-headed vector -- the scalar product of D with a vector A yields a vector... 

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