Dyadic notation

Dyadic notation is used in this equation, so X.i- denotes the outer product... 

The dyadic notation has been used to express the terms arising from the kinetic energy part of the Lagrangian. The explicit form for one such term is... 

Show that (a rfc (-iV ) + r -iVk) >) = ii ab a rkr b) for two energy eigenfunctions o) and b), where Uab = Ea Eb in the units with Planck s constant = 27t, and electron mass=l. Note the relation to the electric quadrupole moment operator for an electronic system f dfrrq(f). Note the dyadic notation ff for the tensor quantity ... 

It is often convenient to employ a symbolic ( dyadic ) notation, writing... 

Here, D is a second-rank tensor (in dyadic notation) that describes the (traceless) quadrupole and 02 is the angle that the principal axes of D form with the coordinate axes e., e, . Without loss of generality, one can write at any point Fh... 

It is clear from (3.7) that the exciting field consists of the incident field and the field contributions of all other volume elements with m. The heart of the discrete dipole approximation is (3.7), while (3.8) is used to express the total field in terms of the exciting field. In dyadic notations, (3.8) read as... 

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. 

In general, the notation AB will be used to denote a multiplication of two vectors to yield a dyadic. A review of the properties of dyadics is given in... 

The principle of coordinate invariance states that constitutive rules must be stated in such a way that they do not depend on the coordinate system. One way to insure that this principle is satisfied is to work with dyadic or invariant notation as is common practice in the more mathematical engineering disciplines. 

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. ... 

As implied by the subscript notation in Eqs. (327)-(328), the translational and coupling diffusivity dyadics vary with choice of origin. This dependence can be quantitatively established. By invoking appropriate kinematic argu-... 

Then, by injecting this scaling factor into expression 11.36 of the basic quantity, one obtains the general solution under the form of a convolution equation, also called convolution product and notated as a multiplication by means of a star operator (dyadic operator) ... 

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