Tensors antisymmetric

Clearly, /, d, and w are spatial tensors with components relative to the current configuration. Since the trace of an antisymmetric tensor vanishes, from (A.9)... 

We shall often have occasion to use the totally antisymmetric tensor (density) e vx>ff which is defined as follows ... 

The corresponding three-dimensional antisymmetric tensor siik is defined by... 

The Hilbert space of pure A -particle fermion states. It is an iV-foId antisymmetric tensor product of the Hilbert space of pure one-particle states. 

After division of the first three equations by ic the set of equations can be reduced to a simpler form in terms of an antisymmetric tensor defined as... 

The dual pseudotensor of any antisymmetric tensor in 4-space arises from the integral over a two-dimensional surface in 4-space [101], in which the infinitesimal element of surface is given by the antisymmetric tensor ... 

The components of this tensor are projections of the element of area on the coordinate planes. In 3-space, it is always possible to define an axial pseudovector element df, dual to the antisymmetric tensor dfk. ... 

The pseudovector element dfi represents the same surface element as dfjL. and, geometrically, is a pseudovector normal to the surface element and equal in magnitude to the area of the element. In 4-space, such a pseudovector cannot be constructed from an antisymmetric tensor such as dfpv. However, the dual pseudotensor can be defined by [10] ... 

In arriving at this conclusion, we have used antisymmetric tensor definitions such as... 

Provided we consider the Euclidean case, = js xp xp, (p, v, X, p = 1,2,3,4), where is the completely antisymmetric tensor, and equations (2) form the system of four real first-order partial differential equations. 

This nonsymmetric second-rank tensor can be decomposed to a symmetric (i.e., ay = o-ji) and an antisymmetric tensor through a symmetrization process (28). 

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

Like any antisymmetric tensor in three dimensions, fy can be expressed in terms of a vector b(r) as... 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

Note that henceforth it will be convenient to use the following notation for the symmetric and antisymmetric tensors of velocity gradients... 

These expressions ought to be transformed to eliminate the dependence on the antisymmetrical tensor of the velocity gradients. We can use the new variable v = s — u to rewrite some of the integrals in the above expressions. So, for example,... 

The application of general thermodynamic theory can be considered, first, in linear approximation. In practice, it is sufficient for the most part of applications. We can use our usual notations for symmetric and antisymmetric tensors of the velocity gradients... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

Since a second-rank cartesian tensor Tap transforms in the same way as the set of products uaVfj, it can also be expressed in terms of a scalar (which is the trace T,y(y), a vector (the three components of the antisymmetric tensor (1 /2 ) Tap — Tpaj), and a second-rank spherical tensor (the five components of the traceless, symmetric tensor, (I /2)(Ta/= + Tpa) - (1/3)J2Taa). The explicit irreducible spherical tensor components can be obtained from equations (5.114) to (5.118) simply by replacing u vp by T,/ . These results are collected in table 5.2. It often happens that these three spherical tensors with k = 0, 1 and 2 occur in real, physical situations. In any given situation, one or more of them may vanish for example, all the components of T1 are zero if the tensor is symmetric, Yap = Tpa. A well-known example of a second-rank spherical tensor is the electric quadrupole moment. Its components are defined by... 

Fortunately, several simplifications can be made (Nye, 1957). Transport phenomena, for example, ate processes whereby systems transition from a state of nonequilibrium to a state of equilibrium Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical, that is, Ty = t, (for antisymmetric tensors, t = t, ). This is known as a reciprocal relation thus transport properties are symmetrical second-rank tensors. The Norwegian physical chemist Lars Onsager (1903-1976) was awarded the 1968 Nobel Prize in chemistry for reciprocal relations. Using the reciprocal relations, Eq. 6.2 can be rewritten as ... 

A set of points M is said to be a -dimensional manifold if each point of M has an open neighborhood, which has a continuous 1 1 map onto an open set of of R , the set of all w-tuples of real numbers. Consider an w-dimensional Riemannian manifold with metric G. In an arbitrary coordinate system x, .. . , x", the volume -form is generally given by u> = dx a a dx . Here, g is the determinant of the metric in this basis, and a denotes the wedge or antisymmetric tensor product. For a flow field on the manifold prescribed by x = x) with density f x, t), a continuity equation for f x, t) can be obtained by considering the number of ensemble members >T t) within a volume Q of phase space given by... 

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