Contravariant Components Research Articles

Equations for fluids with internal magnetic and mechanical moments

Here p is the fluid density, Pe = ep is the density of the free electric charge of the fluid, e is a constant, u i are the contravariant components of the imaginary dimensionless velocity vector of the individual points of the fluid, Fij = 3iAj -3jA. are the electromagnetic field tensor components, 3i = 3/Oxi is the s)nnbol of! the derivative with respect to the variables x i (i = I, 2, 3, 4) of the observed Cartesian coordinate system in a pseudo-Euclidean space of events with the metric tensor gij (g~ = g22 = gaa = _ g44 = i, gij = 0 for i # j), d/dr = cui3 i is the symbol of the derivative with respect to the intrinsic time, c is the velocity of light in a vacuum, S is the entropy, A m = Am(p, S) is a given function, KzJ = vpz] are the antisymmetric tensor components of the internal mechanical moment of the fluid, which are proportional to the volume density tensor components pij of the internal electromagnetic moment, v is a constant (the gyromagnetic ratio). The quantities p, u ~, ~lj in (i) are connected by the relationships (p is a constant)

Singular matrix representations of point groups

Starting from a comparison of the properties of cartesian coordinate functions and plane waves as basis functions, a discussion is given of the use of contravariant and covariant components in setting up representation matrices for group operators. The use of covariant components together with a linearly dependent basis is shown to lead to singular matrix representations of a point group. A general theory of such singular representations is given, together with some examples.

Physical Applications of Vectors and Tensors

H Teichman (transl. C W Kilmister) London: George G Harrap 1970 pp 235 price £3.25 cloth £2.60 paper This book provides a uniform treatment of a wide variety of applications of vector and tensor analysis within the fields of newtonian mechanics, differential geometry, the theory of crystal lattices, electromagnetic theory and relativity. A valuable feature is the early introduction of nonorthogonal coordinate axes, which permits a discussion of covariant and contravariant components in an elementary setting.

An Introduction to the Rudiments of Tensor Analysis

The notion of a tensor is developed in terms presumed familiar to the reader; tensors are shown to be a natural outgrowth and extension of vectors and matrices. The paper begins with an elementary discussion of affine vector spaces in a way which presupposes no prior contact with linear algebra on the part of the reader. The notions of the contravariant and covariant components of a vector are introduced early and the vector is characterized as a tensor of rank one so that the reader may readily generalize the results to tensors of higher ranks. Roughly, the discussion is divided into two major headings, Tensor Algebra and Tensor Analysis; a brief introduction to differential geometry (where tensor analysis achieves its greatest power and beauty) is included under the latter heading.

Stationary gravitational fields due to single bodies

In previous papers it has been shown how stationary gravitational fields due to single bodies may be calculated by a method of successive approximations. The method is now improved and made more compact by using the contravariant components of the energy tensor instead of the covariant components.

Rheological Equations of Viscoelastic Liquid

Assuming that the elastic displacement of a material particle is expressible by the displacement from θei to Θj in general curvilinear coordinate system or that from xei to Xi in Cartesian coordinate system, the contravariant component of the elastic strain tensor will be defined byeij=1/2(∂Θi/∂xeα∂Θj/∂xeα-∂Θi/∂Xβ∂Θj/∂Xβ) (1)The contravariant component of the actual strain rate tensor will be given byfij=1/2(Vi/j+Vj/i) (2)where Vi represents the contravariant component of velocity vector and Vi/j denotes the contravariant differentiation of Vi by Θj.We assume that the stress tensor τ is a function of both the elastic strain tensor e and the actual strain rate tensor f, and may be expressible as a polynomial in e and f. If the material is isotropic in its rest state, this relation must be invariant to the transformation of the coordinate system. Then, considering the symmetric property of τ, the relation will be written as follows:τ=(-P+a0)1+α1e+α2f+α3e2+α4f2+α5(ef+fe)+α6(e2f+fe2)+α7(ef2+f2e)+α8(e2f2+f2e2) (3)where P is the hydrostatic pressure, 1 is the metric tensor and α0, α1, …… α8 are polynomials of ten invariants such as tre, tre2, tre3, trf, trf2, trf3, tref, tre2f, tref2 and tre2f2. Considering that the principal axes of e are equivalent to those of τ but not to f, Eq. (3) will be reduced toτ=(-p+α0)1+α1e (4)This equation assures the coincidence of the principal axes of τ with those of e in every case, but in some cases it may happen that the principal axes of the other term of Eq. (3) become equal to those of τ. In simple shearing flow, such term is α7(ef2+f2e). Then, Eq. (3) will beτ=(-p+α0)1+α1e+α7(ef2+f2e) (5)which may be expressed in terms of the contravariant components of tensors as follows:τij=(-P+α0)Gij+α1eij+α7(eiαfβαfβj+fiαfβ;αeβj) (6)The applications of our theory to simple shearing flow and steady flow through pipe have been made with satisfactory results; especially it has been shown that two normal stresses in the direction perpendicular to the stream are equivalent to each other.

La solution générale des équations d’Einstein {fx1141-1}

The rigorous calculation of the general solution of Einstein’s equations {ie1141-1} leads, for the affine connexion, to a simple univocally determinated and completely explicited expression, in function of the contravariant components ({ie1141-2} and {ie1141-3}) of the fundamental tensor {ie1141-4}. The existence conditions of this general solution are also explicited.