Tensor analysis

In an isotropic medium, vectors such as stress S and strain X are related by vector equations such as, 5 = kX, where S and X have the same direction. If the medium is not isotropic the use of vectors to describe the response may be too restrictive and the scalar k may need to be replaced by a more general operator, capable of changing not only the magnitude of the vector X, but also its direction. Such a construct is called a tensor. 

An important purpose of tensor analysis is to describe any physical or geometrical quantity in a form that remains invariant under a change of coordinate system. The simplest type of invariant is a scalar. The square of the line element ds of a space is an example of a scalar, or a tensor of rank zero. 

In a space of v dimensions two coordinate systems may be defined in such a way that the same point has coordinates. .., x ) and. .., x ) 

Suppose that an infinitesimal displacement moves point A (coordinates x ) to position B (coordinates x + dx ). To describe the same displacement in the other coordinate system, it is necessary to differentiate the expression for 

To simplify the notation, it is customary to omit the summation sign and sum over indices which are repeated on the same side of the equation. An index which is not repeated is understood to take successively the values 

---

APPENDIX - SUMMARY OF VECTOR AND TENSOR ANALYSIS The scalar (dot) product of two vectors is a number found as... 

Simmonds, J. G. A Brief on Tensor Analysis. Springer-Verlag (1994). 

The triple product of three noncolinear line elements in the reference configuration provides a material element of volume dV. Another well-known theorem in tensor analysis provides a relation with the corresponding element of volume dv in the current spatial configuration... 

D. Tinet, A. M. Faugere, and R. Prost, Cd NMR chemical shift tensor analysis of cadmium-exchanged clays and clay gels, J. Phys. Chem. 95 8804 (1991). 

C. Young, Vector and Tensor Analysis Second Edition (1993)... 

R. Abraham, J. E. Marsden, T. Ratiu 1988, Manifolds, Tensor Analysis, and Applications, 2nd edn, Springer Verlag, New York. 

Wrede, R. G, Introduction to Vector and Tensor Analysis, John Wiley Sons, New York (1963). Dover Publications, New York (1972). 

Borisenko, A. I. and Taropov, 1. E., Vector and Tensor Analysis with Applications, Prentice-Hall, Englewood Cliffs, New Jersey (1968). 

Spiegel, Murray R., Schaum s Outline of Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis, McGraw-Hill Book Company, New York (1959). 

Placement of indices as superscripts or subscripts follows the conventions of tensor analysis. Contravariant variables, which transform like coordinates, are indexed by superscripts, and coavariant quantities, which transform like derivatives, are indexed by subscripts. Cartesian and generalized velocities and 2 thus contravariant, while Cartesian and generalized forces, which transform like derivatives of a scalar potential energy, are covariant. 

See, e.g., 1. V. Sokolnikoff, Tensor Analysis, Theory and Applications to Geometry and Mechanics of Continua (Wiley, New York, 1964) L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields (Pergamon Press, New York, 1979). 

Several of Tesla s patented circuits exhibit this effect, as analyzed and rigorously shown by Barrett [41]. However, this can only be seen when the circuits are examined in a higher-topology electrodynamics. Barrett s analysis is in quaternionic electrodynamics. Tensor analysis will not show it. 

The approach to finding the transformation metric factors can be found in most books that discuss vector-tensor analysis (an excellent reference is Malvern [257]). For orthogonal coordinate transformations, metric factors are given generally as... 

For many types of flow, calculations are complex, Where they can be made at all. Ihey require the methods of tensor analysis of the stresses and strains involved. One important relationship that is widely useful in die systematic study of fluid flow in the equation of continuity. 

Although we are concerned with linear algebra it will be convenient to use a notation slightly different from the usual matrix notation and to adopt the summation convention from tensor analysis. Thus asr will denote the element in the rth row and 5 th column of the matrix a = (a ). The product a fi of a with a matrix ft = (, ) is the matrix (otsrfi s), where summation over s = 1, 2,. .., s is implied. The range of a particular affix (i.e., s = 1,2, s) is given in the table of nomenclature. In this notation the order of any matrix is apparent and the use of partitioned matrices can be represented rather easily. 

Since the publication of some prolegomena to the rational analysis of systems of chemical reaction [1] other cognate work has come to light and some earlier statements have been made more precise and comprehensive. I would like first to advert to an earlier work previously overlooked and to mention some recent publications that partially fill some of the undeveloped areas noticed before. Secondly, I wish to extend the theorem on the uniqueness of equilibrium to a more general case and to establish the conditions for the consistency of the kinetic and equilibrium expressions ( 2, 3). Thirdly, the conception of a reaction mechanism is to be reformulated in a more general way and the metrical connection between the kinetics of the mechanism and those of the ostensible reactions clarified. The notation of the earlier paper ([1], hereinafter referred to as P) will be followed and augmented where necessary. In particular the reader is reminded that the range of each affix is carefully specified and the summation convention of tensor analysis is employed. 

The mathematical detail of TGR depends on complicated tensor analysis which will not be considered here. The important result for purposes of the present discussion is the relationship, which is found to exist between two fundamental tensors1 The symmetric Riemann curvature tensor Rjlv (with... 

TENSOR ANALYSIS FOR PHYSICISTS, J.A. Schouten. Concise exposition of the mathematical basis of tensor analysis, integrated with well-chosen physical examples of the theory. Exercises. Index. Bibliography. 289pp. 55 x 85. 

---