Jeffreys H. Cartesian tensors

Cambridge: At the University Press. – 1931. – 101 p.It is widely felt that when the equations of mathematical physics are written out in full Cartesian form the structural simplicity of the formulae is often hidden by the mechanical labour of writing out every term explicitly. Attempts have been made to reduce this labour by one form or another of vector algebra; but it has always seemed to me that this method both introduces new difficulties and is insufficiently general. Thus the product of two vectors, in vector language, means one of two things, either the scalar or the vector product, and it is not physically obvious why just these functions of the vectors should arise and no others. The use of tensor notation, with the summation convention, carries out as great a simplification of the writing as does vector notation. The notation has actually attracted attention owing to its applications in the theory of relativity, but for ordinary purposes two great abbreviations may be made. We use rectangular Cartesian axes; the result is that the distinction between соvariant and contravariant vectors disappears, and with it the terms arising from curvature of the surfaces of reference. The formidable character of most of the formulae of the theory of relativity is absent from the formulae of tensors referred to Cartesian axes. The tensor method is a necessity for relativity; for applications in dynamics, electricity, elasticity, and hydrodynamics it is a great convenience.Cartesian Tensors
Geometrical Applications
Particle Dynamics
Dynamics of Rigid Bodies
Equivalence of Systems of Forces
Continuous Systems
Isotropic Tensors
Elasticity
Hydrodynamics