Section 2k replacement

Section 2 (k) replacement PhL 10.3.12 ready to install. // Installed on 10.19.12 at 8:30 PM. (k) Definition of the words "scalar", "vector" and "tensor" In section (a) a "scalar" was defined as something that is invariant under some transformation F, and this was identified with a "rank-0 tensor". Similarly, a "vector" is either a contravariant vector or a covariant vector and both of these are "rank-1 tensors". In Section 5 (e) certain "rank-2" tensors will appear -- they are matrices that transform in a certain way under a transformation F. In Section 7 (j) tensors of rank-n will appear, and these are objects with n indices which transform in a certain manner under F. To be more precise and to provide protection against the vagaries of "the literature", these objects probably should have been defined with the word "tensorial" in front of them. "tensorial scalar" ≡ rank-0 tensor with respect to some transformation F "tensorial vector" ≡ rank-1 tensor with respect to some transformation F "tensorial tensor" ≡ rank-n tensor with respect to some transformation F As has been emphasized several times, a "tensorial tensor" is linked to a particular underlying transformation F, and one should really use the more precise term "tensorial tensor under F". In this paper, we generally omit the word "tensorial" when discussing the above objects. This brings us into conflict with the following definitions which are often used: (Here, we use the term "expression" to indicate a number, a variable, or some combination of same.) A "scalar" is a single expression, a 1-tuple. No invariance under any transformation is implied. A "vector" is an N-tuple of expressions. No transformation rule is implied. A "second order tensor" is a matrix of expressions. No transformation rule is implied. A "tensor" is an object with n indices, n = 2,3,4... which includes the previous item. A tensor is therefore a collection of expressions which are labeled by n indices each of which goes 1 to N. No transformation rule is implied. To these definitions we can add another list: A "scalar field" is a single function of x ( the x-space coordinates). No implication of invariance. A "vector field" is an N-tuple of functions of x -- an N-tuple of scalar fields. No transform implied. A "tensor field" of order n is a set of Nn scalar functions, for example, Tabc...(x). Same. In any discussion which includes relativity (special or general), the words scalar, vector and tensor would always imply the tensorial definitions of these words. Continuum mechanics, however, seems to use the above alternate list of definitions, so that any matrix is called a tensor. Usually such matrices are functions of space and should be called tensor fields, but everybody knows what is meant. In Section 7 (u) we shall discuss the notion of an equation being "covariant", which means it has the exact same form in different frames of reference which are related by a transformation. For example, one might have F = ma in frame S, and F' = m'a' in frame S', where these frames are related by a static rotation. F and a are tensorial vectors with respect to this rotation, and m and m' are tensorial scalars, and m = m' for that reason. Both sides of F = ma transform as tensorial vectors. Since rotations are an invariance of Newtonian physics, any valid equation of motion must be "covariant", and this applies of course to particle, rigid body and continuum mechanics. In the latter field, continuum mechanics, one naturally seeks out model equations which are covariant. In order to do this properly, one must know which tensors are tensorial tensors, and which tensors are just tensors with either no transformation rule, or some transformation rule that does not match the tensor. Continuum mechanics has evolved special words to handle this situation. If a tensor is a tensorial tensor, it is said to be objective, or indifferent. In continuum mechanics an equation which is covariant is said to be frame-indifferent. Possible definitions of "tensor" are examined further in Appendix E (j). In this document we shall follow the time-honored tradition of being inconsistent in our use of the words scalar, vector and tensor, but the reader is now at least warned. The notion of tensor densities described in Appendix D further complicates the nomenclature. One can have scalar densities and vector densities of various weights, for example. Appendix J explores a few commonly used tensors in continuum mechanics and determines which of these tensors actually transform as tensors (are objective), and which tensors do not transform as tensors (are non-objective).