ON GEOMETRY |
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numbers, and bo the geometry of surfaces becomes a theory of invariants of a very general type.
On curved surfaces there are in general no straight lines - there are
shortest lines, which are called 'geodetic lines'. To find them, we divide any arbitrary lines joining two points into small elements, which we measure, and select the line for which the sum of these elements is less than for any other line between the two points.* Analytically we can calculate them, when the g's are given, by the aid of the generalized Pythagorean theorem. The geodetic lines, and also the curvature, are given by invariant formulae, which represent intrinsic characteristics of the surface, independent of any co-ordinates. All higher invariants are obtained from these invariants.^{4}We shall not attempt to give an explanation of the tensor calculus, as at present there is no elementary means of presenting a brief explanation; short of a small volume - at least the writer does not know of any.
^{5}The name 'tensor' originally came from the Latin word
tendere - to stretch, whence tensio = tension. Nowadays, however, it is used in a more general way; namely, to express the relation of one vector to another, and not necessarily to imply stress or tension. As an example, we can give the representation for stresses occurring in elastic bodies, which originally led to the name.^{8}As we have already seen, when we deal with
relations of vectors our expressions become additionally independent of units. Such equations, independent of the measure code, are called tensor equations.^{7}As we are interested in equations which are invariant under arbitrary transformations, certain functions, called tensors, are defined, with respect to any system of co-ordinates by a number of functions of these co-ordinates, called the components of the tensor, from which we can calculate them for any new system of co-ordinates. If two tensors of one kind are equal in one system, they will be equal in any other system. If the components vanish in one system, they vanish in all systems. Such equations express conditions which are independent of the choice of co-ordinates. By the study of structural laws of the formation of tensors we acquire means of formulating structural laws of nature in generally invariant forms. Obviously, such methods and language are uniquely appropriate for physics and the formulations of the laws of nature. If a law cannot be formulated in some such form, there must be something wrong with the formulation and it needs revision.
The tensor calculus is also peculiarly fitted to describe processes in a
plenum. We do not use it to describe the metrical conditions but to describe the field which expresses the physical states in a metrical plenum.Eddington gives an excellent example of the fact that it is definitely necessary to look into the way we build up our formulae (structure) and the method of handling them.
♦More generally, the geodetic represents a track of minimum or maximum interval-length between two distant events, either of them being unique (one-valued) in a given case. 40 |
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