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1991 | Buch | 2. Auflage

Fundamental Not(at)ions Kapitel lesen Erstes Kapitel lesen

Tensor Geometry

The Geometric Viewpoint and its Uses

verfasst von: Christopher Terence John Dodson, Timothy Poston

Verlag: Springer Berlin Heidelberg

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

We have been very encouraged by the reactions of students and teachers using our book over the past ten years and so this is a complete retype in TEX, with corrections of known errors and the addition of a supplementary bibliography. Thanks are due to the Springer staff in Heidelberg for their enthusiastic sup­ port and to the typist, Armin Kollner for the excellence of the final result. Once again, it has been achieved with the authors in yet two other countries. November 1990 Kit Dodson Toronto, Canada Tim Poston Pohang, Korea Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI O. Fundamental Not(at)ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Physical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 I. Real Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Subspace geometry, components 2. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Linearity, singularity, matrices 3. Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Projections, eigenvalues, determinant, trace II. Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1. Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Tangent vectors, parallelism, coordinates 2. Combinations of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Midpoints, convexity 3. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Linear parts, translations, components III. Dual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1. Contours, Co- and Contravariance, Dual Basis . . . . . . . . . . . . . . 57 IV. Metric Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1. Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Basic geometry and examples, Lorentz geometry 2. Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Isometries, orthogonal projections and complements, adjoints 3. Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Orthonormal bases Contents VIII 4. Diagonalising Symmetric Operators 92 Principal directions, isotropy V. Tensors and Multilinear Forms 98 1. Multilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Tensor Products, Degree, Contraction, Raising Indices VE Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 1. Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Metrics, topologies, homeomorphisms 2. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Convergence and continuity 3. The Usual Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Inhaltsverzeichnis

Frontmatter
0. Fundamental Not(at)ions
Abstract
Please at least skim through this chapter; if a mathematician, your habits are probably different somewhere (maybe f −1 not f ←) and if a physicist, perhaps almost everywhere.
Christopher Terence John Dodson, Timothy Poston
I. Real Vector Spaces
Christopher Terence John Dodson, Timothy Poston
II. Affine Spaces
Abstract
Our geometrical idea (I.1.01) of a vector space depended on a choice of some point 0 as origin. However, just as for bases, there may be more than one plausible choice of origin. Similarly, it may be useful to avoid committing oneself on the question (a fact discovered by Galileo). For this purpose, and for the sake of some language useful even when we have an origin, we shall consider affine spaces.
Christopher Terence John Dodson, Timothy Poston
III. Dual Spaces
Abstract
Throughout this chapter X and Y will denote finite-dimensional real vector spaces, and n and m their respective dimensions.
Christopher Terence John Dodson, Timothy Poston
IV. Metric Vector Spaces
Abstract
So far, we have worked with all non-zero vectors on an equal footing, unconcerned with the idea of their length except in illustration (as in I.3.14), or of angles between them. All the ideas we have considered have been independent of these concepts, and for instance either of the bases in Fig 1.1 can be regarded as an equally good basis for the plane. Now, the notions of length and angle are among the most fruitful in geometry, and we need to use them in our theory of vector spaces. But this means adding a “length structure” to each vector space, and since it turns out that many are possible we must choose one — and define what we mean by one.
Christopher Terence John Dodson, Timothy Poston
V. Tensors and Multilinear Forms
Abstract
Starting with a vector space X, we already have several spaces derived from it, such as the dual space X* and the spaces L 2(X;R) and L 2(X*;R) of bilinear forms on X and X*. We shall now produce some more. Fortunately, rather than adding more spaces to an ad hoc list, all as different as, say X* and L 2(X; R) seem from each other, our new construction gives a general framework in which all the spaces so far considered occur as special cases. (This gathering of apparently very different things into one grand structure where they appear as examples is common in mathematics — both because it is often a very powerful tool and because many mathematicians have great difficulty in remembering facts they can’t deduce from a framework, like the atomic weight of copper or the date of the battle of Pondicherry. This deficiency is often what pushed them to the subject and away from chemistry or history in the first place, at school.)
Christopher Terence John Dodson, Timothy Poston
VI. Topological Vector Spaces
Abstract
When we use logarithms for practical calculations, we rarely know exactly the numbers with which we are working; never, if they result from any physical operation other than counting. However if the data are about right, so is the answer. To increase the accuracy of the answer, we must increase that of the data (and perhaps, to use this accuracy, refer to log tables that go to more figures). In fact for any required degree of accuracy in the final answer, we can find the degree of accuracy in our data which we would need in order to guarantee it — whether or not we can actually get data that accurate. The same holds for most calculations, particularly by computer. Errors may build up, but sufficiently accurate data will produce an answer accurate to as many places as required. (The other side of this coin is summarised in the computer jargon GIGO — “Garbage In, Garbage Out ”.)
Christopher Terence John Dodson, Timothy Poston
VII. Differentiation and Manifolds
Abstract
Throughout this chapter X, X′ will be affine spaces of (finite) dimensions n, m respectively, with difference functions d, d′ and vector spaces T, T′.
Christopher Terence John Dodson, Timothy Poston
VIII. Connections and Covariant Differentiation
Abstract
We have remarked (VII.5.02) that any vector in TM can arise as a tangent vector to a curve. It can moreover be defined in this way; Exercises 1–3 outline this construction of the tangent bundle. This way of looking at tangent vectors is central to the notation and thinking of this chapter, so if you do not do these exercises in full, at least be sure you are clear what is asserted in them. The tangent bundle is like compactness: not to be grokked in fullness from any one point of view.
Christopher Terence John Dodson, Timothy Poston
IX. Geodesics
Abstract
The ancient custom in the Eastern Mediterranean of the straight, royal road for the exclusive use of the semi-divine ruler (cf. Aristotle telling Alexander there was no royal road to geometry — he had to go the same way as everyone else) involved a clear, if unformulated, idea of “straight”. With the rigid formalisation of geometry into the Euclidean system, “straight” became a more restricted notion which clearly would not fit a road that bent over the horizon, as a long enough road must. Hence a new word was needed. Earth had been considered a perfect sphere since early Greek times, and on such if you keep “straight on”, deviating neither to the left nor to the right, for long enough you return to your starting point and your starting direction. Your path, then, unambigously divides the earth into two parts, to its left and to its right: hence the chosen word for such a path was “geodesic” or “divides the earth”. This name has become fixed for an undeviating path, though only on a perfect sphere does such a path always have this dividing property (and the earth is not such thing).
Christopher Terence John Dodson, Timothy Poston
X. Curvature
Abstract
In treating the geometry of manifolds that were not simply nice flat affine spaces we have paid major attention to parallel transport along curves; the feature of general spaces that distinguishes them most dramatically is the disappearance of “absolute” parallelism. This prompts
Christopher Terence John Dodson, Timothy Poston
XI. Special Relativity
Abstract
In this chapter and the next we examine the specific models of physical phenomena that grew from the considerations discussed in Chap. 0.§3.
Christopher Terence John Dodson, Timothy Poston
XII. General Relativity
Abstract
Aristotle and Newton held that things fall because they are pulled to the earth; Nâgasana the sage and Einstein, that they fall because nothing stops them from falling. The difference is a profound one.
Christopher Terence John Dodson, Timothy Poston
Backmatter
Metadaten
Titel
Tensor Geometry
verfasst von
Christopher Terence John Dodson
Timothy Poston
Copyright-Jahr
1991
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-10514-2
Print ISBN
978-3-662-13117-6
DOI
https://doi.org/10.1007/978-3-642-10514-2