Friday, October 4, 2013

Scalars, Vectors, and Tensors -- Reloaded

In 300 B.C., Euclid created (or some would say documented) Geometry as a deductive, axiomatic system. In 1637 René Descartes introduced the coordinate system in Euclidean Geometry, combining algebra and geometry into a single mathematical structure. A logical procession of this line of thinking ultimately led to the development of the Tensor.

One can say that tensors came of age with the appearance of the remarkable paper by the famous mathematicians Ricci and Levi-Civita called Methods de Calcul diferential absolu et leurs applications, Mathematische Annalen, 1901. But it is primarily due to Einstein's use of tensors in his General Theory of Relativity that was responsible for the sudden emergence of the tensor calculus as a popular field of mathematical activity.

Tensor calculus is concerned with the study of abstract objects, called tensors, whose properties are independent of the reference frames used to describe the objects. A tensor is represented in a particular reference frame by a set of functions, termed its components, just as a vector is determined in a given reference frame by a set of components and if a new coordinate system is introduced, the same tensor is determined by a new set of components and the new components are related to the old ones, in a different way known as the covariant or covariant way.

Since tensor analysis deals with entities and properties that are independent of the choice of reference frames it forms as an ideal tool for the study of natural laws. In particular, Einstein found it an excellent tool for the presentation of his General Theory of Relativity, and he spent a couple of years in correspondence with Levi-Civita to get the math right. (We all need a little help from our friends.) As a result tensor calculus came into general prominence and in now invaluable in its applications to most branches of Theoretical Physics, it is also indispensable in the Differential Geometry of curves and surfaces in Euclidean space.

Tensor is a generalization of the term "vector" and tensor calculus is a generalization of vector calculus.

Reference frames are important in Newtonian physics as well as Special Relativity, but it is the distortion and curving of space that Einstein's General Theory predicts that make tensors such important tools.

In mathematics, a manifold is a topological space that near each point resembles Euclidean space. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighboring maps (in the context of manifolds they are called charts), the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity.

As a kid interested in cars and souped up cars called "hot rods," I find a good example of a manifold in the metal parts of an engine that share the term. Both the intake manifold and the exhaust manifold are good examples of the twists and turns that are part of the geometric representation of mathematical manifolds.

The notion of a fiber bundle first arose out of questions posed in the 1930s on the topology and geometry of manifolds. By the year 1950 the definition of fiber bundle had been clearly formulated, the homotopy classification of fiber bundles achieved, and the theory of characteristic classes of fiber bundles developed by several mathematicians, Chern, Pontrjagin, STiefel, and Whitney. Steenrod's book, which appeared in 1950, gave a coherent treatment of the subject up to that time.

Around 1955 Milnor gave a construction of a universal fiber bundle for any topological group. At that same time, Hirzebruch clarified the notion of characteristic class and used it to prove a general Riemann-Roch theorem for algebraic varieties. This work led to methods for mappings between various manifolds and ultimately led to a mathematics course that combined Fiber Bundles and Cobordism with topics of principal bundles, vector bundles, classifying space, connections on bundles, curvature, topology of gauge groups and gauge equivalence classes of connections, characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators, spectral sequences of Atiyah-Hirzebruch, Serre, and Adams, Cobordism theory, Pontryagin-Thom theorem, and calculation of unorientated and complex Cobordism. I am now the freshest graduate of that class, and ready to map the reference frames until my slide-rule melts from the friction.

Now I'm ready to tackle that General Theory. I don't have Levi-Cevita to help me, but I do have my trusty Dale Husemoller "Fibre Bundles" text, even if the spelling is British. Good old Springer-Verlag, New York, Heidelberg, and Berlin. Das ist gut.

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