CapitalistImperialistPig

Enter the main characters in his drama: Differential Manifolds and Tensors We have seen that the possibility of defining continuous maps (or functions) is the key ingredient that makes topological spaces interesting. A differentiable manifold is a space on which differentiable functions can live. The physical space in which we live is thought to be such a space, and so are many other interesting mathematical objects, such as the space of solutions to differential equations. Fundamentally, a differentiable manifold is a space which looks locally like R^n. Each point in the manifold has an open neighborhood which has a continuous 1-1 map onto an open set of R^n for some n. 2.1 Definition of a Manifold The key point here is that for a general manifold M, no single map from M->R^n will do. In general, multiple overlapping maps (an Atlas ) will be needed, and for M to be a diffentiable manifold, those maps must overlap smoothly, permitting differentiable coordinate transformations ...

Sections 1.1 and 1.2 are devoted to introducing some ideas of topology needed for the definition of a manifold. R^n is the prototypical manifold, and Schutz sets about defining its open sets with a Euclidean Metric and some variations. Much or most of modern mathematics can trace its ancestry back to Decartes' marriage of geometry and algebra. The idea of identifying points in space with pairs (or triples, for 3-D) of numbers was the key element, and has a natural generalization to n-tuples of numbers representing points in n dimensional space. Topology represents space stripped down to an essence - that essence that preserves continuity. What do we mean by continuity anyway, and how do we represent it? One key attribute is that we expect that there should be points arbitrarily close to any other point. Topologists found that they could boil that essence down to the notion of a set and the behavior of its open subsets - but I won't spoil any punchlines here - if you want t...