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Unformatted text preview: A very brief introduction to differentiable manifolds Tom Carter http://cogs.csustan.edu/~ tom/diff-manifolds Santa Fe Institute Complex Systems Summer School June, 2001
1 Our general topics: Why differentiable manifolds Topological spaces (ex) Examples of topological spaces (ex) Coordinate systesm and manifolds (ex) Manifolds (ex) References (ex): exercises. 2 Why differentiable manifolds Differentiable manifolds can generally be thought of as a generalization of Rn. They are mathematical objects equipped with smooth (local) coordinate systems. Much of physics can be thought of as having a natural home in differentiable manifolds. A particularly valuable aspect of differentiable manifolds is that unlike traditional flat (Euclidean) Rn, they can have (intrinsic) curvature. 3 Topological spaces We need a way to talk about "nearness" of points in a space, and continuity of functions. We can't (yet) talk about the "distance" between pairs of points or limits of sequences we will use a more abstract appro...
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