Manifolds occur throughout mathematics -- and hence other sciences -- as spaces in which something interesting happens or spaces in which something interesting is to be found. Often the set of solutions to some problem forms a manifold, or the set of possible configurations of some system forms a manifold. By studying manifolds and by developing a set of tools with which to study particular manifolds, we can gain considerable insight into those problems where manifolds occur.

The key property of a manifold is that it locally looks like ordinary Euclidean space. Therefore, things that work on small patches of Euclidean spaces can often be made to work on manifolds. The most important of these things is calculus. Indeed, one can regard manifolds as the right places to do calculus and many of the questions involving manifolds have their origin in considering a particular theorem of calculus.

This course is designed to be an introduction to the theory of manifolds, looking at particular examples and developing simple techniques for studying them.

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