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TheMathematicsLab
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Transition Maps
A map between two neighbouring local charts of a topological manifold. If it is a diffeomorphism, then the two charts are said to be smoothly compatible. We develop the transition maps between coordinate systems of finite-dimensional vector spaces and show that it is equivalent to the change of basis transformation from elementary linear algebra.
Topological Manifolds
A second-countable Hausdorff space that is locally homeomorphic to open subsets of Euclidean space. At each point in the manifold, a local homeomorphism exists to endow the region with a Euclidean coordinate system. Globally, manifolds may exhibit more complex topologies that make them incompatible with the usual tools of calculus that we otherwise use in real or complex spaces.
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