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cotangent

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The term "microlocal" implies localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point. en.wikipedia.org

The cotangent space will then consist of all the possible first-order behaviors of a function near x. en.wikipedia.org

The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms. en.wikipedia.org

The ratio of the lengths of the catheti defines the trigonometric functions tangent and cotangent of the angles in the triangle. en.wikipedia.org

See also tangent bundle and cotangent bundle. en.wikipedia.org

Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces. en.wikipedia.org

Likewise, the phase space can be defined more deeply using topological manifolds and the cotangent bundle. en.wikipedia.org

Cotangent complexes were originally defined in special cases by a number of authors. en.wikipedia.org

A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates. en.wikipedia.org

Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. en.wikipedia.org

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