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Examples ofcotangent space
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Typically, thecotangentspaceis defined as the dual space of the tangent space at "x", although there are more direct definitions (see below).
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This example is from Wikipedia and may be reused under a CC BY-SA license.For mechanical systems, the phase space usually consists of all possible values of position and momentum variables (i.e. thecotangentspaceof configuration space).
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This example is from Wikipedia and may be reused under a CC BY-SA license.Likewise,cotangentspaceis a contravariant functor, essentially the composition of the tangent space with the dual space above.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecotangentspacewill then consist of all the possible first-order behaviors of a function near x.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The tangent space is, the dual vector space to thecotangentspace.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The tangent space and thecotangentspaceat a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecotangentspaceat a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The term "microlocal" implies localisation not only with respect to location in the space, but also with respect tocotangentspacedirections at a given point.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
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