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Examples oftangent space
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In this bifurcation the geometrical splitting of thetangentspaceis preserved but the expansion in the unstable direction degenerates near a periodic orbit.
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One piece of useful notation concerns the natural basis for atangentspaceinduced by a set of coordinates.
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In the geometric way of looking at things, the kinetic energy metric is a smooth assignment of an inner product to eachtangentspaceof the configuration space.
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Notice that the identification process allows one to define atangentspaceat each point of the local model, including on the singular locus.
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If hyperbolic currently means the existence of some splitting of thetangentspacein one expanding direction and one contracting direction, the notion of 'expansion' is not clear.
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Of course, in a manifold, the notion of a derivative being diagonal does not make sense, since it maps onetangentspaceinto a completely different one.
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In this description the linearly independent vectors of thetangentspaceof the groups near the vicinity of the unity element (the so-called generators) play a key role.
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In this case, it is not unusual to call the elements of thetangentspacethe "generators" of the manifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Once the set of points generates thetangentspacethe tangent algebra and its pseudoscalar follow immediately.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The elements of thetangentspaceare called tangent vectors at "x".
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This example is from Wikipedia and may be reused under a CC BY-SA license.More strictly this defines an affinetangentspace, distinct from the space of tangent vectors described by modern terminology.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The bundle of differential forms, at each point, consists of all totally antisymmetric multilinear maps on thetangentspaceat that point.
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This example is from Wikipedia and may be reused under a CC BY-SA license.At each point in, thetangentspaceto can be considered as a subspace of thetangentspaceof, which is just.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A smooth path in a smooth manifold has (at every point) the tangent vector, belonging to thetangentspace(attached to this point).
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This example is from Wikipedia and may be reused under a CC BY-SA license.Typically, the cotangent space is defined as the dual space of thetangentspaceat "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.Likewise, cotangent space is a contravariant functor, essentially the composition of thetangentspacewith the dual space above.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The map assigning to itstangentspacedefines a map from to.
FromThis example is from Wikipedia and may be reused under a CC BY-SA license.
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This example is from Wikipedia and may be reused under a CC BY-SA license.For example, the blowup of a point in a plane replaces the point with the projectivizedtangentspaceat that point.
FromThis example is from Wikipedia and may be reused under a CC BY-SA license.
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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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