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manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics and augmented-reality given the need to associate pictures with coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
The study of manifolds requires working knowledge of calculus and topology.

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  2. D

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    • Forum: Plumbing Jobs | The Job-board
  3. S

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  4. A

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  5. M

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  6. N

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  7. P

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  8. R

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  9. phil hair

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  10. adam99

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  12. A

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  13. M

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  14. M

    Hello and a question.

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  15. S

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  16. J

    Part installed system

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  17. W

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  18. F

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  19. J

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  20. leesparkykent

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