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.
View More On Wikipedia.org
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.
View More On Wikipedia.org
Creating content since 2001. Untold Media.
Newest Plumbing Threads
-
Cistern filling one third of the way
- Started by dwaindigital
- Replies: 2
-
Boiler losing pressure only on central heating- Started by SnosRap75
- Replies: 2
-
tanking. Is it ok to just seal everything with plastic.
- Started by Staggersandfalls
- Replies: 3
-
Pop quiz, does anyone recognise this radiator valve?
- Started by dimuc
- Replies: 3
-
SUsing coil to clear pipe leading to gutter
- Started by Sonya K
- Replies: 1
-
Replacing outdoor P-Trap Gully with Bottle Gully
- Started by DIwhYdidIstartThis
- Replies: 0
-
-
-
-
-
Mixer mixer shower, compatible with tanked supply
- Started by Matt0029
- Replies: 2
-
-
No Hot water, Fixed and Broken Again Ideal Logic 24
- Started by Filthbags
- Replies: 2
-
-
What's happening to this radiator, paint flaking off?
- Started by mikmikmikmik2024
- Replies: 3
