Definition, Betydelse & Synonymer | Engelska ordet MANIFOLD

Definition av MANIFOLD

1. (matematik) mångfald

Exempel på hur man kan använda MANIFOLD i en mening

• All cotangent spaces at points on a connected manifold have the same dimension, equal to the dimension of the manifold.
• It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
• More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold.
• A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction).
• A metric tensor, in differential geometry, which allows defining lengths of curves, angles, and distances in a manifold.
• For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
• In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold.
• For example, in general relativity, the gravitational field is described through the metric tensor, which is a tensor field with one tensor at each point of the space-time manifold, and each belonging to the tensor product of the cotangent space at the point with itself.
• Specifically, they are a transcendental bijection of the spacetime continuum, an asymptotic projection of the Calabi–Yau manifold manifesting itself in anti-de Sitter space.
• An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold.
• In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1.
• The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.
• In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing geodesic curvature.
• In mathematics, a degenerate distribution (sometimes also Dirac distribution) is, according to some, a probability distribution in a space with support only on a manifold of lower dimension, and according to others a distribution with support only at a single point.
• Mixing cylinders have ground glass joints instead of a spout, so they can be closed with a stopper or connected directly with other elements of a manifold.
• This characterization of symmetry is useful, for example, in differential geometry, for each tangent space to a manifold may be endowed with an inner product, giving rise to what is called a Riemannian manifold.
• Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold.
• A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point.
• For smooth mappings h:M→TN from any smooth manifold M, the connector K:TTN→TN satisfies : ∇ h = K○Th:TM→TN where Th:TM→TTN is the differential of h.
• In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are.

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