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%ما هو (من)٪ 1 - تعريف

GEOMETRIC STRUCTURE THAT GENERALIZES THE EUCLIDEAN SPACE

Affinely independent; Affine subspace; Affine independence; Affinely dependent; Affine dependence; Affine line; Linear manifold; Affine set; Affine point; Affine coordinate system; Affine Coordinates; Barycentric coordinates; Linear Manifold; Affine frame; Affine basis; Affine spaces; Affine coordinates; Point–vector distinction; Affine space (algebraic geometry); Draft:Affine space (algebraic geometry); Affine property; Point-vector distinction

$In <math>\mathbb{R}^3,</math> the upper plane (in blue)<math>P_2</math> is not a vector subspace, since <math>\mathbf{0} \notin P_2</math> and <math>\mathbf{a} + \mathbf{b} \notin P_2;</math> it is an ''affine subspace''. Its direction (the linear subspace associated with this affine subspace) is the lower (green) plane <math>P_1,</math> which is a vector subspace. Although <math>\mathbf{a}</math> and <math>\mathbf{b}</math> are in <math>P_2,</math> their difference is a ''displacement vector'', which does not belong to <math>P_2,</math> but belongs to vector space <math>P_1.</math>$

Piecewise linear manifold

TOPOLOGICAL MANIFOLD WITH A PIECEWISE LINEAR STRUCTURE ON IT.

Piecewise-linear manifold; PL manifold; PL structure; PL-manifold; Piecewise linear structure; Piecewise-linear structure; PL-structure; PL homeomorphism; Piecewise linear homeomorphism

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions.

https://en.wikipedia.org/wiki/Piecewise_linear_manifold

G2 manifold

SEVEN-DIMENSIONAL RIEMANNIAN MANIFOLD WITH HOLONOMY GROUP CONTAINED IN G2

Joyce manifold; G2-manifold

In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G_2 is one of the five exceptional simple Lie groups.

https://en.wikipedia.org/wiki/G2_manifold

Differentiable manifold

MANIFOLD UPON WHICH IT IS POSSIBLE TO PERFORM CALCULUS (ANY DIFFERENTIABLITY CLASS)

Differential manifold; Smooth manifold; Smooth manifolds; Differentiable manifolds; Manifold/rewrite/differentiable manifold; Differental manifold; Sheaf of smooth functions; Geometric structure; Ambient manifold; Non-smoothable manifold; Curved manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas).

https://en.wikipedia.org/wiki/Differentiable_manifold

ويكيبيديا

Affine space

In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.

Any vector space may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrast, always contain the origin of the vector space.

The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane.

https://en.wikipedia.org/wiki/Affine space