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In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. 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 tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of is twice the dimension of . Each tangent space of an n-dimensional manifold is an n-dimensional vector space
The tangent space has an interpretation in terms of K[t]/(t 2), the dual numbers for K; in the parlance of schemes, morphisms from Spec K[t]/(t 2) to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x. [3] Therefore, one also talks about tangent vectors. See also: tangent space to a ...
Smoothness of X means that the dimension of the Zariski tangent space is equal to the dimension of X near each point; at a singular point, the Zariski tangent space would be bigger. More generally, a scheme X over a field k is smooth over k if each point of X has an open neighborhood which is a smooth affine scheme of some dimension over k.
An atlas for a topological space is an indexed family {(,):} of charts on which covers (that is, =). If for some fixed n , the image of each chart is an open subset of n -dimensional Euclidean space , then M {\displaystyle M} is said to be an n -dimensional manifold .
For a set of (non-singular) coordinates x k local to the point, the coordinate derivatives = define a holonomic basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the tangent bundle, whose dimension is 2n. The tangent bundle is where tangent vectors lie, and is itself a differentiable ...
for tangent vectors v, w (the inner product makes sense because dn(v) and w both lie in E 3). [c] The right hand side is symmetric in v and w, so the shape operator is self-adjoint on the tangent space. The eigenvalues of S x are just the principal curvatures k 1 and k 2 at x.
The torus of dimension is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2 , {\displaystyle n=2,} and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point.