Tangent Space

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A tangent space is a vector space of all possible tangent vectors to a point on a manifold.

Tangent space is a graduate-level concept that would be first encountered in a topology course.


Jacobian: The Jacobian of a function consists of its partial derivatives arranged in matrix form and arises when performing a change of variables in multivariable calculus.
Manifold: A manifold is a topological space that is locally Euclidean, i.e., around every point, there is a neighborhood that is topologically the same as an open unit ball in some dimension.
Tangent Vector: A tangent vector is a vector pointing in the direction of the tangent line to the graph of a function.
Topology: (1) As a branch of mathematics, topology is the mathematical study of object's properties that are preserved through deformations, twistings, and stretchings. (2) As a set, a topology is a set along with a collection of subsets that satisfy several defining properties.
Vector Space: A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space.

Classroom Articles on Topology (Up to Graduate Level)

  • Closed Set
  • Möbius Strip
  • Differential Topology
  • Neighborhood
  • Dimension
  • Open Set
  • Homeomorphism
  • Point-Set Topology
  • Homology
  • Projective Plane
  • Homotopy
  • Projective Space
  • Knot
  • Subspace
  • Link
  • Topological Space
  • Metric
  • Torus
  • Metric Space
  • Vector Bundle