Tangent Space

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.