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

Let x be a point in an n-dimensional compact manifold M, and attach at x a copy of R^n tangential to M. The resulting structure is called the tangent space of M at x and is denoted T_x M. If γ is a smooth curve passing through x, then the derivative of γ at x is a vector in T_x M.

chart tangent space | submanifold tangent space | tangent | tangent bundle | tangent plane | tangent vector

graduate school level