Manifold

Tangent space

Associated with every point on a differentiable manifold is a tangent space and its dual, the cotangent space.

Related Topics:
Tangent space - Dual - Cotangent space

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The former consists of the possible directional derivatives, and the latter of the differentials, which can be thought of as infinitesimal elements of the manifold.

Related Topics:
Directional derivative - Infinitesimal

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

These spaces always have the same dimension n as the manifold does.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle, whose dimension is 2n.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~