Tangent Plane

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and 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 manifold.
In differential geometry, one can attach to every point $x$ of a differentiable manifold a tangent space—a real vector space... plus