A tangent vector to a point P on a smooth curve is any vector parallel to the tangent line. If r(s) is a parameterization of the curve by arc length, then t = r´(s) is a unit tangent vector (see also Serret-Frenet formulae). More generally, a tangent vector to a point P in a smooth manifold X is a tangent vector to a smooth curve in the manifold passing through P.

The tangent space at P is the vector space whose elements are the tangent vectors at P. So, if r(u,v) is a parameterization of a surface about P = r(u₀,v₀), then the tangent space is spanned by ∂r/∂u(u₀,v₀) and ∂r/∂v(u₀,v₀). Note that the tangent space is parallel to the tangent plane but passes through the origin. The tangent space at P is denoted T_P(X).