tangentummet

Tangentummet, in the language of differential geometry, refers to the tangent space TpM of a differentiable manifold M at a point p ∈ M. This real vector space encodes the possible directions in which one can move from p along curves on M, capturing the first-order behavior of M near p.

One common definition describes tangentummet as the set of equivalence classes of smooth curves γ:(-ε,ε)→M with γ(0)=p, where two curves are equivalent if their velocity at t=0 agrees. An equivalent and often more practical definition identifies TpM with the space of derivations at p, i.e., linear maps v: C∞(M)→R satisfying Leibniz’s rule v(fg)=f(p)v(g)+g(p)v(f). A third view uses coordinate charts: if (U,φ) is a chart with p∈U and φ(p)=x, the tangentum is spanned by the coordinate vectors ∂/∂x^i|_p, and any v∈TpM has components v^i that describe its action on smooth functions via velocity in the chart coordinates.

The collection of all tangent spaces forms the tangent bundle TM, with a natural projection π: TM→M sending each tangent vector to its base point. For a smooth map f:M→N, the differential df_p: TpM→Tf(p)N is the tangent map (pushforward), providing a functorial relationship d(g∘f)=dg_f∘df.

Examples: TpR^n ≅ R^n for any p, and for a smooth surface S⊂R^3, TpS is the 2-dimensional plane tangent to S at p. Tangentummet plays a central role in theories of vector fields, connections, and differential forms, as a foundation for describing first-order geometry on manifolds. See also cotangentummet, tangent bundle, and differential forms.