Definition:Tangent Vector/Definition 1
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Definition
Let $M$ be a smooth manifold.
Let $m \in M$ be a point.
Let $V$ be an open neighborhood of $m$.
Let $\map {C^\infty} {V, \R}$ be defined as the set of all smooth mappings $f: V \to \R$.
A tangent vector $X_m$ on $M$ at $m$ is a linear transformation:
- $X_m: \map {C^\infty} {V, \R} \to \R$
which satisfies the Leibniz law:
- $\ds \map {X_m} {f g} = \map {X_m} f \map g m + \map f m \map {X_m} g$
Also known as
A tangent vector is also known as a derivation.
Also see
- Results about tangent vectors can be found here.
Sources
- 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 3$: Tangent Vectors. Tangent Vectors
- 2013: Gerd Rudolph and Matthias Schmidt: Differential Geometry and Mathematical Physics: $\S 1.4$: Tangent Space