information:classification, linguistic:paraphrasing
Vector fields on manifolds

Given a differentiable manifold M {\displaystyle M}, a vector field on M {\displaystyle M} is an assignment of a tangent vector to each point in M {\displaystyle M}. More precisely, a vector field F {\displaystyle F} is a mapping from M {\displaystyle M} into the tangent bundle T M {\displaystyle TM} so that p ∘ F {\displaystyle p\circ F} is the identity mappingwhere p {\displaystyle p} denotes the projection from T M {\displaystyle TM} to M {\displaystyle M}. In other words, a vector field is a section of the tangent bundle.

An alternative definition: A smooth vector field X {\displaystyle X} on a manifold M {\displaystyle M} is a linear map X: C ∞ (M) → C ∞ (M) {\displaystyle X:C^{\infty }(M)\to C^{\infty }(M)} such that X {\displaystyle X} is a derivation: X (f g) = f X (g) + X (f) g {\displaystyle X(fg)=fX(g)+X(f)g} for all f, g ∈ C ∞ (M) {\displaystyle f,g\in C^{\infty }(M)}.

If the manifold M {\displaystyle M} is smooth or analytic —that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold M {\displaystyle M} is often denoted by Γ (T M) {\displaystyle \Gamma (TM)} or C ∞ (M, T M) {\displaystyle C^{\infty }(M,TM)} (especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by X (M) {\textstyle {\mathfrak {X}}(M)} (a fraktur "X").