A vector field assigns an infinitesimal direction to every point. Its integral curves assemble into a local flow, converting differential equations into geometric transformations of the manifold.
Vector fields and derivations
Definition. A smooth vector field is a smooth section \(X:M\to TM\), \(\pi\circ X=\operatorname{id}_M\). It acts on functions by \[ Xf(p)=X_p(f). \] In coordinates, \(X=\sum_iX^i\partial_i\).
Theorem. Every derivation \(D:C^\infty(M)\to C^\infty(M)\) is induced by a unique smooth vector field.
Proof. Define \(X_p(f)=D(f)(p)\). The Leibniz rule makes \(X_p\) a tangent vector. In coordinates, \[ X=\sum_iD(x^i)\partial_i, \] whose coefficients are smooth. This field induces \(D\), and uniqueness is pointwise.
Integral curves and maximal flows
Definition. An integral curve of \(X\) through \(p\) is a curve \(\gamma\) satisfying \[ \dot\gamma(t)=X_{\gamma(t)},\qquad \gamma(0)=p. \] The maximal flow is the map \(\Phi:\mathcal D\subset\mathbb R\times M\to M\) with \(\Phi_t(p)=\gamma_p(t)\).
Theorem (flow theorem). Every smooth vector field has a unique maximal flow. Its domain is open, \(\Phi_0=\operatorname{id}\), and whenever both sides are defined, \[ \Phi_{t+s}=\Phi_t\circ\Phi_s. \] Each \(\Phi_t\) is a diffeomorphism with inverse \(\Phi_{-t}\).
Proof. In a chart the equation is a smooth Euclidean ODE, so local existence, uniqueness, and smooth dependence on initial conditions apply. Uniqueness lets overlapping local solutions patch into maximal ones and gives the group law. The group law also proves that \(\Phi_{-t}\) is the inverse of \(\Phi_t\).
A field is complete if \(\mathcal D=\mathbb R\times M\). Every smooth field on a compact manifold is complete, since a maximal integral curve cannot escape all compact subsets in finite time.
Lie brackets
Definition. The Lie bracket is the vector field \[ [X,Y]f=X(Yf)-Y(Xf). \] In coordinates, \[ [X,Y]^k=\sum_i\left(X^i\partial_iY^k-Y^i\partial_iX^k\right). \]
Proposition. The local flows of \(X\) and \(Y\) commute if and only if \([X,Y]=0\).
Infinitesimally, the commutator loop \(\Phi_{-t}^Y\Phi_{-t}^X\Phi_t^Y\Phi_t^X\) differs from the identity by \(t^2[X,Y]+o(t^2)\). Thus the bracket measures the failure of the two directions to integrate independently.
Example. On \(\mathbb R^2\), let \(X=\partial_x\) and \(Y=x\partial_y\). Then \([X,Y]=\partial_y\). Moving in \(x\), then in the \(x\)-dependent \(y\)-direction, produces a second-order displacement unavailable from either field alone at \(x=0\).