Editing Tangent bundle (section)

==Higher-order tangent bundles==
Since the tangent bundle <math>TM</math> is itself a smooth manifold, the [[double tangent bundle|second-order tangent bundle]] can be defined via repeated application of the tangent bundle construction:

:<math>T^2 M = T(TM).\,</math><!-- "\," improves the display of this formula. Do not delete.-->

In general, the <math>k</math>th order tangent bundle <math>T^k M</math> can be defined recursively as <math>T\left(T^{k-1}M\right)</math>.

A smooth map <math> f: M \rightarrow N</math> has an induced derivative, for which the tangent bundle is the appropriate domain and range <math>Df : TM \rightarrow TN</math>.  Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives <math>D^k f : T^k M \to T^k N</math>.

A distinct but related construction are the [[jet bundle]]s on a manifold, which are bundles consisting of [[jet (mathematics)|jets]].