Tangent vectors, vector fields, integration and derivations
Continued discussion of calculus in domains lf .
- Tangent vector: vector attached to a point, formally a pair . Tangent space .
- Differential of a smooth map at a point : the linear map from to .
- Vector field: a smooth map . Vector fields as a module over .
- Special features of . Special role of functions as maps and curves as maps .
- Integral curves and derivations.
- Algebra of smooth functions . Contravariant functor which associates with each smooth map a homomorphism of algebras . Composition of maps vs. composition of morphisms.
- Derivation: a -linear map which satisfies the Leibniz rule .
- Vector fields as derivations, . Action of diffeomorphisms on vector fields (push-forward ).
- Flow map of a vector field: a smooth map (caveat: may be undefined for some combinations unless certain precautions are met) such that each curve
is an integral curve of at each point . The “deterministic law” .
- One-parametric (commutative) group of self-homomorphisms . Consistency: is a derivation (satisfies the Leibniz rule). If is associated with the flow map of a vector field , then .
Update The corrected and amended notes for the first two lectures can be found here. This file replaces the previous version.