## 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.

Hi Segei,

I missed your class on Monday. Are there lecture notes for this lecture? If not, could you direct me to a book / chapter / other notes that cover the material of this lecture?

Comment by Renan — Wednesday, November 16, 2016 @ 2:57 |

Shalom Renan,

The notes written (and attached) to Lecture 1 were intended to cover the first two lectures. I am now amending and extending them to reflect better what I covered in the class: stay tuned, I hope to post the edited version here during the weekend.

Comment by Sergei Yakovenko — Thursday, November 17, 2016 @ 4:14 |