# Sergei Yakovenko's blog: on Math and Teaching

## Vector fields on manifolds and what they are good for

Various ways to formally introduce tangent vectors to an abstract object that apriori sits nowhere (via charts, derivations, etc.) How the tangent spaces get their structure of vector spaces. Flows and related stuff.

Commutator of vector fields. Mystery of the depressed order.

Flows and the Lie derivative. Lie bracket. Commutator revisited.

[Not covered in the class]: Commuting vector fields, commuting flows, common integral surfaces, involutive distributions, Frobenius theorem*.

The last couple of lectures deviated slightly from the lecture notes from the past years. For your convenience, here is a more digestible (?) version.

*The topic is briefly covered in the revised lecture notes. Please let me know (in comments), if you are interested in more explanations. Otherwise chances are that I will include the detailed proof (split into doable steps) as a problem for the exam (in which case you will learn the subject much better 😉 ).

### Lecture 4 (April 15, 2018)

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:09
Tags: , ,

## Smooth manifolds: howto and whatfor

Topological spaces with an extra smooth structure. Examples: subsets of the Euclidean spaces, Lie groups, quotient spaces. Examples.

How to work with manifolds. Charts, atlases.

How to simplify your thoughts once you know what they are about. Structural algebra of smooth functions, various definitions of the vector fields, functoriality.

### Lecture 3 (April 8, 2018)

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:04
Tags: , ,

## Vector fields in open domains of $\mathbb R^n$

Ordinary differential equations, differential operators of the first order, local analysis and global consequences.

Create a free website or blog at WordPress.com.