# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, November 15, 2016

### Lecture 2 (Nov. 14, 2016).

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 5:07
Tags: , ,

## Tangent vectors, vector fields, integration and derivations

Continued discussion of calculus in domains lf $\mathbb R^n$.

• Tangent vector: vector attached to a point, formally a pair $(a,v):\ a\in U\subseteq\mathbb R^n, \ v\in\mathbb R^n$. Tangent space $T_a U=\ \{a\}\times\mathbb R^n$.
• Differential of a smooth map $F: U\to V$ at a point $a\in U$: the linear map from $T_a U$ to $T_b V,\ b=F(a)$.
• Vector field: a smooth map $v(\cdot): a\mapsto v(a)\in T_a U$.  Vector fields as a module $\mathscr X(U)$ over $C^\infty(U)$.
• Special features of $\mathbb R^1\simeq\mathbb R_{\text{field}}$. Special role of functions as maps $f:\ U\to \mathbb R_{\text{field}}$ and curves as maps $\gamma: \mathbb R_{\text{field}}\to U$.
• Integral curves and derivations.
• Algebra of smooth functions $C^\infty(U)$. Contravariant functor $F \mapsto F^*$ which associates with each smooth map $F:U\to V$ a homomorphism of algebras $F^*:C^\infty(V)\to C^\infty(V)$. Composition of maps vs. composition of morphisms.
• Derivation: a $\mathbb R$-linear map $L:C^\infty(U)\to C^\infty(U)$ which satisfies the Leibniz rule $L(fg)=f\cdot Lg+g\cdot Lf$.
• Vector fields as derivations, $v\simeq L_v$. Action of diffeomorphisms on vector fields (push-forward $F_*$).
• Flow map of a vector field: a smooth map $F: \mathbb R\times U\to U$ (caveat: may be undefined for some combinations unless certain precautions are met) such that each curve
$\gamma_a=F|_{\mathbb R\times \{a\}}$ is an integral curve of $v$ at each point $a$. The “deterministic law” $F^t\circ F^s=F^{t+s}\ \forall t,s\in\mathbb R$.
•  One-parametric (commutative) group of self-homomorphisms $A^t=(F^t)^*: C^\infty(U)\to C^\infty(U)$. Consistency: $L=\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0}A^t=\lim_{t\to 0}\frac{A^t-\mathrm{id}}t$ is a derivation (satisfies the Leibniz rule). If $A^t=(F^t)^*$ is associated with the flow map of a vector field $v$, then $L=L_v$.

Update The corrected and amended notes for the first two lectures can be found here. This file replaces the previous version.

1. 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

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