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 | Reply

    • 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 | Reply

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