Sergei Yakovenko's blog: on Math and Teaching

Friday, April 27, 2018

Lecture 3 (April 8, 2018)

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:04
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Vector fields in open domains of \mathbb R^n

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

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Sunday, March 25, 2018

Lecture 2. March 25, 2018

Differentiable maps

  • Definition of differentiability at a point. Maps f:U\to W between open subspaces of the Euclidean spaces U\subseteq \mathbb R^n,\ W\subseteq\mathbb R^m smooth on their domain.
  • Tangent spaces T_a U, tangent bundle TU=\bigcup_{a\in U}T_a U\simeq U\times\mathbb R^n.
  • Differential of a smooth map: \mathrm df:TU\to TW.
  • What is the derivative? (answer: exists only when n=m=1). Partial derivatives.
  • How do we define functions “having more than one derivative”?

Algebraic formalism:

  • Algebra C^\infty(U) of functions infinitely smooth in a domain U\subseteq\mathbb R^n
  • Pullback morphism of algebras f^*:C^\infty(W)\to C^\infty(U).

Vector fields: smooth maps v:U\to TU, such that v(a)\in T_a U.
Lie (directional, flow) derivations L_v:C^\infty(U)\to C^\infty(U). The Leibniz rule (algebra) and its meaning (“Any Leibniz linear map of C^\infty(U) to itself is a Lie derivative along some vector field).
Commutator of two vector fields (to be discussed more in the future).
Push-forward of vector fields by smooth invertible maps.

Monday, December 12, 2016

Lecture 6, December 12, 2016

Exterior derivation

The differential \mathrm df of a smooth function f is in a sense container which conceals all directional derivatives L_Xf=\left\langle\mathrm df,X\right\rangle along all directions, and dependence on X is linear.

If we consider the directional Lie derivative L_X\omega for a form \omega\in\Omega^k(M) of degree k\ge 1, then simple computations show that L_{fX}\omega is no longer equal to f\cdot L_X\omega. However, one can “correct” the Lie derivative in such a way that the result will depend on X linearly. For instance, if \omega\in\Omega^1(M) and X is a vector field, we can define the form \eta_X\in\Omega^1(M) by the identity \eta_X=L_X\omega-\mathrm d\left\langle\omega,X\right\rangle and show that the 2-form \eta(X,Y)=\left\langle\eta_X,Y\right\rangle is indeed bilinear antisymmetric.

The 2-form \eta is called the exterior derivative of \omega and denoted \mathrm d\omega\in\Omega^2(M). The correspondence \mathrm d\colon\Omega^1(M)\to\Omega^2(M) is an \mathbb R-linear operator which satisfies the Leibniz rule \mathrm d(f\omega)=f\,\mathrm d\omega+(\mathrm df)\land \omega and \mathrm d^2 f=0 for any function f\in\Omega^0(M).

It turns out that this exterior derivation can be extended to all k-forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra \Omega^\bullet(M)=\bigoplus_{k=0}^n\Omega^k(M).

The lecture notes are available here.

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