Sergei Yakovenko's blog: on Math and Teaching

Sunday, September 2, 2018

What is logarithm?

Filed under: lecture — Sergei Yakovenko @ 3:31
Tags: ,

The way logarithms, exponential and trigonometric functions are defined in the high school, is concealing more than explaining. In these notes, delivered in the framework of the annual Ulpana de Shalit, I try to explain how these functions (and their complex and matrix generalizations) naturally appear as solutions of ordinary differential equations.


Friday, July 6, 2018

Calculus on manifolds 2017/8 – Exam

Filed under: Calculus on manifolds course,problems & exercises — Sergei Yakovenko @ 5:12

Finally the summer is here, and with it the exams.

The revised lecture notes are available here. The problems for the exam are at the end of the notes.

The rules of the game are extremely simple.

    1. The grading is by the pass/fail scheme.
    2. There is no predefined minimum of problems/items to get “pass”. Everything is decided in a purely subjective matter. Please avoid writing nonsense just for the sake of writing something: if accumulated beyond certain critical mass, it may result in the unlikely “fail”.
    3. All textbooks are at your disposition: actually, you are encouraged to consult as many of them as you feel like it. Solutions of the form “this is theorem 123 from [XXX]” are accepted, provided that they indeed are coherent (e.g., not based on a different set of definitions).
    4. You are welcome to talk to each other during preparation, but please spare me from checking identical submissions. The art of writing math texts is in itself important and you can learn it only by trial and error/corrections.
    5. The deadline for submission is August 3, 2018 as set by the Feinberg Graduate school. I can cover you for some extra time in case you have unexpected circumstances, but please don’t try to stretch the deadline beyond reasonable limits.
    6. If you find an error, invalidating one of the exam problems (errare humanum est), please post a comment to this post, and I will try to correct the formulation (or cancel the problem outright if it is beyond repair, – things happened…)
    7. All questions concerning the exam/notes can be posted here in the comments or (if you prefer) in an email to聽聽(that’s me, in case you didn’t notice). I am essentially around here until the deadline, so if you prefer a personal communication, you are welcome to lay an ambush when I am in my office (Zyskind 150, usually every day between 15:30 and 18:00). But in any case, I’d prefer to be notified about request for a meeting in advance.
    8. The choice of the language for submission is yours. My preference is the pidgin English typeset in LaTeX, but this is by no means mandatory. On the other hand, if you plan to write in Mandarin or Basque, I’ll have to disappoint you. Please double check if your the language of your choice is covered.
    9. Problems marked with asterisks are considerably more difficult then the rest. If you love challenges, this is for you to try, but don’t get frustrated if you fail. If not, just register them as subtle math results which are so close to a general math course.

Good luck, and … and good luck!


Sunday, May 6, 2018

Lecture 7, May 6, 2018

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:05
Tags: , ,

Integration of differential 1-forms. Differential k-forms and their integration

Integration of a 1-form over a smooth curve: use pull-back or approximate by the Riemann sums? Independence of the parameterization.

\displaystyle \int_\gamma\mathrm df=f(\text{end})-f(\text{start}).

Integral of y\,\mathrm dx along a smooth closed curve: area bounded by the curve.

Multilinear forms on a linear n-space. Tensor product.

Symmetric and antisymmetric multilinear forms. Symmetrization and alternation. Determinant. Wedge product. Algebraic properties of the wedge product, action by adjoint operators.

Differential k-forms and their integration along smooth k-dimensional “pieces” (images of a cube).

Lecture 6, Apr 29, 2018

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 3:55
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Differential 1-forms

Vectors vs. covectors, covector field = differential 1-form. Functoriality: how to pull them back by smooth maps. Examples: differentials of smooth functions. Module of differential 1-forms over the structure algebra of smooth functions. Lie derivation along a flow of vector field.

Friday, April 27, 2018

Lecture 5 (April 22, 2018)

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 馃槈 ).

Lecture 4 (April 15, 2018)

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

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.

Lecture 1, March 18, 2018

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 5:13

Crash course on linear algebra

  • Fields: from natural numbers to \mathbb {Q,R,C}
  • Linear spaces: shake but not stir add/subtract but do not multiply except by numbers.
  • Linear maps: respect the linear structure.
  • One-dimensional case: the protean duality (either a field with a distinguished element 1, the multiplicative neutral element, or a 1-dimensional vector space over itself without a distinguished basis vector).
  • Dual space: linear maps \xi\colon V\to\mathbb R themselves form a linear space, which in the finite-dimensional case has the same dimension as V.
  • General linear transformations: the group that naturally acts on everything linear.

Analysis on Manifolds – old-new course announcement

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 5:02


Yielding to the popular demand, the course Analysis on Manifolds will be repeated in the spring semester of this year. It will be largely the same course as was taught the past year, although some changes will be inevitable. You are welcome to bookmark the web page (bear in mind that this is a weblog, so the most recent entries will appear first).

The set of lecture notes from the past year is available here, so that you may have an idea of what expects you. I will edit these notes so that towards the end of the semester you will have (hopefully) a better set.

I will post (mainly for my own records) the brief syllabus of things covered in each lecture. This might be helpful to those who missed a class or two and want to catch up.

Every year I urge students who take this course to profit from the interactive nature of the blog: asking questions (even “stupid” questions, though there is no such thing!) stimulates the digestive tract and helps even those who keep silence.

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