Sergei Yakovenko's blog: on Math and Teaching

Sunday, May 6, 2018

Lecture 7, May 6, 2018

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:05
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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).

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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
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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.

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
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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
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Welcome!

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.

Monday, January 29, 2018

Exam

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 5:05
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Exam

The exam is posted online on Jan 30, 2017, and must be submitted on March 16, 2018 (the first day of the new semester).

Its goals are, besides testing your newly acquired聽skills in the Analysis, to teach you a few extra things and check your ability聽for logical reasoning, not your proficiency in performing long computations.
If you find yourself mired in heavy computations, double check whether聽you understand the formulation of the problem correctly. Remember, small聽details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not聽forgetting explicit or implicit quantifiers. They really may change聽the meaning of what you write!

Problems are subdivided into items. The order of these items is by far non-random, you have to solve them from the first through the last, (solution of one item may be a building block for the next one). A complete solution of a problem is a proof of some important theorem in Analysis, so you will be discovering these results on your own. The Remarks will help you to place the freshly proved theorem on the general picture.

To get the maximal grade 100, it is not necessary to solve all problems. Problems are of varying length, variable complexity, various level of abstraction. No apriori points are assigned for solution of each problem, no summation at the end. You can get extra points for short and elegant argument or have some points removed for writing an obviously stupid things (honest errors will simply bring you zero points). You have all the time, try to solve as many problems as you can, we will appreciate and assess the results as objectively and honestly as possible.

You are assumed to work individually, which is, of course, impossible to verify, but please in any case avoid submitting isomorphic solutions: this is a bad taste for take-home exams.

You are absolutely free to write in English (easier for me) or Hebrew, submit handwritten pages or compuscripts, in a hard copy or by email (even scans will work). I we will encounter difficulties reading your submission, we’ll let you know.

If you believe you found an error or crucial omission in the formulation聽of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions)聽or cancelled (on my account). In case of any doubt don’t hesitate to leave your questions as talkbacks to this post, so that other people will be able to follow. Asking questions is never penalized!

Good luck to everybody!

Lectures 11-13 (Jan 16, 23, 30)

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 4:53

Infinite series

These three lectures were devoted to the general theory of numeric series and the issues of their convergence. Then we switched to studying power series (both formal and convergent). Finally, we used the convergent power series to move into the complex domain and discover fascinating phenomena there.

There two separate sets of lecture notes, on infinite series and operations on them and on functions of complex variable.

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