# Sergei Yakovenko's blog: on Math and Teaching

## Continuity and limits

In these four lectures (sorry for the delay with posting the notes) we have introduced and discussed the notions of continuous functions. Contrary to the usual practice, we switch immediately to the case of functions of several variables, where pictures are much more illustrative.

We discuss the first topological notions: open/closed sets, accumulation/interior points, limits of functions as a way to extend functions continuously beyond the “natural” way of their definition by formulas.

Then we pass to more involved notions as compactness, connectivity (in two flavors) and finally end up by our first really nontrivial topological result, the fixed point theorem for 2-dimensional disk.

The (preliminary) lecture notes are available here: please note that there are over 30 problems approximately of the same sort that will appear on the exam. A more extended version will appear later, besides, you are always invited to recycle the lecture notes from the past years, available from this blog.

## Real numbers as solutions to infinite systems of equalities

In the past we already extended our number system by adding “missing” elements which are assumed to satisfy certain equations, based only on knowing what these equations are. It turns out that we may extend the set of rational numbers $\mathbb Q$ to a much larger set of real numbers $\mathbb R$ by adding solutions to (infinite numbers) of inequalities. As before, the properties of these new numbers could be derived only from the properties of inequalities between the rational numbers.

On one leg, the idea can be explained as follows. Since for any two rational numbers $r,s\in\mathbb Q$ one and only one relation out of three is possible, $r$ < $s$, $r=s$ or $r$ > $s$, we can uniquely define any, say, positive rational unknown number $x$ by looking at the two sets, $L=\{l\in\mathbb Q: 0\le l\le x\}$ and $R=\{r\in\mathbb Q: x\le r\}$. (You don’t have to be too smart at this moment: $x$ is the only element in the intersection $L\cap R$ 😉

However, sometimes the analogous construction leads to problems. For instance, if $L=\{l\in\mathbb Q: l\ge 0, l^2\le 2\}$ and $R=\{r\in\mathbb Q: r^2\ge 2\}$, then $L\cap R=\varnothing$, since the square root of two is not a rational number, but $L\cup R=\mathbb Q_+$, i.e., for any positive rational number we can say whether is smaller or larger the missing number $\sqrt 2$. This allows to derive all properties of $\sqrt 2$, including its approximation with any number of digits.

Proceeding this way, we introduce (positive) real numbers by indication, what is their relative position to all rational numbers. This allows to describe the real numbers completely.

The details can be found here.

## A didactic digression

Some of you complained about insufficient number of problems that are discussed during the tutorials. Everybody knows that problems and questions for self-control are the most important elements of study mathematics, especially in comparison with other disciplines. The rationale behind is the assumption that a student who understands the subject, should be able to answer these questions immediately or after some reflection. Composing such problems is an easy thing: you any mathematical argument you can stop for a second and ask yourself: “why I can do as explained?” or “under what conditions are my actions justified?”. In the lecture notes (see the link above) tens of such problems are explicitly formulated. Similar problems will await you on the exam.

However, remember one simple thing. If you already know how to solve a problem, this is not a problem but rather a job. Unless you solve these problems yourselves, there is no sense in memorizing their solutions: knowing solution of one such problem won’t help you with solving another problem unless you really understand what’s going on. There are no “typical problems”: each one of them is of its own sort, though, of course, some problems can be solved by similar methods.

A practical advice: you should not expect that all problems that appear on the exam will be discussed at length at the tutorials. There are no ready recipes to memorize. Only to understand honestly. Believe me, this is easier than memorize by heart endless formulas and algorithms.

## Numbers

The basic set theory allows us to construct a set $\mathbb N=\{|,||,|||,||||,|||||,\dots\}$ with a function “next”, denoted by $\mathrm{Succ}:\mathbb N\to\mathbb N\smallsetminus\{|\}$, which is bijective. This set describes the process of counting objects and is the most basic structure. Starting from a distinguished element denoted by 1, we construct an infinite number of elements $2=\mathrm{Succ}(1),\ 3=\mathrm{Succ}(2),\ 4=\mathrm{Succ}(3)$ etc. There are two axioms guaranteeing that the set $\mathbb N$ indeed coincides with what we call the set of natural numbers:

1. $\forall x\in\mathbb N\ \mathrm{Succ}(x)\ne 1$
2. Any element $x\in\mathbb N$ is obtained by the iteration of $\mathrm{Succ}$: $x=(\mathrm{Succ}\circ\cdots\circ\mathrm{Succ})(1)$.

Using this function and its partial inverse  one can define on $\mathbb N$ the order and the operations of addition (as repeated addition of 1 which is just evaluation of $\mathrm{Succ}$) and multiplication (repeated addition).

However, not all equations of the form $x+a=b$ or $x\cdot a=b$ are solvable. One can enlarge $\mathbb N$ by adding solutions of all such equations, obtaining the set of integer numbers $\mathbb Z$ which is a commutative group with respect to the operation of addition, and finally the set of rational numbers $\mathbb Q$ in which division is available by any nonzero number.

Division by zero is impossible: if we add “solution of the equation $0\cdot x=1$” as a new imaginary element, then we will not be able to do some arithmetic operations on it. Still, if we are ready to pay this price, then the rational numbers can be extended by a new element so that, say, the function $f(x)=1/x$ would be everywhere defined and continuous.

Details are available in the lecture notes here.

# שלום כיטה א!

The main feature that distinguishes the Calculus (or Mathematical Analysis) from other branches of mathematics is the repeated use of infinite constructions and processes. Without infinity even the simplest things, like the decimal representation of the simple fraction $\frac13=0.333333\dots$ becomes problematic.

Yet to deal with infinity and infinite constructions, we need to make precise our language, based on the notions of sets and functions (maps, applications, – all these words are synonymous).

Look at the first section of the lecture notes here.

You are most welcome to start discussions in the comments to this (or any other) post. Don’t be afraid of asking questions that may look stupid: this never harms! Write in any language (besides Hebrew/English, I hope that Ghadeer will take care of questions in Arabic, and I promise to deal with French/Spanish/Catalan/Italian/Russian/Ukrainian questions) 😉 Subscribe for updates on this site with your usual emails, to be independent from any dependence 😉

Looking forward for a mutually beneficial interaction in the new semester!

## Saturday, February 11, 2017

### Exam

Filed under: Calculus on manifolds course,lecture,problems & exercises — Sergei Yakovenko @ 5:22
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## Problems for the take-home exam

Here is the file with the problems for the exam.

The rules are simple:

1. The submission is due on the last day of the exam period (including the vacations) as per the FGS rules, that is, March 26, 2017.
2. English is preferred over Hebrew, typeset solutions to the handwritten ones, although no punishment for the deviant behavior will ensue. Hardcopy solutions should be put into my mail box in the Zyskind building,  otherwise feel free.
3. Nobody is perfect: if you believe you find an error and the problem as it is stated is wrong, don’t hesitate to write a talkback to this post. All bona fide errors will be corrected or the problem cancelled outright.
4. I tried to make the exam as instructive as the course was. Most of the problems are things that I planned to include, but didn’t have time to. To simplify your life, they were split into what I believe are simple steps. Don’t hesitate to consult textbooks, but let me see that you indeed read and digested them. The presumptively harder items are marked by the asterisk.
5. To get the perfect grade 100, you don’t have to submit solutions to all problems. The grade will be based on my purely subjective assessment of your exam and in any case will not be additive neither multiplicative.  Please be aware that writing patently stupid things may be more detrimental to the outcome than just skipping an item that you cannot cope with.
6. I hope to post on this site the aggregated and slightly polished lecture notes in hope they might help you.
7. I hope to be able to answer any questions you might have concerning the problems, better posted here than emailed to me. Moreover, I encourage open discussions here as long as they don’t result in posting complete solutions. Sometimes one stumbles over the most stupid things and needs to talk to other to overcome that. That’s fairly normal. To enter math formulas, you type the dollar sign $immediately followed by the word “latex”, and after the blank space type in your formula. Don’t forget to close with another$.
8. If you cannot meet the deadline for serious reasons, write me. Everything is negotiable.

Good luck and merry ט”ו בשבט!

## Calculus on complex manifolds

If $V$ is a complex vector space, then it is naturally also a real vector space (if you allow multiplication by complex numbers, then that by real numbers is automatically allowed). However, forgetting how to multiply by the imaginary unit results in the fact that the dimension $\dim_{\mathbb R}V$ of the space over the real numbers is two times higher. If we regret our decision to forget the complex multiplication, we still can restore it by introducing the $\mathbb R$-linear operator $J\colon V\to V$ such that $J^2=-E$, where $E$ is the identity operator.

An even-dimensional real vector space with such an operator is called an almost complex space, and it obviously can be made into a complex vector space (over $\mathbb C$). However, if we consider an even-dimensional manifold $M$ with the family of operators as above, it is somewhat less than a complex analytic manifold (a topological space equipped with an atlas of charts with biholomophic transition functions). For details, follow the lecture notes that will be available later.

## Symplectic manifolds

In parallel with the Riemannian manifolds equipped with a positive definite (symmetric) scalar product on each tangent space, it is interesting to consider manifolds equipped with an antisymmetric scalar product on each tangent space, i.e., with a differential 2-form $\omega\in\Omega^2(M)$. This form is called a symplectic structure, if $\mathrm d\omega=0$ and an additional nondegeneracy condition is met.

It turns out that this structure naturally arises on the cotangent bundle $M=T^*N$ of an arbitrary smooth manifold $N$. Moreover, this structure is intimately related with the mechanics of frictionless systems: the Hamiltonian differential equations can be naturally described by vector fields $X$ which satisfy the Hamiltonian condition $\mathrm i_X\omega=\mathrm d H$, where $H$ is a function (Hamiltonian, or full energy) on the symplectic manifold. Thus each Hamiltonian vector field is “encoded” by a single function, rather than by a tuple of functions. The commutator of Hamiltonian vector fields is again Hamiltonian: this is the invariant definition of the Poisson bracket.

There are two instant ramifications from this point. One can discuss integrability of the Hamiltonian vector fields. Another, less physically motivated direction is to study the symplectic geometry, first locally, then globally. It is a surprising twisted counterpart of the Riemannian geometry, which has no intrinsic curvature but nevertheless is very rich globally.

The lecture notes will be available later.

### Lecture 12 (Jan 23, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 4:51
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## Lie groups and Lie algebras

A Lie group is a smooth manifold with carries on it the structure of a group which is compatible with the smooth structure (i.e., the multiplication by an element of the group is a smooth self-map, necessarily a diffeomorphism, of the manifold).

This group structure means very high “homogeneity” of the manifold, in particular, existence of a flat connexion. On the other hand, there is a distinguished point on the manifold, corresponding to the group unit.

It turns out that the tangent space at the group unit is equipped with a natural operation, the antisymmetric bilinear bracket, closely related to the commutator of vector fields on the Lie group. This algebraic structure is called the Lie algebra, and it in a sense “encodes” the group structure.

The notes will be available later.

## Monday, January 30, 2017

### Lecture 11 (Jan 16, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 3:54
Tags: , ,

## Geodesics

Definitions of geodesic curves on a Riemmanian manifold. Differential equations of the second order. Local existence of solutions. Geodesic map. Geodesic spheres, orthogonality. Local minimality of geodesic curves. Metric and geodesic completeness of Riemannian manifolds.

Survey of adjacent areas. Behavior of the nearby geodesics. Jacobi field. On surfaces: conjugated points, the role of the Gauss curvature, global properties of manifolds with positive and negative curvature (comparison). Hyperbolic plane $\mathbb H$, geodesics on it. Realization of the hyperbolic geometry on a surface in $\mathbb R^3_{++-}$. Impossibility of global embedding of $\mathbb H$ into $\mathbb R^3_{+++}$.

## Introduction to the Riemannian geometry

1. Flat structure of the Euclidean space and coordinate-wise derivation of vector fields.
2. Axiomatic definition of the covariant derivative and its role in defining the parallel transport along curves on manifolds. Connexion.
3. Covariant derivative $\overline\nabla\text{ on }\mathbb R^n$ and its properties (symmetry, flatness, compatibility with the scalar product).
4. Smooth submanifolds of $\mathbb R^n$. The induced  Riemannian metric and connection. Gauss equation.
5. Weingarten operator on hypersurfaces and its properties. Gauss map.
6. Curvatures of normal 2-sections (the inverse radius of the osculating circles). Principal, Gauss and mean curvatures.
7. Curvature tensor: a miracle of a 2-nd order differential operator that turned out to be a tensor (“0-th order” differential operator).
8. Symmetries of the curvature and Ricci tensors.
9. Uniqueness of the symmetric connexion compatible with a Riemannian metric. Intrinsic nature of the Gauss curvature.

The lecture notes are available here.

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