In this post, we will see the book from the series Little Mathematics Library titled The Euler Characteristic by Yu. A. Shashkin.

About the book

This booklet gives proofs of Euler’s famous formula for convex polytopes and of its analogues for other figures (planes, spaces and polygons). The formulas bring the reader naturally to the notion of Euler’s characteristic. Two definitions of the notion are given and their equivalence is proved. The part played by the Euler characteristic in different geometrical problems, i.e. in the decomposition of planes and spaces, in calculating areas, in coverings of spheres, is discussed.

The book is intended for senior pupils, junior college and university students and all lovers of mathematics.

Translated from the Russian
by Vladimir Shokurov
First published 1989
Revised from the 1984 Ruasian edition

Many, many thanks to the Anonymous uploader who commented on the Mirtitles blog!
This is a cleaned/optimised version from the original scan.

You can get the book here and here

Versión en español aqui and aqui

This was the last remaining volume of the Little Mathematics Library, it is now complete!

Twitter: @MirTitles
Mastodon: @mirtitles@mastodon.world
Mastodon: @mirtitles@mastodon.social
Bluesky: mirtitles.bsky.social
GitLab: https://gitlab.com/mirtitles
Internet Archive: https://archive.org/details/mir-titles

Contents

Preface 6
1. Euler’s Formulas for a Straight Line and a Plane 8
2. What is the Euler Characteristic? 14
3. The Euler Characteristic for Polygons 21
4. The Euler Characteristic and the Sum of the Exterior Angles of a Polygon 31
5. Applying the Euler Characteristic to Calculation of Areas 36
6. Euler’s Formula for Space 42
7. Euler’s Formula for Convex Polytopes and Its Consequences 47
8. Axioms of the Euler Characteristic 55
9. Proof of the Existence of the Euler Characteristic 62
10. The Equivalence of the Two Definitions of the Euler Characteristic 68
11. Elementary Figures on the Sphere and Their Euler Characteristics 78
12. Further Applications of the Euler Characteristic 82
Solutions, Hints and Answers 89
References 96

Follow us on The Internet Archive: https://archive.org/details/@mirtitles

Follow Us On Twitter: https://twitter.com/MirTitles

Write to us: mirtitles@gmail.com

Fork us at GitLab: https://gitlab.com/mirtitles/

Add new entries to the detailed book catalog here.

Contents

Preface 6

1. Systems of Linear Equations and Their Solutions 7
2. Matrices and Their Elementary Transformations 11
3. A Method for Solving Systems of Linear Equations 22
4. The Rank of a Matrix 31
5. The Theorem on Principal Unknowns 39
6. Fundamental Systems of Solutions 48

Answers 56
Solutions 60