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# Basic Analysis: Introduction to Real Analysis

**By: Jiří Lebl **(website #1 (personal), website #2 (work: OSU), email: jiri.lebl@gmail.com)

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[Download the book (volume I) as PDF] [volume II as PDF]

[Buy paperback (volume I) on Amazon] [volume II on Amazon]

This free online textbook (OER more formally) is a course in undergraduate real analysis (somewhere it is called "advanced calculus"). The book is meant both for a basic course for students who do not necessarily wish to go to graduate school, but also as a more advanced course that also covers topics such as metric spaces and should prepare students for graduate study. A prerequisite for the course is a basic proof course. An advanced course could be two semesters long with some of the second-semester topics such as multivariable differential calculus, path integrals, and the multivariable integral using the second volume. There are more topics than can be covered in two semesters, and it can also be reading for beginning graduate students to refresh their analysis or fill in some of the holes.

This book started its life as my lecture notes for Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009. It was later enhanced to teach the Math 521/522 sequence at University of Wisconsin-Madison (UW-Madison) and the Math 4143/4153 sequence at Oklahoma State University (OSU).

The book (volume I) starts with analysis on the real line, going through sequences, series, and then into continuity, the derivative, and the Riemann integral using the Darboux approach. There are plenty of available detours along the way, or we can power through towards the metric spaces in chapter 7. The philosophy is that metric spaces are absorbed much better by the students after they have gotten comfortable with basic analysis techniques in the very concrete setting of the real line. As a bonus, the book can be used both by a slower-paced, less abstract course, and a faster-paced more abstract course for future graduate students. The slower course never reaches metric spaces. A nice capstone theorem for such a course is the Picard theorem on existence and uniqueness of ordinary differential equations, a proof which brings together everything one has learned in the course. A faster-paced course would generally reach metric spaces, and as a reward such students can see a streamlined (but more abstract) proof of Picard.

Volume II continues into multivariable analysis. Starting with differential calculus, including inverse and implicit function theorems, continuing with differentiation under the integral and path integrals, which are often not covered in a course like this, and multivariable Riemann integral. Finally, there is also a chapter on power series, Arzelà-Ascoli, Stone-Weierstrass, and Fourier series. Together the two volumes provide enough material for several different types of year-long sequences. A student who absorbs the first volume and the first three chapters of volume II should be more than prepared for graduate real and complex analysis courses.

I have tried (especially in recent editions) to add many diagrams and graphs to graphically illustrate the proofs and make them more accessible. Usually, these are precise and more in-depth versions of the drawings I attempt on the board in class. Together, the two volumes have over a hundred figures.

The aim is to provide a low cost, redistributable, not overly long, high-quality textbook that students will actually keep rather than selling back after the semester is over. Even if the students throw it out, they can always look it up on the net again. You are free to have a local bookstore or copy store make and sell copies for your students. See below about the license.

One reason for making the book freely available is to allow modification and customization for a specific purpose if necessary (as the University of Pittsburgh has done for example). If you do modify this book, make sure to mark them prominently as such to avoid confusion. This aspect is also important for the longevity of the book. The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks.

Furthermore, errata are fixed promptly, meaning that if you teach the same class next term, all errata that are spotted are most likely already fixed. No need to wait several years for a new edition. Every once in a while I make some major addition and a new major version (edition), and then in between as errata are fixed I make minor version updates (like a corrected printing) usually once or twice a year, depending on the errata discovered. Exercise, chapter, and section numbers are preserved as much as humanly possible. What's added is added at the end with new numbers, so the book is generally compatible even if students (or the instructor) have an older printed copy. The minor updates are totally interchangeable and have very minimal changes, essentially nothing new.

MAA published a review of the book (they looked at the December 2012 edition of Volume I, there was only the first volume then).

## Table of contents:

Introduction

1. Real Numbers

2. Sequences and Series

3. Continuous Functions

4. The Derivative

5. The Riemann Integral

6. Sequences of Functions

7. Metric Spaces

Volume II:

8. Several Variables and Partial Derivatives

9. One Dimensional Integrals in Several Variables

10. Multivariable Integral

11. Functions as Limits

There are 528 exercises and 65 figures in Volume I (version 5.3, that is, June 10th 2020 edition).

There are 263 exercises and 43 figures in Volume II (version 2.3, that is, June 10th 2020 edition).

Please let me know at jiri.lebl@gmail.com if you find any typos or have corrections, extra exercises or material, or any other comments.

There is no solutions manual for the exercises. This situation is intentional. There is an unfortunately large number of problems with solutions out there already. Part of learning how to do proofs is to learn how to recognize your proof is correct. Looking at someone else's proof is a far less effective way of checking your proof than actually checking your proof. It is like going the gym and watching other people work out. The exercises in the book are meant to be a gym for the mind. If you are unsure about the correctness of a solution, then you do not yet have a solution. Furthermore, the best solution for the student is the one that the student comes up with on their own, not necessarily the one that the professor or the book author comes up with.

## Adoption:

Do let me know (jiri.lebl@gmail.com) if you use the book for teaching a course! The book was used, or is being used, as the primary textbook at (other than my courses at UIUC, UCSD, UW-Madison, and OSU) University of California at Berkeley, University of Pittsburgh, Vancouver Island University, Western Illinois University, Medgar Evers College, San Diego State University, University of Toledo, Oregon Institute of Technology, Iowa State University, California State University Dominguez Hills, St. John's University of Tanzania, Mary Baldwin College, Ateneo De Manila University, University of New Brunswick Saint John, and many others. See below for a more complete list.

The book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

See a **list of classroom adoptions** for more details.

## Download:

Download the volume I of the book as PDF

(Version 5.3, June 10th, 2020, 282 pages, 1.8 MB download)

Download the volume II of the book as PDF

(Version 2.3, June 10th, 2020, 195 pages, 1.4 MB download)

Check for any errata (volume I) (volume II) in the current version.

Look at the change log (volume I) (volume II) to see what changed in the newest version.

I started numbering things with version numbers starting at 4.0 for volume I, and version 1.0 for volume II. The first number is the major number and it really means "edition" and will be raised when substantial changes are made. The second number is raised for corrections only.

## Buy paperback:

I get a bit of money when you buy these (depending on where exactly they are bought). Probably enough to buy me a coffee (as long as it is not a fancy coffee), so by buying a copy you will support this project. You will also save your toner cartridge.

Lulu always has the most up to date version more quickly than amazon, the difference is usually in terms of days or weeks. The paperback copy is on Crown Quatro size (7.44x9.68 inch), and the two versions of it (amazon and lulu) are essentially identical except for cover art (there are those who like the blue). I tested both and they both print quite well, so the quality is approximately the same, and I have seen some of them take quite a bit of beating by students.

Lulu also allows me to make a larger (US letter size) coil bound version which I prefer to get when teaching, as it can easily be opened and kept on a certain page. It may be easier to read, and take notes in as it has larger font and wider margins, though a little less portable. It's only a few dollars more.

### Volume I:

Or buy the larger coil-bound copy at lulu.com for $15.60.

This copy is the version 5.3 (June 10th, 2020) revision of volume I.

No ISBN for the lulu version.

This copy is the version 5.3 (June 10th, 2020) revision of volume I.

ISBN-13: 978-1718862401

ISBN-10: 1718862407

### Volume II:

Or buy the larger coil-bound copy at lulu.com for $13.00.

This copy is the version 2.3 (June 10th, 2020) revision of volume II.

No ISBN for the lulu version.

This copy is the version 2.3 (June 10th, 2020) revision of volume II.

ISBN-13: 978-1718865488

ISBN-10: 1718865481

## Web version:

Try a beta of a web version of both volumes put together. This still requires some testing. Do send me any errata regarding this conversion.

## Source:

The source is hosted on **GitHub**: https://github.com/jirilebl/ra (both volumes).

You can get an archive of the source of the released version on github, look under https://github.com/jirilebl/ra/releases, though if you plan to work with it, maybe best to look at just the latest working version as that might have errata fixes or new additions. On the other hand, this might be a work in progress. Just ask me if unsure.

Volume I is realanal.tex and volume II is realanal2.tex (those are the "driver files" text is in separate files for each chapter). I compile the pdf with pdflatex. You need to compile the first volume first before the second volume. You might need to run makeindex (for the index) and makeglossary (for the list of notations) as well, though theoretically it should now be handled automatically. There are scripts publish.sh and publish2.sh, that run everything an obnoxious number of times to make sure it all works.

During the writing of this book, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.

## License:

This work is dual licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License. You can use, print, copy, and share this book as much as you want. You can base your own book/notes on these and reuse parts if you keep the license the same (that is, as long as you use at least one of the two licenses).

## Useful links:

- Notes on Diffy Qs: Differential Equations for Engineers

Another free undergraduate textbook. This one is a bit more basic, it is an application-oriented introduction to differential equations (no proofs there). - Guide to Cultivating Complex Analysis: Working the Complex Field

A graduate complex analysis course for incoming graduate students. After completing this real analysis course, the student should be ready for this complex analysis course. Again, a free textbook. - Wikipedia: Mathematical Analysis
- Introduction to Real Analysis by William Trench
- List of approved free textbooks from the American Institute of Mathematics
- Online Mathematics Textbooks
- Free Online Textbooks, Lecture Notes, Tutorials, and Videos on Mathematics
- Math Books
- OnlineCourses.com, a directory of online courses

## Further reading:

- Robert G. Bartle, Donald R. Sherbert,
*Introduction to real analysis*, 3rd ed., John Wiley & Sons Inc., 2000. - John P. D'Angelo, Douglas B. West,
*Mathematical Thinking: Problem-Solving and Proofs*, 2nd ed., Prentice Hall, 1999. - Joseph E. Fields,
*A Gentle Introduction to the Art of Mathematics*, http://giam.southernct.edu/GIAM/. - Richard Hammack,
*Book of Proof*, http://www.people.vcu.edu/~rhammack/BookOfProof/. - Maxwell Rosenlicht,
Introduction to analysis, Reprint of the 1968 edition, Dover Publications Inc., 1986. ISBN:0-486-65038-3 - Walter Rudin,
*Principles of mathematical analysis*, 3rd ed., McGraw-Hill Book Co., 1976. - William F. Trench,
*Introduction to real analysis*, Pearson Education, 2003, http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF.

# Notes on Diffy Qs: Differential Equations for Engineers

**By: Jiří Lebl **(website #1 (personal), website #2 (work: OSU), email: jiri.lebl@gmail.com)

Jump to: [Download book as PDF] [Buy paperback (amazon)] [Web version] [Search]

A first course on differential equations, aimed at engineering students. The prerequisite for the course is the basic calculus sequence. This free online book (OER more formally) should be usable as a stand-alone textbook or as a companion to a course using another book such as Edwards and Penney, *Differential Equations and Boundary Value Problems: Computing and Modeling* or Boyce and DiPrima, *Elementary Differential Equations and Boundary Value Problems* (section correspondence to these two is given). I developed and used this book to teach Math 286/285 at the University of Illinois at Urbana-Champaign (one is a 4-day-a-week, the other a 3-day-a-week semester-long course). I also taught Math 20D at the University of California, San Diego with this book (a 3-day-a-week quarter-long course). There is enough material to run a two-quarter course, and even a two-semester sequence depending on lecturer speed.

The aim is to provide a low cost, redistributable, not overly long, high-quality textbook that students will keep rather than selling back after the semester is over. Even if the students throw it out, they can always look it up on the net again. You are free to have a local bookstore or copy store make and sell copies for your students. See below about the license.

Another aim of the book is to allow modification and customization for a specific purpose if necessary. If you do modify the book, make sure to mark it prominently as such to avoid confusion. This aspect is also important for the longevity of the book. The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks.

Furthermore, errata are fixed promptly, meaning that if you teach the same class next term, all errata that are spotted are most likely already fixed. No need to wait several years for a new edition. Every once in a while I make some major addition and a new major version (edition), and then in between as errata are fixed I make minor version updates (like a corrected printing) usually once or twice a year, depending on the errata discovered. Exercise, chapter, and section numbers are preserved as much as humanly possible. What's added is added at the end with new numbers, so the book is generally compatible even if students (or the instructor) have an older printed copy. The minor updates are totally interchangeable and have very minimal changes, essentially nothing new.

The graphs in the book were created using the Genius software.

MAA published a review of the book (they looked at the December 2012 edition).

## Table of contents:

Introduction

1. First order equations

2. Higher order linear ODEs

3. Systems of ODEs

4. Fourier series and PDEs

5. More on eigenvalue problems

6. The Laplace transform

7. Power series methods

8. Nonlinear systems

A. Linear algebra

There are currently 740 exercises throughout the book (November 7th 2019 edition), 247 of which have a solution in the back (those numbered 101 and above). A few exercises are within the section text, but most are in their own subsection at the end of every section. Each section should have enough exercises for homework even for a demanding class.

Please let me know at jiri.lebl@gmail.com if you find any typos or have corrections, extra exercises or material, or any other comments.

## Adoption:

Do let me know (jiri.lebl@gmail.com) if you use the book for teaching a course! The book was used, or is being used (other than my courses at UIUC, UCSD, and OSU), at over a dozen universities including Dartmouth College, University of Tennessee, University of Toledo, University of British Columbia, University of California at Irvine, University of Kentucky, University of Hawaii, and many others. The Saylor Foundation is using it as one of the books for their online Math 221 course.

The book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

See a **list of classroom adoptions** for more details.

## Download:

Download the book as PDF

(July 21st, 2020, version 6.1, 466 pages, approximately 4.1 MB download)

*Version 6.0 added Appendix A, and sections 1.9 and 6.5.*

Look at the errata in the current revision (if any).

Look at the change log to see what changed in the latest version.

If you need the old version, version 5.5, for some reason, you can find it here.

## Buy paperback:

I get a bit of money when you buy these (depending on where exactly they are bought). Probably enough to buy me a coffee, so by buying a copy you will support this project. You will also save your toner cartridge. The difference between these two versions is essentially just the cover art. I have seen printed versions from both and they are both good quality.

ISBN-13: 978-1706230236

ISBN-10: 1706230230

(The older edition 5 has ISBN-13: 978-1541329058, if you really need the old version)

## Web version:

Browse the web version of the book (for easier reading on the web). This version uses PreTeXt and so should be easier to browse and read. The PDF version is the canonical version and should be the one used for printing.

## Search:

Search this site, including the web version:

## Interactive SAGE demos:

Section 1.6: Several interactive demos on autonomous equations in one variable.

Section 1.7: An interactive demo of Euler's method.

Section 2.4: Several animations of mechanical vibrations.

Section 2.6: Interactive demo of forced oscillations and resonance.

Section 3.5: Interactive demos of two-dimensional autonomous systems.

Section 3.6: A second order system (two carts with springs between them) interactive demo.

## Instructor Resources

### Online homework (WeBWorK / Edfinity):

**WeBWorK:** I put together a set of problems for WeBWorK, containing currently 487 problems. Download the set as a tgz file (or see the github repository). You just upload the tgz file to your course, and it should automatically unpack in your templates directory. You'll have a directory called diffyqs-webwork. It has predefined problem sets for the relevant sections as "def" files, that you can "import" as new homework. There are currently problems for chapters 0, 1, 2, 3, 4, 6, 7, 8, A. Majority of the problems come from OPL (Open Problem Library), but have been edited to fit the course or generally improved, some have been edited or changed quite heavily, and new problems added. Hopefully there should be enough problems for most types of courses but let me know if anything is missing, or if there are other OPL problems that I think I should include.

**Edfinity:** Essentially the same set of problems is available on Edfinity. Edfinity is quite a bit easier to use than WeBWorK, you don't have to install anything, it is hosted by Edfinity, and is student-paid (on the order of $2-$4 per month). The Edfinity will be updated with the working copy above every once in a while. You can still add other OPL problems into your class, or create your own.

Let me know (jiri.lebl@gmail.com) if you use the problem set. I'm also interested to know any feedback on what's missing, what should be changed, etc.

### Discussion/Announcement forum:

I set up a discussion forum for the book on Google Groups. I expect this to be a low volume forum, but it might be a good place for instructors to interact, where to post extra material, ask a question of the other instructors, discuss the WeBWorK problems, and I will send announcements there, such as when a new version is out.

### Other instructor resources:

I put together all the figures as PDFs as one big zipfile. This should make it easier to create computer slides using the figures, without messing with the source tarball. If a figure appears in multiple places, its filename only refers to the first such place.

There's tons of extra materials (including longer modeling projects) at SIMIODE.

The IODE software is a free software package for experimenting with basic ODEs developed at University of Illinois specifically for teaching this kind of course. IODE works both with Matlab (proprietary) and Octave (free, but no GUI). The IODE website has several extra projects for the students to work through as homework.

## Translations and derivative versions:

Prof. Martin Weilandt of Universidade de Santa Catarina has prepared a partial Portuguese translation. See his class page.

Prof. Charles Bergeron has created a modified version of the book based heavily on Notes on Diffy Qs. The title is Differential Equations: Including Linear Algebra Topics And Computer-Aided Problem-Solving. The book removes some topics (e.g. PDEs), and adds its own linear algebra chapter (this was before appendix A). The book covers the use of the computer algebra system Maxima in the context of the material.

The department at University at Buffalo (Brian Hassard, James Javor, John Ringland, Asela Viraj) have created their own edition without the PDE content and including some extra content on using python.

The text (slightly older version) has been entered into the libretext.org platform run by UC Davis.

## Source:

The source is hosted on **GitHub**: https://github.com/jirilebl/diffyqs

You can get an archive of the source of the released version on github, look under https://github.com/jirilebl/diffyqs/releases, though if you plan to work with it, maybe best to look at just the latest working version as that might have any errata or new additions. Though these might be a work in progress. Perhaps best is to let me know.

The main file is diffyqs.tex, which includes the chapters that are in separate files ch-*.tex. I compile the pdf with pdflatex diffyqs. You also want to run makeindex to generate the index (I generally run pdflatex diffyqs three times, then makeindex diffyqs, and then finally pdflatex diffyqs again). The setup file with all the preamble you may want to edit is diffyqssetup.sty.

The github 'master' version is the current working version, so it will have whatever new changes I make in my tree.

During the writing of this book, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.

## License:

This work is dual licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License. You can use, print, copy, and share this book as much as you want. You can base your own book/notes on these and reuse parts if you keep the license the same (that is, as long as you use at least one of the two licenses).

## Useful links:

- My other free undergraduate textbook: Basic Analysis: Introduction to Real Analysis
- My graduate textbook: Tasty Bits of Several Complex Variables
- Differential equation - Wikipedia, the free encyclopedia
- List of approved free textbooks from the American Institute of Mathematics
- Online Mathematics Textbooks
- Free Online Textbooks, Lecture Notes, Tutorials, and Videos on Mathematics
- Math Books
- SIMIODE, lots of resources including longer projects especially focusing on modeling.
- IODE software
- OnlineCourses.com, a directory of online courses
- WeBWorK, free software online homework system with lots of questions on differential equations (mainly ODE) in the standard problem library, some from this book.