most recent 30 from http://mathoverflow.net 2013-06-19T21:53:15Z http://mathoverflow.net/feeds/question/761 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/761/undergraduate-level-math-books person 2009-10-16T16:47:52Z 2011-05-10T23:02:38Z

<p>What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?</p> <p>EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive &amp; Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)</p> <p><strong>Please post only one book per answer</strong> so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/773#773 Michael Lugo 2009-10-16T18:14:38Z 2009-10-16T18:14:38Z

<p>The <a href="http://www.ocf.berkeley.edu/~abhishek/chicmath.htm" rel="nofollow">Chicago undergraduate mathematics bibliography</a> is a nice annotated list of books.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/778#778 Grétar Amazeen 2009-10-16T18:58:46Z 2009-10-16T18:58:46Z

<p>For undergraduate level topology (mostly point set topology) I recommend "Topology" by Munkres. I learned topology from this book as an undergrad and I remember this being one of my favorite books at the time.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/780#780 Gerald Edgar 2009-10-16T19:03:58Z 2009-10-16T19:03:58Z

<p>Spivak, <em>Calculus</em></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/781#781 bhwang 2009-10-16T19:10:29Z 2009-10-16T19:10:29Z

<p>There is a good list <a href="http://hk.geocities.com/mathphyweb/puremath.htm" rel="nofollow">here</a>, divided by subject, that also contains many links to freely available textbooks and lecture notes. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/782#782 Gerald Edgar 2009-10-16T19:10:55Z 2009-10-16T19:10:55Z

<p>Kelley, <em>General Topology</em></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/783#783 Andreas Holmstrom 2009-10-16T19:11:45Z 2009-10-16T19:11:45Z

<p>Ok, this is not a single book, but I have often found books from the Springer Undergraduate Mathematics Series (SUMS) to be excellent. Here is a <a href="http://www.springer.com/series/3423?detailsPage=titles" rel="nofollow">list of titles</a>.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/784#784 Grétar Amazeen 2009-10-16T19:13:45Z 2009-10-16T19:13:45Z

<p>Another one I like is "An introduction to Lie algebras." by Erdmann and Wildon.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/785#785 Justin 2009-10-16T19:17:57Z 2009-10-16T19:17:57Z

<p>Algebraic Topology by Hatcher (available online <a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html" rel="nofollow">here</a>).</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/787#787 Andreas Holmstrom 2009-10-16T19:39:29Z 2009-10-16T19:39:29Z

<p>Searcóid: Elements of Abstract Analysis. I loved this book as an undergraduate, for many reasons, but mainly because it gave me an idea of the unity of mathematics. It starts from the axioms of set theory and takes you all the way to C*-algebras and the Gelfand-Naimark theorem. Here's the <a href="http://books.google.com/books?id=c3ChND-I42wC&amp;dq=%22elements+of+abstract+analysis%22&amp;source=gbs%5Fnavlinks%5Fs" rel="nofollow">Google Books page</a>.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/788#788 David Zureick-Brown 2009-10-16T19:55:30Z 2009-10-16T19:55:30Z

<p><em><a href="http://books.google.com/books?id=G%5FA8HciIro4C&amp;pg=PP1&amp;dq=ian+galois+theory&amp;ei=PM%5FYSsYqiIaVBKGQ4JIB#v=onepage&amp;q=&amp;f=false" rel="nofollow">Galois Theory</a></em> by Ian Stewart is excellent. The third edition is quite different from the second and includes many more problems.</p> <p><hr /></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/795#795 Clemens Koppensteiner 2009-10-16T20:23:51Z 2009-10-16T20:23:51Z

<p>Not exactly one of the topics in the question, but I particularly liked Silverman and Tate's <em>Rational Points on Elliptic Curves</em>.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/796#796 Kevin P. Costello 2009-10-16T20:29:55Z 2009-10-16T20:29:55Z

<p>For discrete mathematics, I would recommend Van Lint-Wilson's "A Course in Combinatorics" as a good introductory text. It consists of 38 (in my edition) chapters that give (often largely self-contained) introductions to various areas of the field. Although it doesn't go nearly as in depth as, say, Stanley's "Enumerative Combinatorics" or a text focused solely on graph theory, I found it excellent for giving a broad overview and indicating to me where I wanted to explore deeper. </p> <p>My one caveat would be that some chapters require background in either linear algebra or basic group theory, though those are easily skippable due to the structure of the book. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/797#797 Qiaochu Yuan 2009-10-16T20:32:46Z 2009-10-16T20:32:46Z

<p>The <a href="http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X" rel="nofollow">Princeton Lectures in Analysis</a> by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis (in that order!).</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/802#802 Henning Arnór Úlfarsson 2009-10-16T21:14:47Z 2009-11-06T09:29:52Z

<p>Alexandre Stefanov keeps an extensive <a href="http://mathbooks.110mb.com/mylist.php" rel="nofollow">list</a> of free math books / lecture notes. The list is divided according to subject and updated frequently. I have found some very nice books there.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/803#803 Harrison Brown 2009-10-16T21:21:54Z 2009-10-16T21:21:54Z

<p><em>Generatingfunctionology</em> by Wilf is fun, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow">free</a>, requires very little in the way of prerequisites, and is as good an introduction to the methods of analytic combinatorics as could be asked for. It's long been one of my favorite textbooks.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/814#814 Max M 2009-10-17T00:19:39Z 2009-10-17T00:19:39Z

<p>Real Mathematical Analysis by Charles Pugh</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/816#816 Max M 2009-10-17T00:22:52Z 2009-10-17T00:22:52Z

<p>Lectures on Linear Algebra by I. M. Gel'fand</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/817#817 Max M 2009-10-17T00:26:04Z 2009-10-17T00:26:04Z

<p>Ordinary Differential Equations by Vladimir I. Arnold</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/837#837 David Speyer 2009-10-17T04:53:31Z 2009-10-17T04:53:31Z

<p>Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/858#858 Elisha Peterson 2009-10-17T11:37:50Z 2009-11-30T05:20:24Z

<p>Introduction to Topology: Pure and Applied, by Adams and Franzosa. The figures in the book are beautiful, the problems are good, and the applications are good (and unusual) to see in an undergraduate text.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/859#859 Elisha Peterson 2009-10-17T11:41:27Z 2009-10-17T11:41:27Z

<p>Real Analysis, by Frank Morgan. The chapters are short and very directed. The proofs are written well. The exercises are well-selected. The book is written at a level accessible for most students.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/888#888 Applied Applied Math guy 2009-10-17T16:13:33Z 2009-12-28T13:07:39Z

<p>I am surprised no one has mentioned Halmos' Naive Set Theory or Finite-Dimensional Vector Spaces or Rudin's Principles of Mathematcal Analysis. There's also Sheldon Axler's Linear Algebra Done Right and Royden's Real Analysis.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/891#891 lhf 2009-10-17T16:45:02Z 2009-10-17T16:45:02Z

<p><em>Basic Algebra</em> by Jacobson.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/927#927 Michael Hoffman 2009-10-17T19:35:54Z 2009-10-17T19:35:54Z

<p>Since Numerical Mathematics has not been covered, I would recommend the following</p> <p>Introduction to Numerical Analysis by Stoer et. al. <a href="http://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X/ref=sr%5F1%5F14?ie=UTF8&amp;s=books&amp;qid=1255807973&amp;sr=8-14" rel="nofollow" title="Introduction to Numerical Analysis, Stoer et. al.">http://www.amazon.com/Introduction-Numerical-Analysis-J-Stoer/dp/038795452X/ref=sr_1_14?ie=UTF8&amp;s=books&amp;qid=1255807973&amp;sr=8-14 </a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/928#928 Michael Hoffman 2009-10-17T19:39:05Z 2009-10-17T19:39:05Z

<p>Also, I just started this book and absolutely love it</p> <p><a href="http://www.amazon.com/Geometry-Euclid-Beyond-Robin-Hartshorne/dp/0387986502/ref=sr%5F1%5F1?ie=UTF8&amp;s=books&amp;qid=1255808262&amp;sr=1-1" rel="nofollow" title="Geometry: Euclid and Beyond, Hartshorne">Geometry: Euclid and Beyond, Hartshorne</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1259#1259 Dmitri Pavlov 2009-10-19T18:46:15Z 2009-10-19T18:46:15Z

<p>Serre, A Course in Arithmetic.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1260#1260 Dmitri Pavlov 2009-10-19T18:50:21Z 2009-10-19T18:50:21Z

<p>Milnor, Topology from the differentiable viewpoint.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1459#1459 Alexander Vorobiev 2009-10-20T17:40:24Z 2009-10-20T17:40:24Z

<p>Linear Algebra and Its Applications by Gilbert Strang. You can also watch his video lectures at MIT <a href="http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/index.htm" rel="nofollow">OpenCourseWare</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1460#1460 Alon Amit 2009-10-20T17:58:13Z 2009-12-21T15:08:59Z

<p>A basic undergrad algebra book which I feel is not as well known as it should be is Michael Artin's <a href="http://www.amazon.com/Algebra-Michael-Artin/dp/0130047635/ref=sr%5F1%5F1?ie=UTF8&amp;s=books&amp;qid=1256061341&amp;sr=1-1" rel="nofollow">Algebra</a>. I have it in soft cover so I hope it's actually the one in this Amazon link. Anyway it's beautifully written, provides context and motivation and is just a pleasure to read or browse. How often do you find a basic text written by a world-class expert? </p> <p>"Always study from the masters".</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1463#1463 Abdó Roig-Maranges 2009-10-20T18:02:51Z 2009-10-20T18:08:11Z

<p>As an undergraduate I loved Shafarevich's book <a href="http://www.amazon.com/Basic-Notions-Algebra-I-R-Shafarevich/dp/3540612211" rel="nofollow">Basic notions of algebra</a>. This is not a textbook, but gives small beautiful tastes of a broad choice of topics in algebra, emphasizing connections with other fields.</p> <p>I found it very stimulating, in the sense that every example or overview of some topic in this book made me want to learn more details about it. In fact I became interested in algebraic geometry because of this book.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1491#1491 Peter Arndt 2009-10-20T21:08:03Z 2009-10-20T21:08:03Z

<p>"Introduction to Mathematical Logic" by Ebbinghaus, Flum and Thomas</p> <p>Careful introduction, addresses many doubts that one might have about why one does logic in this way and not some other, e.g. whether one is doing something circular when formulating set theory in 1st order logic, or e.g. it proves Lindstroem's Theorem, that says that classical 1st order logic has the highest power of expressability among the logics with completeness and Loewenheim-Skolem.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1494#1494 Peter Arndt 2009-10-20T21:13:09Z 2009-10-20T21:13:09Z

<p>Jaenich: "Topology"</p> <p>Introduces the concepts of point set topology ("paracompact" and all this stuff) motivating each via examples which are rigorously defined but also drawn. Other advantage: It is short!</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1524#1524 Kim Greene 2009-10-20T23:49:40Z 2009-10-20T23:49:40Z

<p>Metric Spaces by Mícheál Ó Searcóid</p> <p><a href="http://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256082496&amp;sr=1-1" rel="nofollow">http://www.amazon.com/Metric-Spaces-Springer-Undergraduate-Mathematics/dp/1846283698/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256082496&amp;sr=1-1</a></p> <p>It's an exhaustive introduction analysis at the level of the metric space that's well worth reading.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1526#1526 Kim Greene 2009-10-20T23:53:39Z 2009-10-20T23:53:39Z

<p>Introduction to Analysis by William R. Wade</p> <p><a href="http://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256082707&amp;sr=1-1" rel="nofollow">http://www.amazon.com/Introduction-Analysis-4th-William-Wade/dp/0132296381/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256082707&amp;sr=1-1</a></p> <p>This is a good transition from undergraduate calculus to analysis.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1528#1528 Kim Greene 2009-10-20T23:56:54Z 2009-10-20T23:56:54Z

<p>An Introduction to Manifolds by Loring W. Tu</p> <p><a href="http://www.amazon.com/Introduction-Manifolds-Universitext-Loring-W/dp/0387480986/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256082981&amp;sr=1-1" rel="nofollow">http://www.amazon.com/Introduction-Manifolds-Universitext-Loring-W/dp/0387480986/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256082981&amp;sr=1-1</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1529#1529 Kim Greene 2009-10-21T00:02:05Z 2009-10-21T00:02:05Z

<p>Riemannian Geometry: A Beginner's Guide by Frank Morgan</p> <p><a href="http://www.amazon.com/Riemannian-Geometry-Beginners-Frank-Morgan/dp/1568810733/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256083041&amp;sr=1-1" rel="nofollow">http://www.amazon.com/Riemannian-Geometry-Beginners-Frank-Morgan/dp/1568810733/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1256083041&amp;sr=1-1</a></p> <p>I love this book!</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1532#1532 Deane Yang 2009-10-21T00:10:57Z 2009-10-21T00:10:57Z

<p>Guillemin and Pollack, "Differential Topology"</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1597#1597 diab 2009-10-21T07:13:56Z 2009-10-21T07:13:56Z

<p>Fraleigh's "A First Course in Abstract Algebra"</p> <p><a href="http://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907" rel="nofollow">http://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1608#1608 samv 2009-10-21T08:13:30Z 2009-10-21T08:13:30Z

<p>Godement "Analysis" (I,II,III,IV) <a href="http://www.amazon.com/Analysis-Convergence-Elementary-functions-Universitext/dp/3540059237" rel="nofollow">http://www.amazon.com/Analysis-Convergence-Elementary-functions-Universitext/dp/3540059237</a></p> <p>"... The content is quite classical ... [...] The treatment is less classical: precise although unpedantic (rather far from the definition-theorem-corollary-style), it contains many interesting commentaries of epistemological, pedagogical, historical and even political nature. [...] The author gives frequent interesting hints on recent developments of mathematics connected to the concepts which are introduced. The Introduction also contains comments that are very unusual in a book on mathematical analysis, going from pedagogy to critique of the French scientific-military-industrial complex, but the sequence of ideas is introduced in such a way that readers are less surprised than they might be.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1645#1645 Alessandro S 2009-10-21T12:55:41Z 2009-10-21T12:55:41Z

<p>Real and Complex Analysis by W. Rudin is a beatiful and extremely well written book which presents the fundamentals of real and complex analysis highlighting the interactions between different results and ideas.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/1671#1671 Gerald Edgar 2009-10-21T14:58:23Z 2009-10-21T14:58:23Z

<p>How about an <strong>anti</strong>-recommendation? Someone in another answer mentioned Steven Axler's <em>Linear Algebra Done Right</em>. My comment, not as someone who has used this book in a class, but as someone who has taught the students from this class during the following term: It doesn't prepare the students to use linear algebra in engineering, in physics, in chemistry, or even in branches of mathematics other than abstract algebra.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/2910#2910 Brad Hannigan-Daley 2009-10-27T22:11:28Z 2009-10-27T22:11:28Z

<p>Dummit and Foote's <em>Abstract Algebra</em> is an excellent book for learning group theory, ring theory, and module theory. There's also a section on basic algebraic geometry and homological algebra.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/2930#2930 Akhil Mathew 2009-10-27T23:08:22Z 2009-10-27T23:08:22Z

<p>Spivak, <em>A Comprehensive Introduction to Differential Geometry.</em> There is a nice geometrical philosophy and plenty of motivation.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3004#3004 AgCl 2009-10-28T05:44:46Z 2009-10-28T05:44:46Z

<p>General Topology, by Stephen Willard</p> <p><a href="http://books.google.com/books?id=-o8xJQ7Ag2cC&amp;lpg=PP1&amp;dq=stephen%20willard&amp;pg=PP1#v=onepage&amp;q=&amp;f=false" rel="nofollow">http://books.google.com/books?id=-o8xJQ7Ag2cC&amp;lpg=PP1&amp;dq=stephen%20willard&amp;pg=PP1#v=onepage&amp;q=&amp;f=false</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3005#3005 AgCl 2009-10-28T05:47:50Z 2009-10-28T05:47:50Z

<p>Differential Geometry of Curves and Surfaces, by Manfredo Do Carmo is an excellent introductory book. </p> <p><a href="http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897" rel="nofollow">http://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3039#3039 Humberto Rafeiro 2009-10-28T12:41:55Z 2009-10-28T12:48:00Z

<p>I didn't see any suggested books from the great Russian school of mathematics, here is a brief list of superb, well written, example oriented books:</p> <ul> <li><em>Elements of the Theory of Functions and Functional Analysis</em> by A. N. Kolmogorov and S. V. Fomin</li> <li><em>Theory of Functions of a Real Variable</em> by I. P. Natanson (this I think, it's hard to find)</li> <li><em>Theory of Functions of a Complex Variable</em>, Second Edition (3 vol. set) </li> <li><em>Elements of Functional Analysis</em> by L.A. Lusternik and V.J. Sobolev</li> <li><em>Problems in Mathematical Analysis</em> by B. Demidovich</li> <li><em>Calcul intégral et differentiel</em> (2 vol. set) by N. Piskounov. In English should be something like <em>Differential and Integral Calculus</em></li> <li><em>A Course of Mathematical Analysis</em> (2 vol. set ) S. Nikols'skii</li> <li><em>Differentialrechnung und Integralrechnung</em> (3 vol. set), by Gregor M. Fichtenholz. Unfortunately, there is no English translation of this book, only the German translation that it's mentioned. I think this was THE calculus book on Russia. THIS BOOK SHOULD BE TRANSLATED INTO ENGLISH, and I suspect that there is no copyright, it appears around 1959 I think.</li> <li><em>Mathematical Analysis</em> (2 vol. set) by Vladimir A. Zorich. This is a very recent book from a great mathematician which is in Moscow university. The book is based upon is lecture.</li> </ul> <p>I have also found a Spanish translation of a book written by S. Banach about "Differential and integral calculus". It is very good as an undergraduate book. The spanish translation (for those who want to search) is <em>Calculo diferencial e integral</em></p> <p>Books that undergraduate should not touch (in my humble opinion) are books written in bourbaki's style. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3408#3408 Isaac Copper 2009-10-30T05:37:41Z 2009-10-30T05:37:41Z

<p>Linear Algebra: With Applications Otto Bretscher </p> <p>Used at Carleton College – nice explanations, and quite a few proofs. Presents information primarily by providing examples, definitions/axioms and then proofs. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3438#3438 vonjd 2009-10-30T12:58:02Z 2009-10-30T12:58:02Z

<p>I had many trials and these are in my opinion the best for an introductory, undergraduate level:</p> <p>ODEs: Holzner: Differential Equations for Dummies <br/> PDEs: Farlow: Partial Differential Equations for Scientists and Engineers</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3463#3463 Agusti Roig 2009-10-30T18:40:02Z 2009-10-30T18:40:02Z

<p>Applied Linear Algebra, by B. Noble and J.W. Daniel, <a href="http://www.amazon.co.uk/Applied-Linear-Algebra-Ben-Noble/dp/0130412600/ref=sr_1_14?ie=UTF8&amp;s=books&amp;qid=1256927879&amp;sr=1-14" rel="nofollow">http://www.amazon.co.uk/Applied-Linear-Algebra-Ben-Noble/dp/0130412600/ref=sr_1_14?ie=UTF8&amp;s=books&amp;qid=1256927879&amp;sr=1-14</a> .</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3501#3501 Jonah Sinick 2009-10-31T01:28:23Z 2009-10-31T01:28:23Z

<p><em>A Field Guide to Algebra</em> by Antoine Chambert Loir. Covers a surprising amount of material considering the short length and minimal prerequisites</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3508#3508 Andrea 2009-10-31T02:46:47Z 2009-10-31T02:46:47Z

<p>Complex Analysis, by Lars Ahlfors</p> <p><a href="http://en.wikipedia.org/wiki/Lars_Ahlfors" rel="nofollow">http://en.wikipedia.org/wiki/Lars_Ahlfors</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3558#3558 Vasile Moșoi 2009-10-31T16:21:10Z 2009-10-31T16:21:10Z

<p>"TOPOLOGY WITHOUT TEARS" by SIDNEY A. MORRIS is a very nice introduction in topological spaces theory, I think.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3588#3588 H A Helfgott 2009-10-31T20:20:59Z 2009-10-31T20:20:59Z

<p>Vinogradov's Elements of Number Theory - the problems more so than the text itself.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3678#3678 Andrea 2009-11-01T15:56:46Z 2009-11-01T15:56:46Z

<p>Subject: FUNCTIONAL ANALYSIS</p> <p>Erwin Kreyszig</p> <p><strong>Introductory Functional Analysis with Applications</strong></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3682#3682 Zach Conn 2009-11-01T16:42:01Z 2009-11-01T16:42:01Z

<p>Needham, <em>Visual Complex Analysis</em>. I read this while in high school, and it's simply beautiful. I recommend this book as a <em>supplement</em> to any first course in complex analysis (a different book should probably be used for the main textbook since Needham's is very pretty, very engaging, but not very rigorous).</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3909#3909 Katie Banks 2009-11-03T06:54:38Z 2009-11-03T06:54:38Z

<p>General geometry: Coxeter, Introduction to Geometry.</p> <p>Not so much a textbook as a collection of essays (in particular, it doesn't have exercises), but all of the essays are instructive and enlightening.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3916#3916 kweinert 2009-11-03T08:14:26Z 2009-11-03T08:14:26Z

<p>G. J. O. Jameson: Topology and Normed Spaces for an introduction to functional analysis from a topological point of view.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/3917#3917 kweinert 2009-11-03T08:19:18Z 2009-11-03T08:19:18Z

<p>E. Hairer, G. Wanner: Analysis by its history for an introduction to real and numerical analysis from a historical point of view.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/4405#4405 vlorbik 2009-11-06T17:20:45Z 2009-11-06T17:20:45Z

<p>lindsey childs <I>a concrete introduction to abstract algebra</I> </p> <p><a href="http://books.google.com/books?id=rUApHgaTVx0C&amp;dq=concrete+introduction+to+higher+algebra&amp;printsec=frontcover&amp;source=bn&amp;hl=en&amp;ei=KVr0Sqe4CcLOlAfs__ClAw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=4&amp;ved=0CBUQ6AEwAw#v=onepage&amp;q=&amp;f=false" rel="nofollow">here maybe</A>.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/4439#4439 Shake Baby 2009-11-06T21:17:28Z 2009-11-06T21:17:28Z

<p>Apostol "Calculus"</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/4440#4440 Shake Baby 2009-11-06T21:19:41Z 2009-11-06T21:19:41Z

<p>Bartle "The Elements of Integration and Lebesgue Measure"</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/4572#4572 mymath 2009-11-08T00:31:44Z 2009-11-08T00:31:44Z

<p>Strichartz, <em>The Way of Analysis</em></p> <p>Herstein, <em>Abstract Algebra</em></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/4838#4838 sdcvvc 2009-11-10T10:23:56Z 2009-11-10T10:23:56Z

<p><a href="http://books.google.com/books?id=wp2A7ZBUwDgC" rel="nofollow">Geroch, Mathematical Physics</a></p> <p>Don't be scared by the title: it teaches algebra, topology and measure theory, using category-theoretic language.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/4888#4888 Sam Nead 2009-11-10T16:58:17Z 2009-11-10T16:58:17Z

<p>Milnor's book "Dynamics in One Complex Variable: Introductory Lectures". An early version is available from his website. Suitable for advanced undergraduates, graduate students, and mathematicians.</p> <p><a href="http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims90-5" rel="nofollow">http://www.math.sunysb.edu/cgi-bin/preprint.pl?ims90-5</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/6251#6251 Martijn 2009-11-20T10:57:06Z 2009-11-20T10:57:06Z

<p>For a thorough introduction on Partial Differential Equations, read <a href="http://www.amazon.com/Partial-Differential-Equations-Graduate-Mathematics/dp/0821807722/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1258714574&amp;sr=8-1" rel="nofollow">L.C. Evans, "Partial Differential Equations"</a>. Features both linear and nonlinear equations.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/6260#6260 Peter Arndt 2009-11-20T13:13:28Z 2009-11-20T13:13:28Z

<p>Kock, Vainsencher: An invitation to Quantum Cohomology.</p> <p>Written in the most friendly and motivating style I have ever seen in a book. Almost has no prerequisites: You should that there exists something like algebraic varieties - without having to know any technical details - and that P^1 is such a thing. Everything else is provided in easy exercises or the text. It gives an excellent intuition about the subject with lots of outlooks on a field of current research, and at the same time manages to be easily undergraduate readable.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/7261#7261 Carsten Schultz 2009-11-30T11:46:50Z 2009-11-30T11:46:50Z

<p>Most readers here will not be able to appreciate them for a simple reason, but my favorite beginners' Analysis text is that by <a href="http://www.mathematik.uni-regensburg.de/broecker/index.htm" rel="nofollow">Bröcker</a>. No-nonsense, concise, with a slight orientation towards topology.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9359#9359 Anirbit 2009-12-19T08:04:13Z 2009-12-19T08:04:13Z

<p>"Differential Calculus on Normed Linear Banach Spaces" by Kalyan Mukherjee.</p> <p>The author is a prof at the Indian Statistical Institute (Kolkata) </p> <p>This is an amazing book which introduces differential calculus in arbitrary finite dimensional spaces (thinking of the derivative as the Jacobian) as the next leap from the Apostol and Rudin level and also build in topological ideas. It then goes into manifold theory and shows how to compute tangents to curves inside lie groups. It has nice sections of things like differentiating the determinant function and the matrix multiplication function and inverse function theorem and idea of equivalence of norms. </p> <p>I would strongly recommend this book to an undergrad after he/she has done the Apostol/Rudin level of calculus.</p> <p>"Calculus on Manifolds" by Spivak and "Differential geometry and Lie groups" by Kumaresan are two other good books which can be read alongside it. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9375#9375 Alexander Woo 2009-12-19T16:07:12Z 2009-12-19T16:07:12Z

<p>Miles Reid, Undergraduate Algebraic Geometry.</p> <p>ps - anyone who thinks one can teach an undergraduate class out of Hatcher's Algebraic Topology (which is a great book) at more than 10 universities in the US is sadly deluded. Ditto for a few more things I've seen here.</p> <p>pps - somewhere between a third and half of the math majors here could handle abstract algebra out of Artin. It would be great for those that could, but we're not going to ditch half our students.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9386#9386 Ilya Grigoriev 2009-12-19T20:03:39Z 2009-12-19T20:03:39Z

<p>For a long time, <a href="http://www.amazon.com/Introductory-Real-Analysis-N-Kolmogorov/dp/0486612260" rel="nofollow">Kolmogorov-Fomin's <em>Introductory Real Analysis</em></a> was my standard for a great mahtematics textbook. I can't imagine a better introduction to serious analysis.</p> <p>The translation I'm linking to is very good, and includes excercises (the original has many fewer), but it is incomplete (it's missing the chapter on Fourier Series). So if you can read Russian, I recommend you get the original.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9387#9387 Ryan Budney 2009-12-19T20:13:21Z 2009-12-19T21:36:30Z

<p>I'm a big fan of John Hubbard's "Vector Calculus, Linear Algebra and Differential Forms" text. I was a TA for the course twice at Cornell and was amazed at how well it went. The text has an extremely pleasant "zest" to it. When Hubbard asked me to take a look at it my first response was the text is "overflowing with the spirit of calculus". I still believe that. I have a hard time containing my praise. </p> <p>The main problem with the text is that it's so engrossing. It places more demands on the student than a traditional service course text would ever consider. But it's also <em>far</em> more rewarding. At Cornell it was taught as a branch of their traditional calculus sequence -- it was a course that was earmarked for keener students, mostly from other departments. </p> <p>In short, if you want to have physics, engineering and economics students appreciating the derivative as a linear approximation, thinking Lipschitz bounds for functions are cool, being interested in the computation of norms of linear operators, etc, this is a great resource. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9445#9445 Adrian Petrescu 2009-12-20T22:39:55Z 2011-05-10T23:02:38Z

<p>I liked <a href="http://books.google.ca/books?id=bj1kOY8gOfcC&amp;dq=elements+of+abstract+algebra&amp;printsec=frontcover&amp;source=bl&amp;ots=sMDMivxMQ6&amp;sig=5XAar4HPSaqw0XRPBwt0Om4ibhA&amp;hl=en&amp;ei=SKcuS4fEJsbilAePleCkBw&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CAoQ6AEwAA#v=onepage&amp;q=&amp;f=false" rel="nofollow">Elements of Abstract Algebra</a> by Allan Clark, which is mainly a problem book with a moderate amount of exposition, but the problems are so well-chosen that a diligent undergraduate student working through all of them will come out with a solid knowledge of group theory, classical ring theory, and Galois theory.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9459#9459 Vinoth 2009-12-21T04:52:02Z 2009-12-21T04:52:02Z

<p>From the quick look I've had, not much representation theory has been mentioned so here goes for undergrad level rep theory (perhaps suitable for 3rd/4th year in a standard sequence of undergraduate study), roughly in the order of difficulty (from easiest to hardest):</p> <ul> <li><p>James &amp; Liebeck - "Representations and Characters of Groups" (a very good introduction)</p></li> <li><p>Sagan - "The symmetric group: representations, combinatorial algorithms, and symmetric functions"; (the first two chapters here at least are representation theory) OR James &amp; Kerber - "Representation Theory of the Symmetric Group" (this one includes some modular representations of $S_n$)</p></li> <li><p>Alperin - "Local Representation Theory" (basically, modular representation theory)</p></li> <li><p>Hall - "Lie groups, Lie Algebras and Representation Theory" (a solid introduction to Lie theory); for a more advanced perspective Harris &amp; Fulton - "Representation Theory: A first course" (but it could be slightly terse at points, but not necessarily)</p></li> </ul> <p>For algebraic geometry, the one book I'd suggest is "Algebraic Geometry: A first course" by Joe Harris, very nice and full of examples. For algebraic number theory, a very good introduction is Janusz - "Algebraic Number Fields" (followed perhaps by Childress - "Class Field Theory", or Silverman - "The Arithmetic of Elliptic Curves" to go in a slightly different direction). </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9896#9896 mathphysicist 2009-12-27T20:11:04Z 2009-12-27T20:11:04Z

<p>Karen Smith et al., <a href="http://books.google.com/books?id=8zQYxx7PjqsC" rel="nofollow">An Invitation to Algebraic Geometry</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9938#9938 Brendan Cordy 2009-12-28T05:34:36Z 2009-12-28T05:34:36Z

<p>I think that <em>Linear Algebra</em> by Friedberg, Insel, and Spence is a spectacular linear algebra book. It gets straight to the point, it provides a worked out example or two exactly when they're needed, and it has lots of interesting exercises.</p> <p>There are way too many gigantic linear algebra books with colour pictures and contrived examples which often seem to obsfucate the concepts being introduced. In my mind this book is 'the mathematician's linear algebra book'; clean and concise.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/9956#9956 Eivind Dahl 2009-12-28T15:10:44Z 2009-12-28T15:18:59Z

<p>Here is an undergraduate level math book recommendation from an early undergrad's position:</p> <p>I like "Linear Algebra Done Right". I've looked at a bunch of books on linear algebra, and the usual matrix approach is to me a big turn-off when what you're really interested in is the abstract machinery of transformations between vector spaces. I'm not a research mathematician. In fact, I don't even study linear algebra yet, but as a student of mathematics that like algebra, spaces, maps and all that good stuff, I find this to be a very readable account of linear algebra.</p> <p>There are more abstract books on the subject, and my impression is that LADR prepares you for the next level way before you're usually "allowed to" by other accounts like Lax etc. The trade-off is that LADR is not a book for engineers, but this would be a sad world for a mathematician if that was something he had to worry about (in his spare time). Great for self-study. Reads like a novel. I'd probably prefer it if Axler used sets for span and bases instead of lists, but that's something you'll probably be able to shake off with the next book you read on the subject.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/14326#14326 kakaz 2010-02-05T23:37:53Z 2010-02-05T23:37:53Z

<p>Hugo Steinhaus <a href="http://en.wikipedia.org/wiki/Hugo_Steinhaus" rel="nofollow">http://en.wikipedia.org/wiki/Hugo_Steinhaus</a> book "Mathematical Kaleidoscope". It is kind of mathematical trivia sometimes very deep;-) It is not for learning math but for learning how to learn math in fun way.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/17367#17367 Christopher Olah 2010-03-07T07:26:21Z 2010-03-07T07:26:21Z

<p><em>Visual Complex Analysis</em> by Tristan Needham is <em>awesome</em>!</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/17394#17394 stankewicz 2010-03-07T17:01:10Z 2010-03-07T21:08:15Z

<p><em>Complex Variables: Harmonic and Analytic Functions</em> by Francis J. Flanigan</p> <p>A nice little Dover paperback which turns the standard course on complex variables on its head. It begins by doing some multivariable calculus in the plane and harmonic functions, then proceeds to talk about complex numbers and to build analytic functions.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25837#25837 Victor Protsak 2010-05-25T02:23:47Z 2010-05-25T02:23:47Z

<p>I am surprised this has not been mentioned before (is it too advanced?):</p> <p>Bott and Tu, <em>Differential forms in algebraic topology.</em></p> <p>The best introduction to de Rham cohomology, spectral sequences, characteristic classes from the algebraic point of view, and countless other topics. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25849#25849 Adeel 2010-05-25T06:24:47Z 2010-05-25T06:24:47Z

<p>I found «<a href="http://www.amazon.com/Introduction-Algebra-Student-Mathematical-Library/dp/0821844199" rel="nofollow">A (terse) introduction to linear algebra</a>» (Katznelson) to be a much better book than Axler. It's part of the Student Mathematical Library and published by the AMS.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25893#25893 Hans Lundmark 2010-05-25T15:32:11Z 2010-05-25T15:32:11Z

<p>For (applied) ODEs: <em><a href="http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/" rel="nofollow">Nonlinear dynamics and chaos</a></em> by Steven Strogatz.</p> <p>A very inspiring book! The explanations are crystal clear with lots of pictures. And it's funny too – the "Romeo and Juliet" illustration of 2-dimensional linear systems (Section 5.3) is a classic.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25935#25935 kdw 2010-05-25T21:05:13Z 2010-05-25T21:05:13Z

<p>Siegfried Bosch, Lineare Algebra</p> <p>It's a very elegant, concise but beautifully written approach to Linear Algebra, and I love it.</p> <p>Unfortunately for people who don't speak German, it has never been translated. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25951#25951 Andrew L 2010-05-25T23:36:39Z 2010-05-25T23:36:39Z

<p>I plan to post a complete reading list for undergraduates and graduate students at my blog this summer with my commentaries,but here's one I think that's available online and doesn't get nearly enough credit despite the fame of it's author: Gilbert Strang's <em>Calculus</em>. I wouldn't use anything else for a regular,non-honors calculus course. Carefully written,beautifully motivated with TONS of creative and SIGNIFICANT applications. I hope one day Strang finds the time to write a second edition-I have a list of improvements to suggest. </p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25953#25953 supercooldave 2010-05-25T23:45:24Z 2010-05-25T23:45:24Z

<p><a href="http://www.amazon.co.uk/Topics-Algebra-I-N-Herstein/dp/0471010901/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1274831031&amp;sr=8-1" rel="nofollow">Topics in Algebra</a> by I. N. Herstein. A new edition will be coming out this year.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/25964#25964 Tom Stephens 2010-05-26T03:45:02Z 2010-05-26T03:45:02Z

<p><a href="http://www.amazon.com/Linear-algebra-differential-equations-Bentley/dp/0030802997" rel="nofollow">Linear Algebra with Differential Equations: Bentley and Cooke 1973</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/30853#30853 kh 2010-07-07T04:18:24Z 2010-07-11T08:00:51Z

<p>Ireland and Rosen, <a href="http://www.amazon.com/Classical-Introduction-Modern-Graduate-Mathematics/dp/038797329X" rel="nofollow"><em>A Classical Introduction to Modern Number Theory</em></a> is a great second course in number theory. In spite of being part of "Graduate Texts in Mathematics" series and unlike Rudin's <em>Real and Complex Analysis</em> (see a comment above), this is a book at the undergraduate level. It only presupposes undergraduate algebra as in Herstein <em>Topics in Algebra</em> or M. Artin's <em>Algebra</em>, undergraduate analysis like in Rudin's <em>Principles of Mathematical Analysis</em> and basic number theory. In fact it recalls or proves many of the necessary results in each of those fields. <em>A Classical Introduction to Modern Number Theory</em> bridges the gap between basic number theory (that covers modular arithmetic, Fermat's little theorem and QR) and books like Lang's <em>Algebraic Number Theory</em> or Cassels and Fröhlich.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/30855#30855 BlueRaja 2010-07-07T04:41:29Z 2010-07-07T04:41:29Z

<p><a href="http://www.amazon.com/Concrete-Approach-Abstract-advanced-mathematics/dp/048663647X/ref=sr_1_3?ie=UTF8&amp;s=books&amp;qid=1278477374&amp;sr=8-3" rel="nofollow">A Concrete Approach to Abstract Algebra</a> by W. W. Sawyer ($6 on Amazon!)</p> <p>Though it goes a bit slow at times, it is by far the simplest, most intuitive book on Abstract Algebra in existence. Written for the non-mathematician, it does a great job of teaching the subject in simple, easy-to-understand prose. I couldn't put it down!</p> <p>There are also two chapters on linear algebra, leading up to the final chapters, <em>"vectors over fields"</em> and <em>"fields regarded as vector-spaces"</em>.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31317#31317 Derek Jennings 2010-07-10T18:33:51Z 2010-07-10T18:33:51Z

<p>Introduction to Analytic Number Theory - Tom M. Apostol.</p> <p>When I bought this I really didn't want to put it down. It's a great book for exciting one's interest in the subject.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31319#31319 Kiochi 2010-07-10T18:45:53Z 2010-07-11T14:33:29Z

<p>Atiyah and MacDonald, <em>Introduction to Commutative Algebra</em></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31324#31324 Mark Schwarzmann 2010-07-10T19:17:00Z 2010-09-15T17:20:19Z

<p><a href="http://www.amazon.com/Linear-Algebra-2nd-Kenneth-Hoffman/dp/0135367972" rel="nofollow">Linear Algebra / Hoffman &amp; Kunze</a> - A book that truly develops linear algebra in a gradual manner. It starts with a basic discussion of systems of linear equations, matrices, Gaussian elimination, etc. and gradually progresses to the more abstract theory. Eventually it even touches upon subjects such as tensor products, the exterior algebra and the Grassmann ring. In short, it manages to cover a lot of linear algebra in a very leisurely and clear manner. I think that this is the quintessential example of a how an undergraduate level math book should be written. The only thing I don't like about it is the fact that quotient spaces aren't mentioned throughout the book (they're mentioned in the appendix, though).</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31325#31325 Mark Schwarzmann 2010-07-10T19:18:07Z 2010-07-10T22:42:12Z

<p><a href="http://books.google.com/books?id=3U7tresTD1AC&amp;printsec=frontcover&amp;dq=lebesgue+integration+on+euclidean+space&amp;source=bl&amp;ots=ZFuDMZPPWg&amp;sig=ypt3ZcVvmdowz8uUXc_aNcHGCA8&amp;hl=en&amp;ei=Psc4TJfmO47vOZ-IsYoK&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=2&amp;ved=0CBoQ6AEwAQ#v=onepage&amp;q&amp;f=false" rel="nofollow">Lebesgue Integration on Euclidean Space / Frank Jones</a> - an extremely readable book on Lebesgue theory on $\mathbb{R}^n$ (lots of figures and geometric intuition). He constructs Lebesgue measure in a very down-to-earth manner, much more explicitly than other more abstract constructions (via Caratheodory's extension theorem or Riesz's representation theorem). In my experience, it's best to first study Lebesgue measure on $\mathbb{R}^n$ and only then point out that it's merely one instance of the general theory of measures, which is the way this book is written. It can't compare with the "tougher" books on measure theory (e.g. Big Rudin) since it doesn't discuss the Radon-Nikodym theorem and many other important theorems in measure theory, but then again the book is clearly intended for an undergraduate audience, and as for Lebesgue theory on Euclidean spaces, it provides a pretty complete picture.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31335#31335 Eivind Dahl 2010-07-10T22:04:41Z 2010-07-10T22:04:41Z

<p>Algebra: Chapter 0, Paolo Aluffi</p> <p>Best book on algebra I've had my hands on yet, and I love how it uses category theory. I wouldn't mind having a course taught from this one. Topics from group theory all the way through field theory, linear algebra, and homology. This book deserves more attention!</p> <p><a href="http://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1278799249&amp;sr=8-1" rel="nofollow">http://www.amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/0821847813/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1278799249&amp;sr=8-1</a></p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31375#31375 Amitesh Datta 2010-07-11T08:23:51Z 2010-07-11T08:23:51Z

<p>I would recommend Walter Rudin's classic text entitled <em>Real and Complex Analysis</em> for the mathematically mature undergraduate student. (Of course, this should really only be read after one has familiarity with most of Rudin's earlier textbook: <em>Principles of Mathematical Analysis</em>.)</p> <p>The beautiful thing about this book is that nearly every example or result stated by Rudin is used in "the big picture". As you progress through the book, you really start to appreciate the magic Rudin has used to weave together analysis in a manner that is rarely done in other textbooks. </p> <p>While Rudin states in his preface that the textbook is intended as a first year graduate course on the subject of analysis, I believe that it is quite possible for a mathematically mature undergraduate to follow it. There is no assumption in the book that the reader has any familiarity with linear algebra, abstract algebra, general topology etc. beyond that which was covered in the first seven chapters of his earlier book, but realistically one would like to have at least mastered the basics of these areas before attempting to delve deeper into the analysis covered in this book. (There is a chapter on Banach algebras, but all the necessary abstract algebra here is developed from scratch.)</p> <p>The textbook is indeed challenging with plenty of exercises, but it is not something with which a student with the correct prerequisites should have tremendous difficulty. Furthermore, if a student can successfully read most of the book, he is well-equipped to go deeper into most branches of modern analysis, and should find the other classic texts in the subject, for instance Royden's <em>Real Analysis</em>, Bartle's <em>The Elements of Integration and Lebesgue Measure</em> etc., very easy to follow.</p> <p>The Amazon page for this book can be found <a href="http://www.amazon.com/Complex-Analysis-International-Applied-Mathematics/dp/0070542341" rel="nofollow">here</a>.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31398#31398 David Corwin 2010-07-11T11:41:35Z 2010-07-11T11:41:35Z

<p>Algebraic Theory of Numbers, by Pierre Samuel</p> <p>Assumes only elementary knowledge of group and ring theory (even less than a complete undergraduate course which covers Galois theory) and develops algebraic number theory, a beautiful subject which puts much elementary number theory into an interesting perspective.</p>

http://mathoverflow.net/questions/761/undergraduate-level-math-books/31401#31401 Amitesh Datta 2010-07-11T12:28:32Z 2010-07-11T12:28:32Z

<p>I would also recommend the book entitled <em>Analysis on Manifolds</em> by James Munkres. I think that this is a good undergraduate textbook in mathematics for any student wishing to pursue multivariable calculus in greater depth. My only complaint is that Munkres often chooses to include details which can be seen easily after a little bit of thought. Perhaps this can be viewed as an effort to show the student how to "properly do analysis": doing analysis, just like doing any other branch of mathematics, requires you to carefully apply definitions and theorems, and it is important for the student to appreciate this early in his/her mathematical learning. </p> <p>That said, the book is an excellent text overall for "advanced calculus". The student will need to be familiar with single variable analysis and perhaps some linear algebra. (Even a rudimentary knowledge of linear algebra will do since Munkres develops most of the necessary theory from scratch.) </p> <p>Roughly speaking, the book splits into two parts. The first part covers most of the results students see when doing multivariable calculus that are stated "without proof" in their texts. For example, "the equality of the mixed partials", "double integrals can be done in any order", "a bounded function is Riemann integral if and only if it is continuous almost everywhere", "the change of variables theorem" etc., are (very) imprecise forms of some of the results Munkres establishes.</p> <p>In the second part of the book, manifolds and their theory are introduced. Thus, for example, a rudimentary introduction to tensors is given, and this is supplemented by the basic theory of differential forms, the De Rham groups (of the punctured plane), Stokes' theorem etc. </p> <p>I think that the exposition could be tightened: if you actually pick up the book and really make an effort to read it, it is quite possible to finish the first half of the book in the space of a week (that is, approximately 200 pages in a week) simply because certain topics are explained in more detail (at least in my opinion) than necessary. (One example is Munkres' proof of the linearity, monotonicity, additivity etc. of the Riemann integral. This is proved in three contexts <em>separately</em>: the case of the integral over a rectangle, that over a bounded set, and that of improper integrals, when essentially the proofs can be left as relatively easy exercises in some cases.)</p> <p>As the above comments suggest, I think that this is an excellent book for undergraduate students, but perhaps less so for graduate students. (Spivak's <em>Calculus on Manifolds</em> is good for both undergraduate and graduate students, in my opinion, but <em>some people</em> may suggest that it is too hard for undergraduates.) And after reading this book, you should have more than enough preparation to read more advanced texts such as William Boothby's <em>An Introduction to Differentiable Manifolds and Riemannian Geometry</em>. </p>