Recommendations: Undergraduate Resources

What To Do as an Undergraduate

I defer to this excellent page from the math department at UC Irvine: https://www.math.uci.edu/math-majors/math-grad-school-resources /

General Notes/Remarks

• Anything lower-division: Khan Academy. Just be sure to actually work the problems!

• On Youtube videos: many lectures and talks are being posted online these days, and these can vary in quality. This is slightly mitigated by the fact that you can watch them at 2x speed.

• On video lectures in general: they are a crutch! Use them to supplement and enrich your understanding, but passively watching a lecture is a low-utility activity when it comes to learning.

  The best resources are always materials from your own instructors and courses. If you do passively watch videos, it helps to engage yourself: write questions in your notes, timestamps/title of important topics, etc.

• MIT OCW: the video lectures (when available) are generally of very high quality. These cover a quite a few areas, and are also a good source of practice problems and exams.

• Look (or ask) for book recommendations for your topic/class on Math StackExchange (MSE) or MathOverflow (MO). Don’t be afraid to consult multiple books on the same topic! I’ve generally had good luck following the reference-request tags.

• If you find an author/lecturer/general source that you particularly like or learn well from, see what other content they have! If they cover other topics, be open to learning those – you never quite know what will be useful where.

• For lower division or introductory courses, try to obtain a textbook paired with a solutions manual. This is a great way to drill problems, check your answers, and identify your weaknesses. Beware typos!

• Don’t bother with sites like Chegg. The solutions there are often incorrect, and this can lead to academic integrity issues.

• For computational or engineering courses: Schaum’s Outlines can be useful, but the quality is very subject-dependent.

• Learn how to use a Computer Algebra System (CAS). Don’t use it as a crutch - just learn enough syntax so that you can quickly run “sanity checks” on your computations. Wolfram Alpha and Symbolab are good for quick checks in lower-division courses.

• Learn something like SageMath, which can do symbolic computations and can be used for quite a bit of number theory, linear algebra, group theory, and more.

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Preparing for Graduate School

• Introduction to Higher Mathematics
  • A great survey that highlights many different areas of advanced mathematics.
• Garrity, All the Mathematics You Missed: But Need to Know for Graduate School
  • If you’re thinking about grad school at all, read this! Even if you’re not, it’s a pretty good collection of mathematics that it’s good to at least be familiar with. You can also use this to get an idea of some of the major theorems and results in a variety of subfields.

Resources by Subject

Below are resources for specific courses/subjects; I’ve tried to roughly organize these by increasing complexity with respect to a typical undergraduate Math degree.

Legend

• – Textbooks
• – Videos or online lectures
• – Websites or other online collections of resources
• – Particularly excellent resources that I highly recommend
• Details – Expand these sections for notes on the mathematical content or subject-specific advice.

Lower Division

Calculus

Single Variable Calculus

• Stewart, Calculus: Early Transcendentals
  • What can I say? It’s a Calculus book, and it covers the standard curriculum. This one’s a good choice because there are a few solution manuals floating around for older editions.
• Spivak, Calculus
  • Exposition is a little more advanced, and closer to introductory real analysis. Good for an honors-level course, or if you want to see a more “rigorous” exposition of the topics from Stewart.
• Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra
  • A good alternative to Spivak.
• patrickJMT
  • Videos of how to solve many specific calculus and engineering problems
• Paul’s Online Notes
  • Follows the standard curriculum very closely, with many examples and in-depth explanations.

Multivariable / Vector Calculus

• Schey, Div, Grad, Curl, and All That: An Informal Text on Vector Calculus
  • Most textbooks introduce these operators in a very formal way; this text expands and motivates these definitions greatly.
• MIT OCW Denis Auroux
  • A very geometric approach, with lots of great imagery!
• Paul’s Online Notes
  • More fantastic notes from Paul!

Many Calculus books cover both single and multivariable, so most of the resources from the single variable section will be applicable here as well. However, these particular resources focus almost entirely on the multivariable setting.

Ordinary Differential Equations

• Goode and Annin, Differential Equations and Linear Algebra
  • Good precisely because it sets up the language of linear algebra first, making many concepts in ODEs much easier to explain and understand (e.g. solutions as eigenfunctions of derivative operators).
• MIT OCW: Differential Equations
  • Very Physics-motivated approach, for better or worse.

Linear Algebra

• Axler, Linear Algebra Done Right
  • This subject is usually taught as a bag of computational tricks and algorithms, which obscures the absolute beauty of the subject – this text motivates the theory nicely and shows how powerful it can be.
• Goode and Annin, Differential Equations and Linear Algebra
  • Good balance of rigor vs. brevity and computation vs. theory. Very concise, gives you what you need to start calculating, but also takes time to list vector space axioms, mentions fields, a nice way of viewing the determinant formula, and (best of all) lists of many conditions that are equivalent to a matrix being singular or non-singular.
• Anton, Elementary Linear Algebra
  • Has an entire chapter on many cool applications of Linear Algebra – things like graph theory, computer graphics, and Google’s Pagerank algorithm. Also has a lot of “historical note” blurbs that are pretty interesting.
• Strang, Introduction to Linear Algebra
  • Strang is a giant in the world of linear algebra, so it’s worth seeing how he approaches the subject.
• MIT OCW: Linear Algebra with Gilbert Strang
  • Again, it’s Strang, so worth checking out!
• JJ’s Nullspace Trick
  • Many computations in linear algebra boil down to computing the nullspace of a matrix, and this is an excellent shortcut that lets your write the basis of the nullspace of a matrix almost directly from its reduced row-echelon form.

Discrete Mathematics and Proofs

• Rosen, Discrete Mathematics and Its Applications
  • Huge variety of topics, good prep for many Math-related computer science courses, also just a good survey of many topics at an introductory level.
• Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction
  • Absolutely excellent presentation of recurrence relations, mirroring how solutions of differential equations are found. Also has a good presentation of how to commute quantifiers.
• Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions
  • Good for a solid introduction to proofs.
• UC Berkeley’s EECS 70 Course
  • A wonderful lower-division course for which excellent notes have been posted for a number of years.

• Bender and Orszag Advanced mathematical methods for scientists and engineers

• Graham, Knuth and Patashnik, Concrete Mathematics
  • An encyclopaedic reference for a huge swath of discrete Mathematics.

Standard Topics

• Basic logic (truth tables, quantifiers, implications, contrapositive and converse, induction)
• Basic set theory (set-builder notation, venn diagrams, cartesian products)
• Functions (injectivity/surjectivity, inverse images)
• Relations (partial orders, equivalence relations)
• Basic combinatorics (permutations and combinations, inclusion/exclusion, pigeonhole principle)
• Recurrence relations
• Graphs (Königsberg problem, Hamiltonian/Eulerian cycles)
• Number theory (divisibility, modular arithmetic, the Euclidean algorithm)
• Generating functions
• Probability (odds for dice rolls and cards, distributions, Stirling’s approximation)

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Upper Division

Combinatorics

• Wilf, Generatingfunctionology
  • Freely provided by Wilf on his site, exposition is excellent and it provides a comprehensive overview of how to work with generating functions.
• Bona, A Walk Through Combinatorics

Algebra

Abstract Algebra

• Dummit and Foote, Abstract Algebra
  • Essentially the de-facto standard, plus it also serves as an encyclopaedic reference.
• Beachy and Blair, Abstract Algebra
  • A good undergraduate-level reference.
• Matthew Salomone Abstract Algebra Series
  • Fantastically well-motivated series, covers the equivalent of an entire year of material that naturally leads into Galois theory.
• Benedict Gross, Lectures at Harvard
  • Very clear with lots of examples.

Category Theory

• Talk on Category Theory by Tom LaGatta

Expected in some classes, but often assumed. Used in Algebraic Topology and Algebraic Geometry heavily. You can also find many full lectures online by people like Steve Awodey, Bartosz Milewski, and Eugenia Cheng.

Galois Theory

• Matthew Salomone, Galois Theory Lectures
  • A portion of his Algebra series, the exposition is fantastic because the series follows a cohesive narrative that introduces some of the major results and benefits of Galois theory early on. Highly recommended.

Analysis

Real Analysis

• Rudin, Principles of Mathematical Analysis

  • Essentially a standard in undergraduate real analysis, written in a very terse style but covers a great deal of material. Often referred to as “Baby Rudin”.

  • Extra notes/commentary/suggested exercises from George Bergman.

• Francis Su, Lectures from Harvey Mudd College
  • Recorded lectures, extremely clear and well-motivated!

Standard Topics

• The reals as an ordered field, construction
• Metric spaces, basic topology
• The Riemann-Stieltjes Integral
• Sequences and series, Cauchy sequences and completeness
• Limits and continuity, pointwise and uniform convergence
• The Mean Value Theorem
• Measure theory and the Lebesgue integral

Complex Analysis

• Brown and Churchill, Complex Variables and Applications
  • Good overview of computational techniques at an undergrad level.
• Needham, Visual Complex Analysis
  • Absolutely phenomenal book! The exposition and imagery is truly excellent, although this is perhaps not the best book for learning computations.

Standard Topics

• Arithmetic with complex numbers, roots of unity, the geometry of $\CC$
• The complex integral
• Residue theorems
• Analytic vs. holomorphic vs. complex differentiable

Numerical Analysis

• Burden, Numerical Analysis
  • Has good info on fixed point theory and root-finding, Newton’s method, least squares.

Topology

Point-Set Topology

• Munkres, Topology
  • A standard - the good stuff starts about 10 chapters in, everything before that is aimed at providing a solid grounding in set theory and proofs.
• Lee, Introduction to Smooth Manifolds
  • The appendix has a great über-compressed review of point-set.

Algebraic Topology

• Munkres, Topology
  • Mostly point-set, but introduces things like the fundamental group in the later chapters.
• Hatcher, Algebraic Topology
  • Love it or hate it, this seems to be the standard reference!
• NJ Wildberger, Introduction to Algebraic Topology
  • A good undergraduate-level series, just be aware that he expresses some extremely non-mainstream views in his other videos!

Differential Geometry / Manifolds

• Spivak, Calculus on Manifolds
  • A good follow-up to Spivak’s Calculus book, the exposition is at an undergraduate level. Worth checking out if you like his style.
• Frederic Schuller, International Winter School on Gravity and Light 2015
  • This guy is just phenomenal!

Number Theory

• LeVeque, Fundamentals of Number Theory
  • Short but good!

Standard Topics

• The prime counting function
• Modular arithmetic, solving equations in rings, multiplicative functions (like Euler’s totient function)
• The Chinese remainder theorem, Euler’s theorem, Fermat’s Little Theorem
• Quadratic reciprocity, the Legendre and Jacobi symbol

Algebraic Geometry

• Reid, Undergraduate Algebraic Geometry
  • Great introduction to the field, weaves in a lot of history and classical results.
• Cox, Little and O’Shea, Ideals, Varieties and Algorithms
• Cox, Little and O’Shea, Using Algebraic Geometry

Probability and Statistics

• Ross, A First Course in Probability

• Wasserman, All the Statistics: A Concise Course in Statistical Inference

It is useful to take (either beforehand or concurrently) introductory classes in both statistics and combinatorics.

Misc / Topics

Dynamics

• Milnor, Dynamics in One Complex Variable

• Arnol’d, Methods in Classical Mechanics

• Strogatz, Nonlinear Dynamics and Chaos

• Hale & Koçak, Dynamics and Bifurcations

Computer Science

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, Introduction to Algorithms (CLRS)
  • MIT OCW: Introduction to Algorithms is a good supplement.
• Sipser, Introduction to the Theory of Computation
  • The standard resource for an introduction to theoretical computer science. Covers things like computability, automata, and Turing machines.
• Cracking the Coding Interview
  • Covers a number of extremely typical CS interview questions, absolutely read if you are preparing to apply for internships or jobs.