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/mathmajors/mathgradschoolresources /
General Notes/Remarks

Anything lowerdivision: 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 lowutility 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 referencerequest 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 subjectdependent.

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 lowerdivision 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.
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 subjectspecific 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 honorslevel course, or if you want to see a more “rigorous” exposition of the topics from Stewart.

Apostol,
Calculus, Vol. 1: OneVariable 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 indepth 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 Physicsmotivated 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 nonsingular.

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 rowechelon form.
Discrete Mathematics and Proofs

Rosen,
Discrete Mathematics and Its Applications
 Huge variety of topics, good prep for many Mathrelated 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 lowerdivision 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 (setbuilder 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)
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 defacto standard, plus it also serves as an encyclopaedic reference.

Beachy and Blair,
Abstract Algebra
 A good undergraduatelevel reference.

Matthew Salomone
Abstract Algebra Series
 Fantastically wellmotivated 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
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 wellmotivated!
Standard Topics
 The reals as an ordered field, construction
 Metric spaces, basic topology
 The RiemannStieltjes 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 rootfinding, Newton’s method, least squares.
Topology
PointSet 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 übercompressed review of pointset.
Algebraic Topology

Munkres,
Topology
 Mostly pointset, 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 undergraduatelevel series, just be aware that he expresses some extremely nonmainstream views in his other videos!
Differential Geometry / Manifolds

Spivak,
Calculus on Manifolds
 A good followup 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.
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