I suppose I can think up a few tomes of eldritch lore that I have found useful (college math specifically):

**Calculus**:

Recommendation: Differential and Integral Calculus

Author: Richard Courant

Contenders:

Stewart, *Calculus: Early Transcendentals*:
This is a fairly standard textbook for freshman calculus. Mediocre overall.

Morris Kline, *Calculus: An Intuitive and Physical Approach*:
Great book. As advertised, focuses on building intuition. Provides a lot of examples that aren't the usual contrived "applications". This would work well as a companion piece to the recommended text.

Courant, *Differential and Integral Calculus* (two volumes):
One of the few math textbooks that manages to properly explain and motivate things *and* be rigorous at the same time. You'll find loads of actual applications. There are plenty of side topics for the curious as well as appendices that expand on certain theoretical points. It's quite rigorous, so a companion text might be useful for some readers. There's an updated version edited by Fritz John (*Introduction to Calculus and Analysis*), but I am unfamiliar with it.

**Linear Algebra**:

Recommended Text: Linear Algebra

Author: Georgi Shilov

Contenders:

David Lay, *Linear Algebra and its Applications*:
Used this in my undergraduate class. Okay introduction that covers the usual topics.

Sheldon Axler, *Linear Algebra Done Right*:
Ambitious title. The book develops linear algebra in a clean, elegant, and determinant-free way (avoiding determinants is the "done right" bit, though they are introduced in the last chapter). It does prove to be a drawback, as determinants are a useful tool if not abused. This book is also a bit abstract and is intended for students who have already studied linear algebra.

Georgi Shilov, *Linear Algebra*:
No-nonsense Russian textbook. Explanations are clear and everything is done with full rigor. This is the book I used when I wanted to understand linear algebra and it delivered.

Horn and Johnson, *Matrix Analysis*:
I'm putting this in for completion purposes. It's a truly stellar book that will teach you almost everything you wanted to know about matrices. The only reason I don't have this as the recommendation is that it's rather advanced and ill-suited for someone new to the subject.

**Numerical Methods**

Recommendation: Numerical Recipes: The Art of Scientific Computing

Author: Press, Teukolsky, Vetterling, Flannery

Contenders:

Bulirsch and Stoer, *Introduction to Numerical Analysis*:
German rigor. Thorough and thoroughly terse, this is one of those good textbooks that only a sadist would recommend to a beginner.

Kendall Atkinson, *An Introduction to Numerical Analysis*:
Rigorous treatment of numerical analysis. It covers the main topics and is far more accessible than the text by Bulirsch and Stoer.

Press, Teukolsky, Vetterling, Flannery, *Numerical Recipes: The Art of Scientific Computing*:
Covers just about every numerical method outside of PDE solvers (though this is touched on). Provides source code implementing just about all the methods covered and includes plenty of tips and guidelines for choosing the appropriate method and implementing it. THE book for people with a practical bent. I would recommend using the text by Atkinson or Bulirsch and Stoer to brush up on the theory, however.

Richard Hamming, *Numerical Methods for Scientists and Engineers*:
How can I fail to mention a book written by a master of the craft? This book is probably the best at communicating the "feel" of numerical analysis. Hamming begins with an essay on the principles of numerical analysis and the presentations in the rest of the book go beyond the formulas. I docked points for its age and more limited scope.

**Ordinary Differential Equations**

Recommended: Ordinary Differential Equations

Author: Vladimir Arnold

Contenders:

Coddington, *An Introduction to Ordinary Differential Equations*:
Solid intro from the author of one of *the* texts in the field. Definite theoretical bent that doesn't really touch on applications.

Tenenbaum and Pollard, *Ordinary Differential Equations*:
This book manages to be both elementary and comprehensive. Extremely well-written and divides the material into a series of manageable "Lessons". Covers lots and lots of techniques that you might not find elsewhere and gives plenty of applications.

Vladimir Arnold, *Ordinary Differential Equations*:
Great text with a strong geometric bent. The language of flows and phase spaces is introduced early on, which becomes relevant as the book ends with a treatment of differential equations on manifolds. Explanations are clear and Arnold avoids a lot of the pedantry that would otherwise preclude this kind of treatment (although it requires more out of the reader). It's probably the best book I've seen for intuition on the subject and that's why I recommend it. Use Tenenbaum and Pollard as a companion if you want to see more solution methods.

**Abstract Algebra**:

Note: I am mainly familiar with graduate texts, so be warned that these books are not beginner-friendly.

Recommended: Basic Algebra

Author: Nathan Jacobson

Contenders:

Bourbaki, *Algebra*:
The French Bourbaki tradition in all its glory. Shamelessly general and unmotivated, this is not for the faint of heart. The drawback is its age, as there is no treatment of category theory.

Lang, *Algebra*:
Lang was once a member of the aforementioned Bourbaki. In usual Serge Lang style, this is a tough, rigorous book that has no qualms with doing things in full generality. The language of category theory is introduced early and heavily utilized. Great for the budding algebraist.

Hungerford, *Algebra*:
Less comprehensive, but more accessible than Lang's book. It's a good choice for someone who wants to learn the subject without having to grapple with Lang.

Jacobson, *Basic Algebra* (2 volumes):
Note that the "Basic" in the title means "so easy, a first-year grad student can understand it". Mathematicians are a strange folk, but I digress. It's comprehensive, well-organized, and explains things clearly. I'd recommend it as being easier than Bourbaki and Lang yet more comprehensive and a better reference than Hungerford.

**Elementary Real Analysis**:

"Elementary" here means that it doesn't emphasize Lebesgue integration or functional analysis

Recommended: Principles of Mathematical Analysis

Author: Walter Rudin

Contenders:

Rudin, *Principles of Mathematical Analysis*:
Infamously terse. Rudin likes to do things in the greatest generality and the proofs tend to be slick (i.e. rely on clever arguments that don't really clarify the thing being proved). It's thorough, it's rigorous, and the exercises tend to be difficult. You won't find any straightforward definition-pushing here. If you had a rigorous calculus course (like Courant's book), you should be fine.

Kenneth Ross, *Elementary Analysis: The Theory of Calculus*:
I'd put this book as a gap-filler. It doesn't go into topology and is rather straightforward. If you learned the "cookbook" approach to calculus, you'll probably benefit from this book. If your calculus class was rigorous, I'd skip it.

Serge Lang, *Undergraduate Analysis*:
It's a Serge Lang book. Contrary to the title, I don't think I'd recommend it for undergraduates.

G.H. Hardy, *A Course of Pure Mathematics*:
Classic text. Hardy was a first-rate mathematician and it shows. The downside is that the book is over 100 years old and there are a few relevant topics that came out in the intervening years.

*9 points [-]