Comment author:Epictetus
10 February 2015 11:23:20AM
*
12 points
[-]

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.

How can baby rudin possibly be recommended in almost all use cases there is something better -_-, less wrong is supposed to give good advice not status-signaling type.

Rudin = Bourbaki and I thought we were anti-bourbaki here

Alternatives: Abbot & Bressoud combo(has mathematica code), Pugh, or Strichartz's book(the one patrick says is good)

Comment author:Epictetus
15 April 2015 02:47:20AM
1 point
[-]

How can baby rudin possibly be recommended in almost all use cases there is something better -_-, less wrong is supposed to give good advice not status-signaling type.

I recommend Rudin because he dives right into the topology and metric space approach. It's a lot easier to pick it up when it's used to develop the familiar theory of calculus. It also helps put a lot of point-set topology into perspective. I appreciated it once I started studying functional analysis and all those texts basically assumed the reader was familiar with the approach. The problems are great to work through and the terseness is a sign of things to come for a reader who wants to go on to advanced texts.

There is a caveat. Rudin is not a good text for a student's first foray into the rigors of real analysis. IF one has already seen a rigorous development of calculus, Rudin bridges the gap with a minimum of fluff. If not, the reader is better served elsewhere.

I'm no expert in undergraduate math texts so maybe there's something else that works better. I read Rudin on my own in undergrad and with my background at the time I got a lot out of it, so I'm recommending it.

Rudin = Bourbaki and I thought we were anti-bourbaki here

Bourbaki has its place. There comes a time when you need a good reference for the general theory and that's where the Bourbaki style shines. It makes for bad pedagogy and is cruel to foist upon beginners, but on the other hand good pedagogical books tend to limit their scope and seldom make good references.

What? Did I miss an anti-Bourbaki fatwa? The one mention of their name in the post does not come close to a general stance on Bourbaki, and in any case there must be someone on the site who likes them. In fact, here's one.

Just use patrick's recs for analysis and use yours for pretty much the other stuff(strang for lin algebra?). No serious person would recommend baby rudin give me a break.

not meant for learning except for stuff like lang, conversations like this deserve a thread. sleep apnea related sleep deprivation is hitting me so i will update this later with more info

if less wrong is to have any aesthetic imo we should be able to keep mathematical orientations like this, i'm interested in Eliezer's opinions on this

Comment author:pinyaka
28 July 2015 08:01:42PM
0 points
[-]

I purchased Shilov's Linear Algebra and put it on my bookshelf. When I actually needed to use it to refresh myself on how to get eigenvalues and eigenvectors I found all the references to preceding sections and choppy lemma->proof style writing to be very difficult to parse. This might be great if you actually work your way through the book, but I didn't find it useful as a refresher text.

Instead, I found Gilbert Strang's Introduction to Linear Algebra to be more useful. It's not as thorough as Shilov's text, but seems to cover topics fairly thoroughly and each section seems to be relatively self contained so that if there's a section that covers what you want to refresh your self on, it'll be relatively self contained.

Comment author:Boogie
19 May 2015 11:15:54PM
0 points
[-]

How about Piskunov? I've tried James Stewart, Thomas Finn and Guidorizzi before but now I'm studying through Piskunov and I think it is a good one. But since I didn't finished already I'm more inclined ti hear what is good and bad with this book.

## Comments (327)

Controversial*12 points [-]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 thingsandbe 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 MethodsRecommendation: 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 EquationsRecommended: Ordinary Differential Equations

Author: Vladimir Arnold

Contenders:

Coddington,

An Introduction to Ordinary Differential Equations: Solid intro from the author of one ofthetexts 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.Updated, thanks!

*1 point [-]How can baby rudin possibly be recommended in almost all use cases there is something better -_-, less wrong is supposed to give good advice not status-signaling type.

Rudin = Bourbaki and I thought we were anti-bourbaki here

Alternatives: Abbot & Bressoud combo(has mathematica code), Pugh, or Strichartz's book(the one patrick says is good)

I recommend Rudin because he dives right into the topology and metric space approach. It's a lot easier to pick it up when it's used to develop the familiar theory of calculus. It also helps put a lot of point-set topology into perspective. I appreciated it once I started studying functional analysis and all those texts basically assumed the reader was familiar with the approach. The problems are great to work through and the terseness is a sign of things to come for a reader who wants to go on to advanced texts.

There is a caveat. Rudin is not a good text for a student's first foray into the rigors of real analysis. IF one has already seen a rigorous development of calculus, Rudin bridges the gap with a minimum of fluff. If not, the reader is better served elsewhere.

I'm no expert in undergraduate math texts so maybe there's something else that works better. I read Rudin on my own in undergrad and with my background at the time I got a lot out of it, so I'm recommending it.

Bourbaki has its place. There comes a time when you need a good reference for the general theory and that's where the Bourbaki style shines. It makes for bad pedagogy and is cruel to foist upon beginners, but on the other hand good pedagogical books tend to limit their scope and seldom make good references.

I agree with this post much more. My concern was more ability to learn the subject & less wrong aesthetic in this direction which I think is correct.

What? Did I miss an anti-Bourbaki fatwa? The one mention of their name in the post does not come close to a general stance on Bourbaki, and in any case there must be someone on the site who likes them. In fact, here's one.

*0 points [-]Just use patrick's recs for analysis and use yours for pretty much the other stuff(strang for lin algebra?). No serious person would recommend baby rudin give me a break.

Echoing Hairyfigment here, what is wrong with Bourbaki?

*1 point [-]not meant for learning except for stuff like lang, conversations like this deserve a thread. sleep apnea related sleep deprivation is hitting me so i will update this later with more info

if less wrong is to have any aesthetic imo we should be able to keep mathematical orientations like this, i'm interested in Eliezer's opinions on this

I purchased Shilov's Linear Algebra and put it on my bookshelf. When I actually needed to use it to refresh myself on how to get eigenvalues and eigenvectors I found all the references to preceding sections and choppy lemma->proof style writing to be very difficult to parse. This might be great if you actually work your way through the book, but I didn't find it useful as a refresher text.

Instead, I found Gilbert Strang's Introduction to Linear Algebra to be more useful. It's not as thorough as Shilov's text, but seems to cover topics fairly thoroughly and each section seems to be relatively self contained so that if there's a section that covers what you want to refresh your self on, it'll be relatively self contained.

How about Piskunov? I've tried James Stewart, Thomas Finn and Guidorizzi before but now I'm studying through Piskunov and I think it is a good one. But since I didn't finished already I'm more inclined ti hear what is good and bad with this book.