Enough endless bickering and let's compile a list of good calculus books at various levels to point people to.
>>7653912
>>Single Variable Calculus
>Weak Students
"Calculus Made Easy" by Silvanus Thompson and Martin Gardner
"Calculus: An Intuitive and Physical Approach" (Dover) by Morris Kline
"Elementary Calculus: An Infinitesimal Approach" by Jerome Keisler (Uses infinitesimals)
"A First Course in Calculus" by Serge Lang
>Strong Students
"Calculus" by Spivak (Good mathematical exposition, poor motivation, no applications)
"Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra" by Apostol (Good motivation and problems)
"Introduction to Calculus and Analysis, Volume I" by Richard Courant and Fritz John (Good motivation and applications, very difficult problems)
>Classic References
"A Course of Pure Mathematics" by G. H. Hardy
"Introduction to Analysis of the Infinite", "Foundations of Differential and Integral Calculus" by Euler
>>Multivariable and Vector Calculus
>Weak
"Div, Grad, Curl, and All That: An Informal Text on Vector Calculus" by Schey
"Calculus of Several Variables" by Serge Lang
>Intermediate
"Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability" by Apostol
"Introduction to Calculus and Analysis, Volume II" by Richard Courant and Fritz John
>Advanced
"Advanced Calculus of Several Variables" (Dover) by C. H. Edwards Jr.
"Advanced Calculus: A Geometric View" by Callahan
"Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach" by Hubbard and Hubbard
"Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards
"Advanced Calculus" by Shlomo Zvi Sternberg and Lynn Harold Loomis (for the utterly fearless)
>>7653917
Is infinitesimals good or bad? What are the benefits/ disadvantages? I've only taken 1 semester of calc back in high school (dropped cause I did poorly.)
>>7653917
>>Analysis
>Single Variable (with Metric Spaces)
"Real Mathematical Analysis" by Pugh (weaker version of Rudin)
"Principles of Mathematical Analysis" by Rudin
"Mathematical Analysis" by Apostol
"Mathematical Analysis I & II" by Zorich
>Multivariable Analysis
"Functions of Several Variables" by Fleming
"Analysis on Manifolds" by Munkres
"Calculus on Manifolds" by Spivak
"Differential Forms and Applications" by do Carmo
>Fourier Analysis
"Fourier Series" (Dover) by Tolstov
"Fourier Analysis: An Introduction" by Stein & Shakarchi
"Fourier Analysis and its Applications" by Folland
"Fourier Analysis" by Körner
"Fourier Series and Integrals" by Dym and McKean
>Complex Analysis
"Visual Complex Analysis" by Tristan Needham (reference)
"Complex Analysis" by Stein & Shakarchi
"Functions of One Complex Variable" by Conway
"Complex Analysis" by Ahlfors
>Real Analysis (Measure Theory)
"Real Analysis: Measure Theory, Integration, and Hilbert Spaces" by Stein & Shakarchi
"Real Analysis" by Royden
"Real Analysis: Modern Techniques and Their Applications" by Folland
"Real and Complex Analysis" by Rudin
>>7654064
Infinitesimals are important to calculus, but you can do calculus without talking specifically about them. Overall, it's another way to think of calculus and worth a try if other methods of teaching didn't stick before.
>>7653917
Where's Rudin?
>>7653917
>poor motivation
>Good motivation
What does it mean?
>>7654064
>Is infinitesimals good or bad? What are the benefits/ disadvantages
Depends
>Benefits:
More natural and matches up with how calculus is informally explained and viewed especially in physics
Greatly simplifies proofs for standard calculus results compared to delta-epsilon approach
Better than just skipping proofs
>Disadvantages:
No calculus class officially uses it (they do appear once the teacher inevitably start hand waving)
Won't help you transition to Analysis but neither would any non-honors calculus class that skips proofs altogether
Still has the stigma from back before analysis saved calculus and they were questionable
>>7654500
It's not a calculus book and squarely in analysis territory.
>>7654064
Initially, prior to the 1900s, people did calculus by using infinitesimals informally. Gauss, Euler, and many of your other heroes worked like this (even Archimedes in his "Method of Mechanical Theorems"). Towards the 1900s there was a big movement to finally formalize logic, mathematics, and computability, in an effort to resolve paradoxes and confusion.
When it came time to formalize analysis it was clear from the get go that formalizing infinitesimals was going to be a problem (something that everyone including Newon and Archimedes had always known). Eventually a work-around was established where one makes arguments about "all real numbers greater than 0" in order to indirectly talk about infinitesimally small values (epsilon-delta proofs). This method became the standard.
Unfortunately no one managed to formalize infinitesimals themselves until the 1960's (Abraham Robinson). Regardless, the 1900s formalization craze (formally referred to as the foundational crisis in mathematics) never really caught on to the same extent outside of mathematics hence why physicists and engineers continue to apply calculus proofs and techniques in ways that many in mathematics consider formally incorrect. Those people are in fact still performing calculus in the infinitesimal style from the 1700s.
(cont.)
>>7654680
Since then there have been a number of new approaches to formalizing infinitesimals. Some of them require intuitionistic logic to work (they contain anti-classical axioms that render the systems inconsistent in classical logic). Non-Standard analysis is still the most popular approach within classical logic and in my opinion is morally better than standard analysis. Not because standard analysis is any harder than non-standard analysis (the proofs are really easy if you're comfortable with nested logical quantifiers) but because standard analysis tends to produce students who don't really understand the essence of the proofs they're writing, instead they've just memorized some process to do it.
>>7654492
This is true, but I would argue that calculus with infinitesimals is better.
>>7654516
>Won't help you transition to Analysis but neither would any non-honors calculus class that skips proofs altogether
This is wrong. The infinitesimals used in Keisler's book are precisely the ones formalized by Abraham Robinson's Non-Standard Analysis. As such they do generalize to analysis, topology, and even other fields like probability theory and calculus of variations (and there are textbooks for this as well). However, it's worth noting that the version of Analysis that most people talk about is the 1900's version that makes use of epsilon-delta proofs. Every proof made in this version of analysis can be made in non-standard analysis (the reals are just a subset of the hyperreals), the benefit of having non-standard analysis is that there exist very trivial, natural, and intuitive to accomplish the same things. A student who only learned infinitesimal analysis may be ill-equipped to facing an epsilon-delta proof but it would also be a waste of time since they have already proven the same things in a much nicer more modern system.
Here's some neat stuff about non-standard analysis in probability theory.
http://mathoverflow.net/a/107966
>>7654519
This is the case with Spivak as well (it's pretty much just a less formal version of Lay's Analysis textbook).
>>7654513
I don't agree with that anon's list but typically when one talks about motivation in this context they mean whether or not the book tells you what you're doing, and more importantly why you're doing it (which may or may not include important applications). This gives the reader a sense of direction and "motivates" them to understand the result. Books that are poorly motivated are typically designed to accompany some course lectures where it becomes the professor's job to motivate the work. I personally consider this very poor form but occasionally you run into books with very well written material that have either poor or absent motivation. I do not consider Spivak's Calculus to be one of those books (I find it severely lacking in the level of formalism).
>>7653917
Where is Stewart?
>>7654695
Spivak's book is not a Calculus book despite the title. A proper Calculus book focuses on a variety of tricks, techniques, and application because the goal of a calculus course is to teach the reader how to use calculus in order to solve other problems (It's basically the dumbed down engineer's version of analysis). For instance a Calculus course spends a lot of time on stuff like optimization and related rates, topics not covered in any analysis course (including Spivak).
>>7654702
The fact that anon is too pretentious to include a textbook used in the majority of universities out there should really tell you something about the quality of that list.
>>7654680
>Towards the 1900s there was a big movement to finally formalize logic, mathematics, in an effort to resolve paradoxes and confusion.
No no no, Cauchy and Weierstrass made calculus rigorous in the early 1800s (ie the 19th century) to resolve mounting number of counterexamples to calculus where the naive methods gave incorrect results. That foundational stuff happen separately after Cantor opened a can of worms about infinitieS.
>never really caught on to the same extent outside of mathematics hence why physicists and engineers continue to apply calculus proofs and techniques in ways that many in mathematics consider formally incorrect
Not true at all. Physicists REVOLTED and threaten to teach the blind methods of calculus in their intro physics classes if freshman calculus didn't pander to them around the mid 1950s. Sloppy physics has been increasing ever since.
>>7654704
>It's basically the dumbed down engineer's version of analysis
No metric spaces -> not analysis. It's the calculus book for people would already have some incomplete calculus exposure like in a non-AP calculus class in high school or are supplementing another book with it at the same time.
>optimization and related rates, topics not covered in any analysis course (including Spivak).
Did you miss
>poor motivation, no applications
Yeah, Spivak is a shittier version of Apostol or Courant/John but its fanboys won't shut up if it's not on the list.
>Stewart
A terrible textbook that teaching you nothing but pattern matching to previous worked problems.
>>7654708
>No no no, Cauchy and Weierstrass made calculus rigorous in the early 1800s (ie the 19th century) to resolve mounting number of counterexamples to calculus where the naive methods gave incorrect results. That foundational stuff happen separately after Cantor opened a can of worms about infinitieS.
You're right. For some reason I was under the impression that Weierstrass didn't formalize this approach until after the reals were formalized in the later 1800s (by many parties). Though the foundational stuff happened for many reasons. Computability for instance (in particular formalizing algorithms) was another reason for needing a stronger level of formalism. My understanding is that there was also some general "atmosphere" of doubt developing which eventually led to Dedekind developing his Dedekind Cuts for the reals.
Could you post the source to your picture, it sounds like an interesting read.
>>7654718
Not all analysis courses cover analysis on general metric spaces. Though I agree that they should. For those unfortunate enough to attend such a course I recommend a quick read through Korner's Metric and Topological Spaces.
https://www.dpmms.cam.ac.uk/~twk/Top.pdf
>no applications
I've often heard Calculus textbooks characterized as "a book containing two pages of concepts and 998 pages of applications". Exaggeration aside, I agree that a proper calculus book should cover a variety of applications and techniques.
>fanboys won't shut up if it's not on the list.
Good point.
>A terrible textbook that teaching you nothing but pattern matching to previous worked problems.
I disagree. Funnily enough, a complaint I've heard on /sci/ about Stewart is that solving the problems all come down to applying some highly specialized clever trick that seems to be impossible to figure out if you haven't seen it before (and thus requires heavy use of the solutions manual). I worked through Stewart about 9 years ago and though I do remember there being an overabundance of such clever tricks I don't remember them requiring the solutions manual to figure out.
In my opinion Spivak's Calculus sits somewhere between a Calculus book and an Analysis book and I wouldn't recommend it for either.
>>7654727
>in particular formalizing algorithms
No, mathematicians really didn't give a shit about algorithms until later on in the 20th century and historically thought it was beneath them most of the time. Gauss is famous for throwing away the first instance of the FFT. The real question was whether 1st order statements could be proved mechanically (a foundational questions) that bled into a question on algorithms.
>My understanding is that there was also some general "atmosphere" of doubt developing which eventually led to Dedekind developing his Dedekind Cuts for the reals.
Actually, it was Cantor that first rigorous defined the reals in 1871. The only real doubt at the time was whether infinitesimals themselves were "physical" and apart of the "continuum".
>Could you post the source to your picture, it sounds like an interesting read.
http://www.maa.org/sites/default/files/pdf/CUPM/pdf/MAAUndergradHistory.pdf
>>7654763
>No, mathematicians really didn't give a shit about algorithms until later on in the 20th century and historically thought it was beneath them most of the time. Gauss is famous for throwing away the first instance of the FFT. The real question was whether 1st order statements could be proved mechanically (a foundational questions) that bled into a question on algorithms.
Hilbert's 10th problem is a problem of computability theory. In particular it required people to formalize the notion of a "process", which is a synonym for an algorithm or computable function. In general though it wasn't just mathematics that was being formalized at the time. Logic was as well and many different bizarre approaches to formalizing logic came up at the time (some also problematic in unexpected paradoxical ways).
>Actually, it was Cantor that first rigorous defined the reals in 1871. The only real doubt at the time was whether infinitesimals themselves were "physical" and apart of the "continuum".
I'm not sure when Cantor came up with Cauchy sequences nit Dedekind actually came up with Dedekind Cuts in 1858 but was hesitant to publish it until 1872. It's actually mentioned in the introduction to his paper on it (he also mentions encountering Cantor's paper while writing his). Check it out.
https://archive.org/stream/cu31924001586282
>http://www.maa.org/sites/default/files/pdf/CUPM/pdf/MAAUndergradHistory.pdf
Thank you!, at first impression it looks like a good read!
>>7654780
To add to this. Computability theory in its early days was known as recursive function theory (nowadays recursive function theory is a subset of computability) and was studied within mathematics. It also has close ties to mathematical foundations. I recommend checking it out, it's actually really cool (unless you hate the prospect of working entirely in the naturals).
You should add Calculus With Analytic Geometry by Simmons under weak students.
>>7654409
>Functional Analysis
>Weak
Introductory Functional Analysis with Applications by Erwin Kreyszig
Elements of the Theory of Functions and Functional Analysis (Dover books) by Kolmogorov and Fomin
>Serious
Functional Analysis - Introduction to Further Topics in Analysis by Stein and Shakarchi
Functional Analysis by Lax
Functional Analysis by Rudin
>>7653917
Hold up mah niggah, Spivak is poorly motivated? Since when?
>>7654681
Keisler really disappointed me. It was very informal, with a halfassed "this is ok because this one guy this one time made this one rigorous formulation" tacked on. I mean, I'm no foe to (rigorously formulated) nonstandard analysis. I read Arkeryd, Cutland and Henson's volume, and Keisler's contribution to THAT was pretty sweet. But his elementary text does nothing the classical alternative can not, and gives up much in the process. To my way of thinking, it's more an indoctrination device than a work of substance.
>>7654763
>An influential 1918 National Education Association report [26] had 14 subject area subcommittees, but mathematics was not one of them. Mathematics became an elective subject in many high schools. (It is worth noting that in Europe at this time, calculus was being made a mandatory subject in college-preparatory high schools.)
America, fuck yeah!
>>7656681
You don't know the half of it:
>With roots going back to Jean Jacques Rousseau and with the guidance of John Dewey, progressive education has dominated American schools since the early years of the 20th century. That is not to say that progressive education has gone unchallenged. Challenges increased in intensity starting in the 1950s (New Math), waxed and waned, and in the 1990s gained unprecedented strength. A consequence of the domination of progressivism during the first half of the 20th century was a predictable and remarkably steady decrease of academic content in public schools.
>The prescriptions for the future of mathematics education were articulated early in the 20th century by one of the nation's most influential education leaders, William Heard Kilpatrick. According to E. D. Hirsch, Kilpatrick was "the most influential introducer of progressive ideas into American schools of education." Kilpatrick was an education professor at Teachers College at Columbia University, and a protege of John Dewey. According to Dewey, "In the best sense of the words, progressive education and the work of Dr. Kilpatrick are virtually synonymous."
>Reflecting mainstream views of progressive education, Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. This point of view toward education comported well with the pedagogical methods endorsed by progressive education. Limiting education primarily to utilitarian skills sharply limited academic content, and this helped to justify the slow pace of student centered, discovery learning, the centerpiece of progressivism. Kilpatrick proposed that the study of algebra and geometry in high school be discontinued "except as an intellectual luxury."
>>7656354
>To my way of thinking, it's more an indoctrination device than a work of substance.
I agree that to some extent you're correct. My understanding is that Keisler wrote his text to promote the use of non-standard analysis as 'the' undergraduate calculus coursework. However, there is hope. Keisler also wrote a more advanced accompanying book that covers foundations rigorously (I believe this one is aimed at professors tasked with teaching the course). The accompanying book is still much more approachable than Robinson's original work (which spends the first large portion of the text doing a bunch of non-trivial work in first order logic) but it's also far more rigorous than his "easy book".
You can find his Foundations book here.
https://www.math.wisc.edu/~keisler/foundations.html
>>7656705
>According to Kilpatrick, mathematics is "harmful rather than helpful to the kind of thinking necessary for ordinary living." In an address before the student body at the University of Florida, Kilpatrick lectured, "We have in the past taught algebra and geometry to too many, not too few."
>Progressivists drew support from the findings of psychologist Edward L. Thorndike. Thorndike conducted a series of ""experiments"" beginning in 1901 that cast doubt on the value of mental discipline and the possibility of transfer of training from one activity to another. These findings were used to challenge the justification for teaching mathematics as a form of mental discipline and contributed to the view that any mathematics education should be for purely utilitarian purposes. Thorndike stressed the importance of creating many "bonds" through repeated practice and championed a stimulus-response method of learning. This led to the fragmentation of arithmetic and the avoidance of teaching closely related ideas too close in time, for fear of establishing incorrect bonds. According to one writer, "For good or for ill, it was Thorndike who dealt the final blow to the 'science of arithmetic.'"
>Kilpatrick's opinion that the teaching of algebra should be highly restricted was supported by other so called "experts". According to David Snedden, the founder of educational sociology, and a prominent professor at Teachers College at the time, "Algebra...is a nonfunctional and nearly valueless subject for 90 percent of all boys and 99 percent of all girls--and no changes in method or content will change that."
>In 1915 Kilpatrick was asked by the National Education Association's Commission on the Reorganization of Secondary Education to chair a committee to study the problem of teaching mathematics in the high schools. The committee included no mathematicians and was composed entirely of educators.
>>7653917
>>7654718
>>7654743
As someone who's been working through Stewart's (pic related), here's the opinions I've been given when I've asked about other textbooks.
On Stewart, I think it's a bad book, and I only plan to read the first volume (I'm already skilled in basic calculus anyway, I'm an English student who passed his A-levels). Here's a tip, a proper maths textbook should only have a couple of basic problems (simple shit like "integrate x^2 with respect to x") and then it should immediately move on to intermediate and then difficult stuff. If you can do the intermediate/difficult stuff, then you can automatically do the easy stuff - there's no need to give you 40 different easy questions on limits and then only 4-5 harder ones when you could just give 5 easy questions, 10 intermediate ones and then 5 hard ones. If you manage to do most of them then I fucking guarantee you know the easy shit anyway.
As for the complaints about the clever tricks. I half agree and half disagree. I completely agree that if you've not seen them before, you'll need to use the solutions manual heavily to be able to work out what's been done, however I also feel it worth pointing out that by majority, the clever tricks are worth fucking knowing and if your algebra is shitty enough to not already know said clever tricks, then you probably need a book like Stewart's that'll just drill you on them until you're forced to have them memorized.
No mention of Thomas' calculus? It's the one that my calculus lecturer recommends, although I don't recall his reasons, I've heard it's somewhere between Stewart and Apostle.
>"Calculus" by Spivak
Opinions on this are very mixed, it's either a God-tier book in general, or it's too much analysis for a calculus textbook and too little analysis for an analysis textbook. I've also heard it criticised for some of its less standard approaches, like defining the trigonometric functions in terms of their inverses.
cont
>>7656711
>"Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra" by Apostol
I can't remember much on this either, if memory serves it's the weakest of Apostol's books, I've heard his analysis is much better.
>"Principles of Mathematical Analysis" by Rudin
>"Real and Complex Analysis" by Rudin
I've heard mixed things about Rudin, it's either a trial by fire or it's pointlessly hard. I suppose it tells you something about the people on /sci/ that we never really see much discussion on textbooks outside of calculus textbooks. A couple of my lecturers have told me that they that keep copies of it handy, but never find anything of value when they want to use it as a reference or when they're setting homework problems.
>"Mathematical Analysis" by Apostol
My lecturers tell me it's God-tier, although I've never hear /sci/ mention it.
What about linear algebra textbooks? I've often heard Strang recommended, although I've also been told that since we worked out how to teach linear algebra about a hundred years ago, it doesn't really matter what book you have.
>>7656710
>Kilpatrick directly challenged the use of mathematics to promote mental discipline. He wrote, "No longer should the force of tradition shield any subject from scrutiny...In probably no study did this older doctrine of mental discipline find larger scope than in mathematics, in arithmetic to an appreciable extent, more in algebra, and most of all in geometry." Kilpatrick maintained in his report, The Problem of Mathematics in Secondary Education, that nothing in mathematics should be taught unless its probable value could be shown, and recommended the traditional high school mathematics curriculum for only a select few.
>It was not surprising that mathematicians would object to Kilpatrick's report as an attack against the field of mathematics itself. David Eugene Smith, a mathematics professor at Teachers College and renowned historian of mathematics, tried to stop the publication of Kilpatrick's report as a part of the Cardinal Principles of Secondary Education, the full report of the Commission on the Reorganization of Secondary Education, and one of the most influential documents for education in the 20th century. Smith charged that there had been no meeting of the math committee and that Kilpatrick was the sole author of the report. Moreover, Kilpatrick's committee was not representative of teachers of mathematics or of mathematicians. Nevertheless, Kilpatrick's report was eventually published in 1920 by the U.S. Commissioner of Education, Claxton, a friend of Kilpatrick.
>In the 1940s it became something of a public scandal that army recruits knew so little math that the army itself had to provide training in the arithmetic needed for basic bookkeeping and gunnery. Admiral Nimitz complained of mathematical deficiencies of would-be officer candidates and navy volunteers. The basic skills of these military personnel should have been learned in the public schools but were not. As always, education doctrines did not sit well with much of the public.
>>7656713
>Nevertheless, by the mid-1940s, a new educational program called the Life Adjustment Movement emerged from the education community. The basic premise was that secondary schools were "too devoted to an academic curriculum." Education leaders presumed that 60% or more of all public school students lacked the intellectual capability for college work or even for skilled occupations, and those students would need a school program to prepare them for every day living. They would need appropriate high school courses, including math programs, that focused purely on practical problems such as consumer buying, insurance, taxation, and home budgeting, but not on algebra, geometry, or trigonometry. The students in these courses would become unskilled or semiskilled laborers, or their wives, and they would not need an academic education. Instead they would be instructed in "home, shop, store, citizenship, and health."
>By 1949 the Life Adjustment Movement had substantial support among educators, and was touted by numerous federal and state education agencies. Some educators even suggested that in order to avoid stigmatizing the students in these programs, non-academic studies should be available to all students. Life Adjustment could meet the needs of all American students.
>However, many schools stubbornly clung to the teaching of academic subjects even when they offered life adjustment curricula as well. Moreover, parents of school children resisted these changes; they wanted their own children educated and not merely adjusted. They were sometimes joined by university professors and journalists who criticized the lack of academic content of the progressivist life adjustment programs.
>>7656712
>What about linear algebra textbooks? I've often heard Strang recommended, although I've also been told that since we worked out how to teach linear algebra about a hundred years ago, it doesn't really matter what book you have.
Shit, I forgot to mention Halmos.
>>7656714
>Changes in society at large also worked against the life adjustment agenda. Through the 1940s, the nation had witnessed tremendous scientific and engineering advances. By the end of the decade, the appearance of radar, cryptography, navigation, atomic energy, and other technological wonderments changed the economy and underscored the importance of mathematics in the modern world. This in turn caused a recognition of the importance of mathematics education in the schools. By the end of the 1940s, the public school system was the subject of a blizzard of criticisms, and the life adjustment movement fizzled out.
>Progressive education was forced into retreat in the 1950s, and even became the butt of jokes and vitriol. During the previous half century, enrollment in advanced high school mathematics courses, and other academic subjects, had steadily decreased, thanks at least in part to progressive education. From 1933 to 1954 not only did the percentage of students taking high school geometry decrease, even the actual numbers of students decreased in spite of soaring enrollments. The following table gives percentages of high school students enrolled in high school math courses.
>The "New Math" period came into being in the early 1950s and lasted through the decade of the 1960s. New Math was not a monolithic movement. According to a director of one of the first New Math conferences, "The inception of the New Math was the collision between skills instruction and understanding ...The disagreements between different entities of the New Math Movement were profound. Meetings between mathematicians and psychologists resulted only in determining that the two had nothing to say to each other."
>http://www.csun.edu/~vcmth00m/AHistory.html
>>7656711
>As someone who's been working through Stewart's (pic related)
The sticky is shit, use the wikia already
>>7656712
The problem with Rudin is that it's both very advanced and very hand-wavy at the same time. For his examples he will give a non-trivial problem and then give a five line solution where so much work has been hand-waved away that you would rather sit down and work through it yourself because you plainly just don't believe he's been careful enough. This isn't bad because it does mean you work much more/harder, but it is as you said "pointlessly hard" and most importantly it takes more time to work through. It is worth noting however that there is a big solutions manual available online (made by some grad student) that has a solution for every problem and many university courses in analysis tend to grab problems from Rudin, so it's worth using as a reference, at least for those reasons.
>>7656719
The picture is from 2011 when /sci/ was young and the sticky was new and growing, but yes you are correct. The sticky has long since been abandoned.
>>7656712
>since we worked out how to teach linear algebra about a hundred years ago
It's a common myth or urban legend that the basics subjects have been nearly the same for at least a century but it isn't true at all. School curricula didn't stabilize until after 1970's-1980's.
In 1920, this was the typical undergrad course offering (math majors just needed to take 6 courses):
Math 7- Elements of Analytic Geometry and Calculus (not open to math majors)
Math 8- Differential and Integral Calculus
Math 9- Advanced Plane Trigonometry and Spherical Trigonometry
Math 10- History of Mathematics
>Math 11- Determinants and the Theory of Equations
>Math 12- Determinants and Elimination
Math 13- Solid Analytic Geometry
Math 14- Infinite Series and Products
Advanced offerings (not all offered annually):
Math 15- Projective Geometry
Math 16- Modern Analytic Geometry
>Math 18- Quaternions and Vector Methods
Math 19- Differential Equations
Math 20- Advanced Calculus (basics of multivariable calculus, some special functions, some functions of a complex variable up to Cauchy’s theorem, and possibly a little partial differential equations)
Math 21- Foundations of Geometry
Math 25- Mathematical Analysis (a deeper look at series, representations of functions and other classical topics)
Math 26- Functions of a Complex Variable.
Linear Algebra we know today only became popular after WWII. In fact, much of Linear Algebra was developed only in the 20th century. For example, LU decomposition wasn't invented until 1948 by Turing.
>>7656733
>but it is as you said "pointlessly hard" and most importantly it takes more time to work through
It is not pointless, it is how you build up mathematical maturity. You ideally shouldn't even read his proofs and try to do them yourself first. Then see how he did them. If you're not at that point yet, filling in the gaps to his proofs is the catalyst that gets you there.
I'm sick of poor math students blaming the book rather than their own lack of skill and maturity when they don't get things. You're not a special snowflake, there are many people better than you out there. And no, they are not merely better because they've already learnt it before, many people learn analysis for the first time with Rudin.
>so much work has been hand-waved away that you would rather sit down and work through it yourself because you plainly just don't believe he's been careful enough
All real math books and papers are like this. They aren't sloppy, they assume the reader can filling in the skipped steps on their own.
>>7656757
Sure, it's important to build up mathematical maturity, but it's impractical to reinvent the wheel every step of the way. The ridiculous amount of mathematics out there means you would only ever scratch the surface if you approached every textbook like Rudin's.
>>7656712
>What about linear algebra textbooks? I've often heard Strang recommended
>>7656716
Depends on what you mean by "Linear Algebra"
>Matrix Algebra
>Weak
"Matrices and Linear Algebra" (Dover) by Schneider and Barker
>Average
"Matrices and Linear Transformations" (Dover) by Cullen
>Honors
"Linear Algebra Done Wrong" by Treil (also an intro to proofs and free online)
>Applied Linear Algebra
"Linear Algebra and Its Applications" by Strang
"Matrix Analysis and Applied Linear Algebra" by Meyer
>Numerical Linear Algebra
"Matrix Computations" by Golub and Van Loan
"Numerical Linear Algebra" by Trefethen and Bau III
"Matrix Analysis" by Horn and Johnson
>Finite Vector Spaces Theory
"Linear Algebra" by Shilov (Dover Book)
"Finite Dimensional Vector Spaces" by Halmos
"Linear Algebra" by Friedberg, Insel, and Spence
"Linear Algebra" by Hoffman and Kunze
"Linear Algebra Done Right" by Axler (best paired with Shilov)
>Advanced
"Linear Algebra and Its Applications" by Lax
"Advanced Linear Algebra" by Roman
>>7656719
>>7656734
Any thoughts for updating that guide then? I've kept myself alert to any advice I've been given on textbooks and I've also been particularly annoyed that the guide is clearly aimed at Americans, who are notorious for having university courses where they are forever trapped in calculus, I've even heard of horror stories where years 1-2 they do calculus with the engineers, only to redo it with only their fellow mathematicians in years 3-4.
As for the wiki, I dropped it as soon as I noticed that 'Proofs for THE BOOK' was listed as high school level, it fucking isn't - it's an insanity of mine that I'll not consider myself any sort of mathematician until I can read (and understand) that book cover to cover and even as a lowly first year bachelors' student (that is, quite some bit above high school), I don't even think I can understand all of its proofs that there are infinitely many prime numbers.
Gonna bump this up, I like threads that keep the underaged plebs out.
>>7657171
This thread is undergrad level.
>>7657073
In the US they do calculus separately from analysis.
Updating the sticky is a pain in the ass. Whoever designed it chose to set it up in the most cumbersome annoying way (editing the page is literally like editing a word document).
I also immediately dropped the wiki because the lists on it were all obviously written by one really pretentious guy.
>>7657073
>'Proofs for THE BOOK' was listed as high school level, it fucking isn't
But it is almost high school level. Here in my university it's often used for the 'Proseminar' for 3. Semester undergrad students.
In the 'Proseminar' everyone gets assigned one chapter of a low level book like this one, self-studies it and then every week someone presents the content of his chapter in a 2 hour talk to the other participants.
>>7657073
Proofs are high school level. "Proof from The Book" is just a reference to that that you will return to later as you learn more.
Do you bitch that 1st semester foreign language courses recommend you buy a dictionary even though you won't fully be able to use until years later too?
>>7657331
>"Proof from The Book" is just a reference to that that you will return to later as you learn more.
It's far from a reference appropriate for high school, most of the book is incomprehensible to college students.
>>7657073
>>As for the wiki, I dropped it as soon as I noticed that 'Proofs for THE BOOK' was listed as high school level, it fucking isn't
Calm down, it's a meme (a joke).
Arithmetic isn't typically Preschool level
Trigonometry isn't typically Primary School level
Ordinary Differential Equations isn't typically Secondary School level
Functional Analysis isn't typically Undergrad level
The original version of the list from before the wikia was fetus level, baby level, child level, teen level, manchild level.
>>7657767
Simple differential equations like exponential growth and the harmonic oscillator get covered in secondary school and (Linear) Functional Analysis is undergrad level.
>>7657797
>Simple differential equations like exponential growth and the harmonic oscillator get covered in secondary school
No one ever means that one day lecture on DE in calculus when they say ODEs.
>10 posters, 51 replies
there has been some new level of samefagging right here
I wonder why differential and integral calculus is still being taught in secondary school and college as "required math". Lambda calculus would be a more appropriate "calculus" to learn, given the prevalence of programming in almost every field.
Here you go.
>>7658036
>prevalence
>lambda calculus
Anon, I like functional programming as much as the next guy but it's just not nearly as prevalent as you claim it is.
> Anon, I like functional programming as much as the next guy but it's just not nearly as prevalent as you claim it is.
I wasn't specifically talking about functional programming. I meant programming in general.
>Calculus for physics and engineering
Courant or Apostol?
>>7656712
Rudin is actually fairly complete in most of his proofs. He's just very terse about it; he'll tell you exactly what you need to prove it and nothing more. He tends to develop exactly what's needed for landmark theorems so he can say use (14) and (13) to show that (a) satisfies (11) and thus by (15) we are done. You may need to work a little at it, but the ideas are very clear, and I usually don't have trouble following them.
If you want to see terseness done wrong, I'd look at Ahlfor's Complex Analysis. I get a headache trying to read that thing because it's not formatted well enough. His proofs are usually just him talking about some subject, mentioning some fact, and saying that ___ is clearly true while giving you little to no idea of how to approach the theorem as most of his rambling is purely descriptive and he hadn't developed the analytic tools to prove it.
On a side note, I like Apostol well enough. I used it as a supplement to Rudin for undergrad analysis because my professor often looked at topics out of Rudin.
Also, one of my favorite introductory analysis books is "Introduction to Topology and Modern Analysis" by George Simmons. It goes over general point set topology in the first 6 chapters, then it does a nice chapter on approximation (and its one of the few books I've seen mention Bernstein polynomials for approximation). After that he covers some algebra so he can develop Banach and Hilbert space theory. Then he finishes the book with a few chapters surveying Banach algebras. It's a nice introduction to some functional analysis.
What's a decent book for ito / stochastic calc?
>>7653917
wtf
who needs this shit
way too much calculus
>>7658375
>you
>in charge of reading
you're obviously not supposed to read every one, but the one that fits your needs
>>7658358
Assuming you know some measure theoretic probability:
Stochastic Calculus Models for Finance II: Continuous Time Models by Steven Shreve
Brownian Motion and Stochastic Calculus by Ioannis Karatzas and Steven Shreve
>>7653912
..why just calculus and not algebra or pre-calculus? if you're not solid in those, you will eat shit in calculus no matter what book you buy.
>>7658405
Because you have to be over the age of 18 to browse 4chan and there really isn't much of a difference between good books at this level nor is there much to do.
>Pre-algebra
The Method of Coordinates (Dover) by Gelfand, Glagoleva, and Kirillov
Functions and Graphs (Dover) by Gelfand, Glagoleva, and Shnol
>Elementary Algebra
Algebra by Gelfand and Shen
Elements of Algebra by Euler (Yeah, it's old but it's still incredible good)
>Trigonometry
Trigonometry by Gelfand and Saul
Advanced Trigonometry (Dover) by Durell and Robson
>Precalculus aka review/amalgamation of the above
Basic Mathematics by Lang (for strong students, the only 'rigorous' text at this level and a prep book for proofs)
Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry by Simmons
>>7658491
You must have a real boner for gelfand desu famitsu
Is his calculus of variations text any good?
>>7658406
There's little worth in doing a honors calculus book after a regular calculus book and there's absolutely no point in doing Apostol or Courant as a third calculus go over. Spend the time learning more math and return to analysis when you're ready.
This order is what I would recommend to go "math monk" but there isn't any canonical order in math so reorder as you see fit.
Step 1: Learn Proofs early. Everyone who learns them in college regrets not learning them sooner, even the Engineering students, and it's easy enough to do so first.
Step 2: Learn Matrix Algebra (>>7656771). It's easier than calculus and mostly doesn't depend on any calculus knowledge (there is one or two applications that do but you can come back to them later).
Step 3: Work through Apostol and/or Courant/John and do both of their volumes. If you find the going too tough then don't panic and switch to a easier calculus book like Lang or Keisler. You'll revisit the material when you get to analysis anyway.
Step 4: Learn ODEs. Tenenbaum/Pollard is great for self study and Simmons is the best for honor students or anyone interested in the history of math and science
Step 5: Read a book on Vector Spaces (>>7656771). This step will improve your ability to write abstract proofs
Break: Pick up Set Theory, Mathematical Logic/Foundations, Combinatorics, Graph Theory, Number Theory, Numerical Analysis, Differential Geometry in 2D/3D, or Probability if any of them interest you here.
Step 6: Suffer though PDEs: Haberman, Strauss, and/or John. You'll start to understand why analysis is so important.
Step 7: Study Abstract Algebra. Artin, Herstein's Topics, or Jacobson Basic Algebra Vol1 are all great. D&F is fine but a bit too encyclopedic.
Step 8i: Topology: Munkres, Lee, Kelley, or Willard (Honors) will suffice
Step 8a: Analysis (>>7654409). The order you go about this is up to you. Learning Topology first makes Analysis easier but Analysis serves as the motivation to go learn Topology
>>7658212
Neither, Stewart.
>>7658761
>Work through Apostol
Calculus?
>>7658405
Fuck OP, I hear you Anon. Most of the books prescribed to me by my college suck, but I can think of one excellent PreCalculus off the top of my head. This is one I found of my own accord, to use in addition to the materials recommended by school.
http://www.amazon.com/gp/product/0618052852?psc=1&redirect=true&ref_=oh_aui_detailpage_o05_s00
>>7658854
I have read Larson, now I want something more rigorous. Apostol or Courant?
>>7658761
This seems reasonable. It's definitely much better advice than the pic that wont die, but there's really no one best way/order to self-study math.
I will say I tried to start with baby rudin around 17 with no background but IB math HL and got baffled by simple concepts like compactness because of the abstraction and terse exposition.
Pic related still haunts my sexualized nightmares
>>7659022
Yeah, his calculus text.
After reading through this thread, I've decided I really need to go through Calculus by Apostal.
>>7659447
>Pic related still haunts my sexualized nightmares
...try opening a copy of Bourbaki.
>>7660126
Well, I've done Rudin cover to cover now that I know my shit. It was just a rather daunting experience for an inexperienced kid.
>>7658761
What parts of this would be most beneficial to a physics major. Currently a second year. I've taken a linear algebra course, single variable calc, multi variable and am doing differential equations currently.
>>7660216
most useful would be pdes and probability i would guess
>>7659447
Metric space = normed vector space minus the vector space
(a) Nr(p) the open ball of radius r around the point p
(b) limit point if there are other points arbitrarily close to it
(c) isolated because I can draw a ball around it and isolate it from the rest of E
(d) just the definition
(e) Let E be a subset of the whole space X and p is a point in E (p∈E⊂X). If you can find a open ball around p that's wholly contained in E then p is said to be an interior point (just like points in some object aren't on the surface if they they are fully surrounded by the object)
(f) just the definition
(g) just the definition
(h) just the definition
(i) E is bounded if I can find a ball of fixed radius that can fit E inside of it
(j) just the definition
What's so scary?
>>7659447
>>7660316
These definitions are bad and you should feel bad.
A metric space is just a set with a metric (a function satisfying a small list of properties) defined on it.
Then given a metric space one can use the metric to induce a topology on the set. This is done through open balls. Call this the metric topology induced by the metric.
Note that it's possible for a metric space to have many different typologies on it (though only one is induced by said metric). Furthermore, it's possible for a set to have many metrics on it and thus several induced metric topologies.
>>7660126
I'm not that guy but Rudin's lack of formalism and rigor is what bothers me most. Bourbaki is the exact opposite since the work is so pedantic that it's pretty much impossible to misunderstand.
>>7660216
Almost all of it. Do steps 1, 5-8. In the break section, Differential Geometry of Curves and Surfaces and Probability would be the most useful.
>>7660469
>A metric space is just a set with a metric (a function satisfying a small list of properties) defined on it.
Yeah, a normed vector space minus the vector space. Or a set with a "distance" defined between points.
>>7654718
The formalization of real analysis using topological concepts like metric spaces occurred after the formalization utilizing strictly epsilon-delta arguments on the real numbers without the general concept of a metric. Both approaches constitute analysis.
>>7660615
But just barely. Newton also used synthetic geometry to formulate his calculus but that doesn't make us learn it that way first.
>>7660589
Not all metric spaces are vector spaces. Your metric doesn't even have to be defined on a set with structure and operations. For instance you can define a metric on graphs.
Furthermore, because you have fallen into this retarded trap where you only see things through the lense of vector spaces, allow me to burst your bubble further and point out that given an inner product you can produce a metric but not all metrics are built out of inner products.
>>7660649
I'm not that guy but there are still universities and text books that don't cover analysis with topological concepts. The topological route is considered much easier but it requires more theory.
Also, true synthetic geometry didn't exist until much later. Newton only used the semi-formal Euclidean geometry that existed in his day.
>>7660714
Interestingly enough though, any metric space can be embedded isometrically into a normed vector space (so it doesn't get a vector space structure but it can be seen as a topological subspace of a normed vector space)
>>7660737
Interesting, from your wording I'm inferring that the embedding fails to form a subspace. Is the mapping injective? Does this functor have a name?
>>7660743
Well yes it's injective since it is an isometry (it is actually a homeomorphism onto its image) but what I meant is that the image is not a vector space.
I think it's easier to be convinced if you see the construction:
Consider a metric space (X,d).
Now let [math]E = C^0_b(X, \mathbb R)[/math] stand for the space of bounded real-valued continuous functions on X. E is a normed linear space for the sup norm.
Now, for each x in X, define [math]f_x: y \mapsto d(x,y)[/math]. Then, for each x and y in X, we have [eqn]||f_x - f_y||_{\infty} = \sup_{z \in X} |d(x,z) -d(y,z)| = d(x,y)[/eqn] by triangle inequality.
Now we're almost there but [math]f_x[/math] has no reason to be bounded. However, as seen above, [math]f_x -f_y[/math] is bounded for all x and y, so we fix a point a in X and we define [math]\phi: x \mapsto f_x - f_a[/math].
Now [math]\phi[/math] is an isometric homeomorphism from X to a subset of E. Now, the image itself [math]\{f_x -f_a, x \in X\}[/math] has no reason to be a linear subspace of E but it's still a neat factoid about metric spaces.
I don't think this thing has a name and I don't think it's a functor either
>>7656711
Me again... I'm trying to come up with a reasonable plan for the next few months.
My uni will be teaching me the following:
Calculus
'Core' algebra
Real Analysis
Complex Analysis and Integral Transforms
Linear Algebra
I'd like to read some books on these beforehand, and I'm trying to work out what and in what order.
I might as well finish what I've started, so I'll do the little of Stewart's first volume that I have left.
I'll then read a book on proofs (...did we not list any?). I'll go with How to Prove It: A Structured Approach by Velleman, any objections?
Afterwards, I suppose my choices are either a harder and more maths student focuses calculus book, or a linear algebra book.
From this thread, and other advice I've had, I suppose the calculus options are Spivak or Apostle. Maybe Thomas, but no one's said anything about it (...wasn't even in >>7653917).
Someone remind me, I'm sure I've heard that Apostle's calculus is his worst book and I think I know the pro/cons of Spivak, but what are the pros/cons of Apostle's calculus?
I don't even know where to fucking start for linear algebra >>7656771. I'm pretty sure Strang and Halmos are the only ones I've heard much about. Oh shit... I just remembered Lay's. Fuck... /sci/ educate me.
What remains is real/complex analysis... so that's Rudin, Apostol or Royden, right?
I'm thinking either Rudin or Apostol, remind me, what does each book cover?
>>7660819
>I don't even know where to fucking start for linear algebra >>7656771.
Do you know what systems of equations, matrix operations, Gaussian elimination (also known as row reduction), LU decomposition, determinants, eigenvectors and eigenvalues, and diagonalization are? If not, best start at matrix algebra.
>I'm pretty sure Strang and Halmos are the only ones I've heard much about. Oh shit... I just remembered Lay's.
Lay is an terribly overpriced matrix algebra book, Strang (the book with "its applications") is an applied book, and Halmos is a theory book. Theory and applied are self contained and can be done in any order. Numerical assumes you know the applied material.
>>7660714
>Not all metric spaces are vector spaces
"minus the vector space"
>>7661259
Not all metrics are realizable as inner products on a vector space, e.g. the discrete metric. You really need to stop trying to force vector spaces into this at all. It's a space with a way to quantify the distance between any two points.
>>7661299
MINUS, remove, without, not
>>7660475
What are good books for differential geometry and probability?
>>7661697
For probability:
The Art Of Probability: For Scientists and Engineers by Hamming (Supplement)
Introduction to Probability by Bertsekas and Tsitsiklis
An Introduction to Probability and Random Processes by Rota and Baclawski
Probability Models by Sheldon Ross (1st or 2nd book)
Probability and Random Processes by Grimmett & Stirzaker (1st, 2nd, or 3rd book)
For statistics:
Probability and Statistics by DeGroot and Schervish
Mathematical Statistics with Applications by Wackerly, Mendenhall, and Scheaffer
Statistical Inference by Casella and Berger
For DG:
Elementary Differential Geometry by Pressley
Differential Geometry of Curves and Surfaces by do Carmo
>>7661863
Thank you!
What is the most autistic and correct calculus book? I have good calculus knowledge but want to revisit everything in a very formal and thorough way
>>7662141
As far as I know there isn't one, but there is a good book on the real numbers.
Landau's Foundations of Analysis.
>>7658491
Why do you recommend Gelfand for basically everything? I can't ever find his shit.
>tfw i literally almost buy every book on math that /sci/ posts on this board.
>tfw i only get passed chapter 2 on each and already not interested
>tfw just want to learn maths and be smart
literally spent over 800$ on amazon search these books.
>>7662141
>I have good calculus knowledge but want to revisit everything in a very formal and thorough way
Baby Rudin
>>7662166
Isn't there something a bit more handholding? From what I've read baby Rudin is advanced and correct but uses the bare minimum to explain. I want something thorough that minuciously explains every detail
>>7662112
You are a god, desu, senpai.
>>7662154
Those 4 books were written as a set directed at passionate kids/teens that wanted to learn on their own and perfect for the automaths the list is intended for. The problems in them are also very good to make poor students finally think in terms of understanding mathematical concepts rather than collecting "arbitrary" rules by posing problems as puzzles.
>>7662187
Apostol's Mathematical Analysis. It's the fleshed out Baby Rudin. Zorich goes even slower but still eventually gets around to the actual analysis.
>>7662166
>Rudin
>Formal
>Thorough
10/10 lol
>>7662163
You know that you can just rent them from the library or [spoiler]pirate them[/spoiler], right?
>>7654500
In the trash
I live in a 3rd world country (Eastern Europe) so we didn't do calculus in HS.
Is it possible to learn calculus from Spivak, if i've never been exposed to it?
>>7662432
Yes but Apostol would be the better choice if you're going to learn on your own.
>>7656771
Why is Serge Lang's Linear Algebra not there? I found it pretty comfy, although a bit hardcore for a beginner
>>7660589
Same way a vector space is just a Hilbert space minus the complete norm-inducing inner product space and topological spaces are just metric spaces minus the metric space?
>>7658036
Because it's isomorphic to first order logic... which is thought in schools.
What's a good calculus book for someone that has never done calculus before. Thinking about getting the schaum books but some of them have been known to be filled with errors.
>>7662912
Baby Rudin
>>7662912
What kind of student are you and how interested in math are you?
>>7661335
You're trying to shoehorn vector spaces into a rather simple conversation, though. Set with a distance between any two points is both simpler and more general than "inner product space without vector space."
>>7654409
>Complex Analysis textbooks
Which one could you recommend for the introductory course (self-study)?
>>7663254
Conway
>>7663258
Is Markushevich so good as it claims in amazon's reviews? I'm able to read in russian.
>>7663254
If you can do real analysis: Conway or Stein (although he references a bit of his earlier Fourier material). Complex Analysis by Bak and Newman is good too if your background is weaker.
If you want something less rigorous and more application oriented then get "Fundamentals of Complex Analysis: With Applications to Engineering and Science" by Saff and Snider
And get Needham as reference regardless.
As someone who has taken calculus courses in the past and simply wants to improve/refresh, what would you guys recommend?
>>7663254
Ahlfors is the only true choice.
>>7653912
Only book OP has actually read
>>7663873
A get a Mechanics or Electrodynamics book.
>>7658491
I dunno why everyone says Basic Math is hard. I always thought the Gelfand books had way harder problems
>>7663926
So something a la Griffiths? Any favorites?
Anyone know an android app where I can view pdfs and organize them, every app I try can't sort the pdfs and it becomes a mess
>>7662163
>i
>>7664383
WUT?
IIIIIIIIIIIIIIIIIIIIIIIIIIIII
>>7663871
>If you can do real analysis
Is Courant's "Introduction to calculus and analysis" sufficient or should I go through a particular book on real analysis?
>>7664253
I don't know of an app that organises pdfs, since I just use MuPDF, which lacks any sorting capability. All it has is a basic, fast file browser that sorts files and folders alphanumerically; when you open it it simply starts by displaying the file browser in the last directory you were in. However, it is *really* lightweight and fast. I quite like it.
This isn't calculus; I wanted to post this in the other textbook thread but it died. So I'm posting here instead:
Rings, Fields and Groups: An Introduction to Abstract Algebra by RBJT Allenby
http://gen.lib.rus.ec/book/index.php?md5=9cdfad7e49e8c789c5851eb9f91086ae
>>7663142
Engineering student. I don't hate math but it makes me nervous since if I fail my life is in jeopardy.
What's your opinion on How to Ace Calculus: the streetwise guide? I personally think it belongs under "weak students" for introductory calc. I personally have thompson, kline and a schaum calc book so all I need is to put in some work
>>7664930
I have never heard of these pleb books.
Hoping this is the correct thread for this question.
I'm taking an introductory Linear Algebra class this semester (really a matrix algebra class with a small number of basic proofs, basically things like "prove this is a subspace of a vector space" or "prove that this defined vector product is an inner product", etc...).
I'm looking for a book that will be useful for reference in the future for matrix methods and basic linear algebra. I'm going into academic finance, and I know that I'll need something as a reference for further mathematical finance and econometric classes.
The sticky is ambiguous for which LA books would be a good reference book. Any recommendations for a good matrix algebra/basic linear reference book?
>>7665089
It's good although the humor is hit or miss sometimes
>>7665219
Meyer or Strang
This guy >>7658761 said first learn proofs, so What's the best proofs book for a freshman math undergraduate?
>>7665476
Also, if I go through Apostol's two volumes, have i got basic linear algebra covered? If not, what basic linear algebra book would you recommend me to complement it?
>tfw calc i/ii course uses stewart calculus
>>7665476
Velleman's How to Prove It: A Structured Approach is generally the go-to book for an introduction to proofs.
>>7665481
Apostol covers everything in matrix algebra and then some.
>>7665476
Handhold-y approach:
How to Prove It: A Structured Approach by Velleman
Book of Proof by Hammack (Free here: http://www.people.vcu.edu/~rhammack/BookOfProof/)
A bit more confident:
A Transition to Advanced Mathematics by Smith, Eggen, and Andre
A Primer of Abstract Mathematics by Ash
Also check out Conjecture and Proof by Laczkovich for a much better variety of proofs to practice on. It's a bit challenging but it will improve your proof and creative problem solving skills.
>>7663165
Because most students already know linear algebra before doing analysis and when you think distances you tend to think geometrically or algebraically.
>>7666045
Usually you take Linear Algebra and Analysis both at the same time namely the very first semester.
>>7666062
Sir, this is 'merika. We're lucky if students can solve 5-3x=2 when they enter college
>>7664562
Bump.
>>7665219
Huffman Kunze
or
Insel, Spence & Friedberg
>>7656771
I'm a research engineer in material science, physical metallurgy and welding processes. Now I want to go into the solid state physics (desu don't know if it uses linear algebra at all). Which category of linear algebra should I chose for a repetitive course?
>Applied Linear Algebra
or
>Finite Vector Spaces Theory
>>7662112
ty anon.
Which Calc book should I use to learn? I don't really know if I'm a strong or weak student? I just want to learn it right and well
Intro-tier
>Calculus Made Easy
>>7664562
Bak/Newman would be a good fit.
Other books assume you've done something on baby Rudin's level already
>>7655786
Yes this book is really good for newbies.
What do you guys think about Understanding Calculus by Edwards
>>7668175
It's almost 70 % of Larson/Edwards' textbook. It's good if you're a newbie and have 30 min while go to work (etc) by public transport or have long-term coffee breaks.
Whar is the difference between "real" analysis (Bartle, Schramm) and "mathematical" analysis (Apostol, Rudin)?
So many options I don't know which to choose. I just want to be able to understand the textbook and Calculus. I just want to learn which one would you recommend. I see the list in the thread I just need a recommendation from that thread. I'm no genius but I wouldn't consider myself dumb.
>>7669430
In this case, nothing, they just went went with whatever sounded better for their title.
>>7666152
https://en.wikipedia.org/wiki/Lighthouse_and_naval_vessel_urban_legend
stfu retarded maple man
>>7654500
Not calc and also Apostol is better desu senpai.
>>7669430
>(Bartle, Schramm)
This is what you get when you take a rigorous calculus course and remove the calculus. All the theorems are done in R. Courses in which that book is used are aim at future math teachers, actuaries, compsci's and others that don't care about learning any more math.
>(Apostol, Rudin)
Actual 1st semester analysis. The theorems of calculus are proved in a more general metric space setting rather than always assuming you're in R. It aimed at students who plan on continuing their studies in grad school for math, science, or engineering.
>>7669636
Simmons, Lang, Keisler, Kline
You can't really go wrong with any of them. Read a bit of them and pick whichever one's style you like best.
Thomas, Weir, Hass; Thomas' Calculus; Pearson.
Spivak; Calculus; Cambridge.
Apostol; Calculus; Wiley.
http://scidiv.bellevuecollege.edu/dh/Calculus_all/Calculus_all.html
Online and free.
>>7666812
VS theory would help with quantum mechanical aspect of it
>>7656771
>"Linear Algebra" by Shilov
>yfw there is literally nothing with such title in russian.
>>7670275
What?
>Шилoв Г. E. Maтeмaтичecкий aнaлиз. Кoнeчнoмepныe линeйныe пpocтpaнcтвa -M.:Hayкa, 1969
>>7660819
Anyone?
>>7670351
>Linear Algebra
>Maтeмaтичecкий aнaлиз. Кoнeчнoмepныe линeйныe пpocтpaнcтвa
Not even close.
![292079[1].jpg](https://i.imgur.com/0DENrjQ.jpg "292079[1].jpg")
All you need.
>>7660819
>'Core' algebra
?
>>7671141
I know, it's pretty fucking vague. It's just basic shit that's assumed knowledge for any uni-level maths student like set theory, basic vectors, functions done with some rigour, etc.
>no mention of fractional calculus
>>7662404
found the dumb guy
>>7671311
Pretty niche
https://mega.nz/#F!N4cGVCaZ!nuh0Ftx02YOylt92iRy64A
>>7667299
ur welcome. share your stuff too.
So... say I finished highschool (but have a background in geo alg trigo, everything but calculus.) And need to do differential and integral calculus from scratch (have a decent level in math), which book would suit me best? I'm thinking the N. Pisnukov's differential and integral calculus, I and II. What are your thoughts?
Paperback or hard cover?
I'm poor so I'd rather paperback but I don't know how the book will arrive. Buying both apostol's books.
>>7664253
I use sumatra
As long as this piece of shit doesnt make it on, ill be happy.
>>7671678
>Pisnukov
It's more popular in non-Anglo countries but it's good too.
>>7671359
It has plenty of applications in physics, like acoustics, flow mechanics and QM.
It should be explored more. At least I'm doing so, gives me special snowflake status.
>>7671678
Integration - Bourbaki
>>7672241
I decided to get both Apostol's books. Thanks a lot!
>>7662141
Wildberger's YouTube series
>>7671723
Why not pirate and print it out at school.
I'm trying to make sense of this thread
can someone just tell me wtf I should do to learn Calculus with self-study?
>>7673700
Get a calculus book
Read it
Understand it
Do some problems to cement your knowledge
>>7673635
Can't do that, school charges per page
>>7674346
Doesn't your school has a library?
I can rent every book I want from it for 1 month with up to 4 extensions (so 5 months total).
For introductury textbooks that the library has many copies of I can even directly rent it for 6 months.
And the best thing all of this is free and legal.
Is it possible to work through a proof-based calculus textbook (Spivak) for a person of average intelligence?
>>7674359
Learning proofs is all about investing time in reading them over and over, and then thinking them over and over.
All the people you see in math, that seem to be super good at it, have spent thousands of hours thinking and reading.
>>7674362
>All the people you see in math, that seem to be super good at it, have spent thousands of hours thinking and reading.
Yeah but people like that also have an innate talent for mathematics, whereas I do not. I'm reluctant to start because I'll probably plateau at some point hitting an insurmountable problem and end up frustrated and angry with even less confidence in my abilities.
>>7674356
My native tongue is spanish, and I'm having the test on english so I need to study in english. Too bad, but thanks for the input anyways.
>>7674359
Ya but it's best to familiarize yourself with proofs before starting.
>>7674399
>Yeah but people like that also have an innate talent for mathematics, whereas I do not.
That's a myth people use to feel good about themselves being lazy. Everyone hits a point where they need to actually work hard (otherwise, research wouldn't be a thing)
What calculus is best calculus?
>>7674438
The thread seems to agree on Apostle or Spivak.
Is Stein's series on analysis good or it's merely a supplement to his courses at university?
>>7675059
I got their Fourier Analysis and Complex Analysis ones, I haven't buckled down with them yet but they look pretty complete and well thought out.
What kind of areas of math should I try digging into as an EE major? Is analysis even useful because it seems almost purely abstract in contrast to stuff like linear algebra.
I mean, I'm sure something like topology would be pretty damn useless for an EE. Am I wrong?
APOSTOL OR SPIVAK???
>>7675106
Stewart.
>>7675105
Definitely look into Fourier analysis
I'm a freshman, and I've taken Calc I but have to retake it cause of some BS. I've looked into some stuff about Calc II on my own, but not very in depth. I'll have to wait probably 2 years before taking vector calculus, but I really want to learn it at least informally right now. Someone recommended to me Div, Grad, Curl, and All That, but it's a little bit too complex for me. I'm pretty capable and driven, so I know given the right resources, maybe some Khan Academy for multivariate, I'll understand this book eventually.
Can anyone recommend a suggest path of study in order to better grasp Div Grad Curl?
just someone passing by, thanks to all that contributed
>>7675105
Fourier and Complex Analysis would be very helpful. You will need some basic real analysis under your belt though to get the most out of those.
>>7675106
courant/john
>>7653912
is khan academy not enough?
>>7675105
Analysis is everywhere in EE once you get to the graduate level. Information theory/communications is built on measure theory, signal analysis is built on functional analysis, complex analysis is fucking everywhere and anywhere you look.
>>7675652
of course it isn't. Khan Academy is supplemental crap that you go to because you need some particular shit explained in a different way. Using KA as a primary source of learning is retarded.
>>7654409
can someone please explain me what is analysis?
>>7675685
damn that's what i've been doing to catch up with all the maths i've done in high-school/undergrad but didn't really understand. there's no way i have enough time to read books from calculus up to where i am now, last year of my CS undergrad
>tfw professor had like 5 slides on fourier analysis
>tfw want to go into a math heavy postgrad program
pray for me /sci/
>>7675686
No, go look it up and stop being turbo lazy.
>>7675105
>I'm sure something like topology would be pretty damn useless for an EE. Am I wrong
No, topology pops up quite a bit in EE
>http://webee.technion.ac.il/people/adler/topology.html
>>7675693
i'm just not sure i get it. is it just functions analysis? it feels like 80% of the maths i've done is analysis
>>7675696
Please.
>>7675686
Short answer: Rigorous and more powerful calculus
>>7675106
Courant/John > Apostol > Hardy > Spivak
>>7675692
>damn that's what [I]'ve been doing to catch up with all the maths [I]'ve done in high-school/undergrad but didn't really understand
That's what KA is for, remedial learning.
I wanna learn Probability really well.
What book should I read that's *not long*, please.
I've looked into "Probability Theory: The Logic of Science", but a lot of people say it's super hard, and I'm not much of a clever man.
I've actually already taken courses on probability, descriptive/inferential statistics, information theory, machine learning, etc. but all my univeristy studies are half-assed and I just learn the formulas and pass the exams without *really* understanding the fundamentals (e.g. oh I'm just gonna use Bayes here and derive a probability table and etc.).
Any rec?
Rudin or Zorich?
>>7676069
Rudin if you can keep up
Zorich if you want to take your time and go slow
>>7653917
Would going down this list in order be recommended? Is weak students supposed to say people who suck at learning or people without a solid foundation in that topic?
>>7676129
It means "person of color".
>>7675788
>I've looked into "Probability Theory: The Logic of Science", but a lot of people say it's super hard, and I'm not much of a clever man.
I am reading it. You can skip the math for the most part and still get a lot of the insights. Fantastic book BTW.
My next one is Bayesian Data Analysis, Third Edition
by Andrew Gelman
Expensive and hard to get though (all my college's copies are out).
>>7676129
No, that anon is just pretentious.
When it comes to mathematics books there's several things that affect the quality of the book.
>Rigor - A measure of how airtight the reasoning is.
>Formalism - A measure of how precisely the material is presented. A highly formal text will present definitions and arguments in raw logic over an axiomatic system. A high informal text will present the material using flowery English paragraphs and analogies.
>Morality - A measure of how clear and intuitive the proofs and presentations are (often times longer arguments are more moral than more concise ones).
>Completeness - A measure of how much of the standard material is covered by the book.
Judging by these criteria those books fall all over the scales. Rudin for example is very concise and complete but not very formal, moral, or rigorous (an overabundance of handwaving in his proofs). Spivaks is very rigorous and moral but severely lacking in formalism and arguably incomplete (depending on if you care about several variables or not). Keisler's lacks some formalism and rigor but all of that is covered in his supplementary text "Foundations of Infinitesimal Calculus". Unfortunately Keisler covers non-standard analysis which is a separate branch of analysis that many people do not study.
You should ask yourself what exactly you care about in a textbook and use that as criteria for choosing a book.
Personally I consider "books for weak students" to be books with a low level of formalism. Incidentally I also consider such books to be shit tier books aimed at retards who can't into mathematical precision.
>>7676281
>Rudin
>Handwaving
Leave and never come back to /sci/, you don't belong on any mathematical forum.
>>7676312
>implying Rudin is detailed.
>>7676348
He is.
>>8x+2=3x+3
>>x=1/5
>OMG, this is so undetailed and handwavy! Impossibly to follow, 0/10
>>7676397
Come back once you get past the introductory chapters. Rudin's lack of detail is literally the only complaint people ever make about the book.
>>7676451
you mean chapters 9-11?
>>7676129
No, just do one single variable calculus book and one (or two) multivariable book(s).
>>7675772
Is courant/John difficult to understand?
>>7654409
What order is best to do analysis in? And which topics in analysis are most useful for biomedical engineering?
>>7676451
Rudin is neither detailed nor handwaving.
Every good high level english textbook like Rudin, Lang's algebra, Atiyah-Macdonald's Commutative algebra etc, are written that way so that non-retarded potential future mathematicians fill in the details with relative ease, while dumb shits that complain about it are turned away as they should be, never to return to serious mathematics.
>>7676281
>retards who can't into mathematical precision
so weak students?
>>7654409
is doing both the zorich books enough for both single variable analysis and multivariable?
>>7675686
Study of some aspects of the real field like sequences in real numbers, series, limits, convergence of sequences, rigorous formulation of calculus. It uses stuff from set theory, algebra and topology.
>>7675105
I was a masters in ECE before getting into math. Learning analysis may not be very useful but understanding limits and calculus intuitively will help. Complex analysis, though, will be good for you (contour integration, residue poles and Cauchy's theorems). You don't need to understand real analysis very good for complex analysis, you can skip some proofs and jump straight to analyticity of functions without understanding sequences, although you do need a very basic understanding of convergence of sequences and stuff.
>>7675202
Khan academy.
>not even joking
>>7677578
Not anymore so than the others
What is a good math methods book for reta-, I mean weak students? I am in first years physics.
I've been looking at Boas and Riley.
>>7678214
Dennery & Krzywicki - Mathematics for Physicists (Dover)
Hassani - Mathematical Physics: A Modern Introduction to Its Foundations
Szekeres - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry
Nakahara - Geometry, Topology and Physics
Frankel - The Geometry of Physics
Reed & Simon - Methods of Modern Mathematical Physics I: Functional Analysis; II: Fourier Analysis, Self-Adjointness; III: Scattering Theory; IV: Analysis of Operators
What is the best Calculus text for future engineer and so I can cruise through my upcoming Calculus classes. I want to learn before I actually start because the school uses shitty texts and I want to teach myself as well. I wouldn't say I'm strong but I wouldn't say I'm weak either. Thanks!
