Best Books for Mathematical Analysis/Advanced Calculus

In this video I talk about three really good books on mathematical analysis which is also known as advanced calculus. There are several other really great books on the subject but I just chose three to discuss briefly in this video. I chose these mainly because I have spent a great deal of time reading and working through these books and I found them really helpful.

I have done full reviews on all of these books, see the links below.

The books are...
Principles of Mathematical Analysis(aka Baby Rudin) by Walter Rudin

Advanced Calculus by Fitzpatrick

Advanced Calculus by Buck

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[محمد-رطل]

hi, i want to ask for a favor
can you make a learning plan for mathematics ? for the undergraduate level
i am a physics major and i really want to learn math but i am lost i don't know where to begin and where to stop so if you did it it will be very helpful
p.s. i LOVE your videos

[edwardgraham]

The Story of Rudin's life is pretty interesting as well! Thank you!

[pinklady]

1. Principles of Mathematical Analysis.
2. Advanced Calculus by Patrick M Fitzpatrick.
3. Advanced Calculus by Bucke.

I will return to revise list above.

[zstew]

I'm starting my third year as an undergrad math student at michigan state and I'll be taking my first analysis course in the fall. I've been reading Understanding Analysis by Stephen Abbott (I saw another comment mention this text; it's great for an intro to analysis) to prepare myself but I'll definitely have to check out the books mentioned in this video. Appreciate the video and love the channel!

[acdude]

Thanks for the review. I will check them out. I liked Wilford Kaplan and Jerrold Marsden's books.

[theEntropologist]

Thank you for this video. I have a little request:
I studied economics in high school, where the math was really simplistic, I learned some basics of calculus, and analysis of progressions (That's what we call it in our country, I don't know if it is correct); however we have not studied complex analysis and many other interesting math fields.
I am getting my degree these days, and I am looking forward to learn math on my own, I enjoy solving problems and I am ready to do every exercise that I can find, but my main concern is that I may neglect certain fields that are interesting. I am currently trying to study the basics of trigonometry as they were not included in the calculus I learned.
I would be grateful if you were to make a video explaining how would you go about learning math from scratch (well, we'll assume that scratch means we can solve equations and do simple arithmetic), and what are the resources that you would use.

I saw a similar request to mine, I just wanted to share my situation and perhaps make your video a little bit more specific for my case.
Thank you again for your videos, although I do not understand most of them, I am looking forward to the day where I learn enough math to be able to understand everything here.

[soyoltoi]

Another book I see recommended absolutely everywhere for beginners is Abbott's Understanding Analysis. The review by the MAA calls it dangerous because of how good its exposition is. For something at the level of Rudin, I think Pugh's Real Mathematical Analysis is a really good book too (tons of problems, and even a section on visualizing 4D).

I also recently came across the three volumes by Amann and Escher, and they're probably the most advanced set of introductory books on analysis available on the market. They do things in a top-down approach regarding abstractness, which is the opposite of how it's usually done.

[Math-Tulane]

what do you think about the book by sherbert and bartle? also how about understanding analysis by abbot?

[learnmathwithme]

I would like to mention that their is a similar "baby rudin" book. It is called Pugh: Real Mathematical Analysis. It has more explanation. For a truly gentle Analysis book which sticks to R, Abbot: Understanding Analysis.

[kelvinalanrojasosorio]

Hi, Sorcerer. I really love studying mathematical analysis and, in fact, I have about 8 or 9 books of it. Just one question, what do you think about the Mathematical Analysis I and II of Vladimir Zorich? That book is one of my favorites. Thanks a lot.

[ShinXiao]

Thanks for another awesome books recommendation video. With proofs, I always wonder how to understand them. Because I see proof based questions in books but I can't understand a lot of the time. I guess its because I'm not at a certain level. How does a mathematicians construct proofs?

[Darknight]

That's neat since I actually just got delivered the first 2 of books, except for the Fitzpatrick I got the Advanced Calculus Second Edition, it looks different from your cover but same title and same author.
I took an into proof course in the winter, where we studied on statements, logic, all the different types of proofs (direct, contrapositive, contradiction, cases, induction, etc...), sets and functions.
*However, I fear that I might not be as understanding of the material since it's been some months since I took that class.*
I plan on taking Real Analysis this upcoming fall quarter, and I have 2.5 months (10 weeks) which is a decent amount of time to go over the material again so I can be prepared. *Could you please go over what basics, or knowledge we should have before we take the first Real Analysis class?*
Should I go over all of the class material? Are sets and functions necessary? Should I stick to just the logic and proof material (which is 1/2 of the material and not all of it).
I hope you can answer this Math Sorcerer as others may potentially want to get a head start for the most difficult undergraduate math course they'll ever take as a brief text summary and potentially a more in depth video on it's on topic.
Thanks! Have a great rest of the day / night!!

[daphenomenalz]

Are they suitable if someone just started calculus 😅

[almenarab]

Can you suggest real life applications of complex numbers outside engineering or physics??

[myexflower]

The book by Buck was our Advanced Calculus course, which was a second-year course. Our first-year Calculus books were by Tom Apostol. And then, we read Metric Spaces by Copson and The Elements of Integration and Lebesgue Measure by Bartle for our third-year Real Analysis courses. Finally, we read Introductory Functional Analysis with Applications by Kreyszig for our Functional Analysis course in our fourth, our senior year. These books were those for our analysis books. Algebra, Differential Geometry, Partial Differential Equations, Fourier Analysis, Algebraic Number Theory, etc. are all different issues...

[gustavocardenas]

Thank you very much for these book reviews! In general your channel is awesome, keep it up :)

[joliettraveler]

I am a little confused. I will be starting Cal III. It mainly consists of chapters 10 to 17 of Stewart's Calculus Early Transcendentals. The topics covered are e.g. vectors, partial derivatives, multiple integrals, second order differential equations. Is this the same as your Advanced Calculus, and would the books you referenced help?

[EzraSchroeder]

How is Introduction to analysis by Mattuck? & An introduction to analysis by Wade. I would be interested in if those are good to read as well even if they aren't in your top 10.

[ModuliOfRiemannSurfaces]

Slightly unrelated question, but how would you recommend going about studying the bare minimum point set topology necessary to tackle something like Hatcher or Fulton's algebraic topology? Preferably prioritizing speed as I'm happy to go back and review things as necessary.

[devaiyer]

Thank you! Buck seems interesting, will check it out!