# Introductory Analysis -- Self Studying

#### opus

Gold Member
Hi all. I'm looking for a little guidance on starting out in self-studying analysis. The reason is two-fold. Firstly, I am very interested in it and want to have a deeper understanding of Calculus rather than the computational plug and chugs I learned in Calculus I last semester. Second, I've noticed that many graduate schools for physics suggest having an understanding of analysis. See http://pma.caltech.edu/information-for-applicants-2 as one example.

I have read the self study guide posted by micromass, but sadly it appears he's no longer a member so I can't ask directly! https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/

My question is, in my first link, it states to have an understanding at the level of T.M Apostol's Mathematical Analysis. Micromass suggests using Numbers and Real Analysis by Bloch.
How do these two compare? I am unequipped to make the necessary comparisons, as right now I'm just starting out on the basics like logic and set theory in How to Prove It by Velleman.

Any tips?

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#### jedishrfu

Mentor
A good aside study for Calculus is to watch the Essence of Calculus video sequence by YouTuber 3blue1brown. They will give a geometrical understanding of calculus beyond what you may have learned in your coursework.

#### opus

Gold Member
A good aside study for Calculus is to watch the Essence of Calculus video sequence by YouTuber 3blue1brown. They will give a geometrical understanding of calculus beyond what you may have learned in your coursework.
Thanks! Yes I have a fiendish addiction to that channel. His section on the fundamental theorem of calculus 100% brought the concept home for me. Best channel on YouTube.

#### FactChecker

Science Advisor
Gold Member
2018 Award
The first thing that you should know, if you don't already, is that the mathematical analysis that has applications in physics includes a great many subjects more than calculus. IMHO, you would be smart to look at some books of mathematics for physics. It would also help if you can get some of the required math textbooks for the classes that are prerequisites for your anticipated physics classes. Unfortunately, I don't know any that would be particularly good for self-study and would worry that the effort may be too intimidating and discouraging.

#### opus

Gold Member
Thanks FactChecker. Some time ago I got a copy of Mathematical Methods in The Physical Sciences by Boas, but since I haven't yet taken a Physics class, and just barely finished Calculus I, I couldn't get any use out of it yet. I don't mean to seem as though I'm putting the cart in front of the horse, I just want to know why a lot of the previously learned Calculus stuff actually works, and also want to learn how to think like a mathematician (like what you were describing in my other thread about proving cases for $n$ and $n+1$ etc).

#### Math_QED

Science Advisor
Homework Helper
I would say that Tom Apostol's book Mathematical analysis is harder than Ethan's Block's suggested book, as it deals with abstract metric spaces and more advanced mathematics (for example immediately Riemann-Stieltjes integral instead of Riemann-integral) where as Ethan Blocks's book only treats real analysis for the real numbers in a single variable. Don't be mistaken though, as Block's book isn't easy too.

I suggest you start with Ethan Block's book, and then when you finished the important parts of it, you start with Apostol. Your prerequisites for Block's book would be that you have some experience writing proofs, that you have mastered the mechanical sides of calculus, and that you are familiar with set theoretic concepts like "set, function, injection, surjection, bijection".

What you will learn in Block's book:

- A very carefully developped construction of the real numbers, starting from the natural numbers (there are also entries to start with the real numbers axiomatically given, which may be more interesting if you want to get to the analysis quickly; do read the introduction of the book for this)
- Rigorous treatments of limits/derivatives/Riemann-integrals/series. On a serious note, the chapter of derivatives and especially Riemann-integrals are very nicely written. I didn't find a book yet that treats those topics better than this book.
- The "analysis" feel.

What you will learn in Apostol's book:

- Abstraction and generalisation of the concepts you learned in Block's book (introduction to abstract metric spaces, with applications in concrete metric spaces like $\mathbb{R}^n$).
- All the topics that appeared in Block's book (sometimes in another order though) will return, often more general, but also much more: Riemann-Stieltjes integral, infinite products, derivatives and integrals in higher dimensions. Note: some say that the treatment of integrals in multiple dimensions in the book is not that good. A good reference for this is Munkres' Analysis on manifolds or the more difficult book Spivak's calculus on manifolds.

When you see how you can easily prove theorems like Bolzano-Weierstrass via compactness arguments, or the intermediate value theorem via connectedness theorems, I think you will see how powerful abstract metric spaces are.

Important note: make a lot of exercises! These help to remember theorems, and make sure you "get used" to the theory. Learning a mathematical theory means that you should be able to prove some results on your own, and the exercises are perfect for this!

If you have any questions left, feel free to ask!

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#### opus

Gold Member
I would say that Tom Apostol's book Mathematical analysis is harder than Ethan's Block's suggested book, as it deals with abstract metric spaces and more advanced mathematics (for example immediately Riemann-Stieltjes integral instead of Riemann-integral) where as Ethan Blocks's book only treats real analysis for the real numbers in a single variable. Don't be mistaken though, as Block's book isn't easy too.

I suggest you start with Ethan Block's book, and then when you finished the important parts of it, you start with Apostol. Your prerequisites for Block's book would be that you have some experience writing proofs, that you have mastered the mechanical sides of calculus, and that you are familiar with set theoretic concepts like "set, function, injection, surjection, bijection".

What you will learn in Block's book:

- A very carefully developped construction of the real numbers, starting from the natural numbers (there are also entries to start with the real numbers axiomatically given, which may be more interesting if you want to get to the analysis quickly; do read the introduction of the book for this)
- Rigorous treatments of limits/derivatives/Riemann-integrals/series. On a serious note, the chapter of derivatives and especially Riemann-integrals are very nicely written. I didn't find a book yet that treats those topics better than this book.
- The "analysis" feel.

What you will learn in Apostol's book:

- Abstraction and generalisation of the concepts you learned in Block's book (introduction to abstract metric spaces, with applications in concrete metric spaces like $\mathbb{R}^n$).
- All the topics that appeared in Block's book (sometimes in another order though) will return, often more general, but also much more: Riemann-Stieltjes integral, infinite products, derivatives and integrals in higher dimensions. Note: some say that the treatment of integrals in multiple dimensions in the book is not that good. A good reference for this is Munkres' Analysis on manifolds or the more difficult book Spivak's calculus on manifolds.

When you see how you can easily prove theorems like Bolzano-Weierstrass via compactness arguments, or the intermediate value theorem via connectedness theorems, I think you will see how powerful abstract metric spaces are.

Important note: make a lot of exercises! These help to remember theorems, and make sure you "get used" to the theory. Learning a mathematical theory means that you should be able to prove some results on your own, and the exercises are perfectly for this!

If you have any questions left, feel free to ask!
Thank you! That was extremely helpful. I will continue working on my "proof book" so I can have a good understanding on how they work, then I'll move to Block. Very exciting stuff

#### Math_QED

Science Advisor
Homework Helper
Thank you! That was extremely helpful. I will continue working on my "proof book" so I can have a good understanding on how they work, then I'll move to Block. Very exciting stuff
Good luck on your mathematical journey! It will not be an easy journey, but definitely one that's worth the struggle.

#### opus

Gold Member
Good luck on your mathematical journey! It will not be an easy journey, but definitely one that's worth the struggle.
Thank you! My semester is loaded this Spring, but I've set aside 90 minutes a day for "extra" studying like this. I figure I'll just start taking small consecutive bites out of it.

#### Math_QED

Science Advisor
Homework Helper
Thank you! My semester is loaded this Spring, but I've set aside 90 minutes a day for "extra" studying like this. I figure I'll just start taking small consecutive bites out of it.
Hmmm, here is a little of advice, which you are free to leave or to take.

First finish the necessary school work. If you have time left, then feel free to extra study.

I had the same attitude as you, but I have too much work during the semesters to do something else. Instead, I study what I want during the vacations :)

#### opus

Gold Member
Hmmm, here is a little of advice, which you are free to leave or to take.

First finish the necessary school work. If you have time left, then feel free to extra study.

I had the same attitude as you, but I have too much work during the semesters to do something else. Instead, I study what I want during the vacations :)
Mostly that’s what I meant. Usually at the end of the night when Im done with all my classes I will read before bed, so was thinking about using that time for extra study. But we’ll see how I feel after 7 hours of Calc 2, Phys I, Chem 2 lol. There’s just not enough time in the day to do everything I want

#### Math_QED

Science Advisor
Homework Helper
Mostly that’s what I meant. Usually at the end of the night when Im done with all my classes I will read before bed, so was thinking about using that time for extra study. But we’ll see how I feel after 7 hours of Calc 2, Phys I, Chem 2 lol. There’s just not enough time in the day to do everything I want
Yeah. Is it your first year or you already have experience?

I want to do much besides what I have to do. I want to study differential geometry, complex analysis and expand my knowledge of real analysis/functional analysis/algebra/measure theory but there is simply not enough time in a day. In the end, we are unfortunately forced to set priorities.

#### opus

Gold Member
Yeah. Is it your first year or you already have experience?

I want to do much besides what I have to do. I want to study differential geometry, complex analysis and expand my knowledge of real analysis/functional analysis/algebra/measure theory but there is simply not enough time in a day. In the end, we are unfortunately forced to set priorities.
Not my first year. Ive been going every semester including summers since Summer 2017. I wanted to start with “Pre-College Algebra” and not test out of it because I wanted the exposure to it. I think it was for the best because in doing so it seems the upper level stuff comes easier in knowing the fundamentals.

#### Math_QED

Science Advisor
Homework Helper
Not my first year. Ive been going every semester including summers since Summer 2017. I wanted to start with “Pre-College Algebra” and not test out of it because I wanted the exposure to it. I think it was for the best because in doing so it seems the upper level stuff comes easier in knowing the fundamentals.
Completely agree with this! One must not walk before one can crawl.

#### WWGD

Science Advisor
Gold Member
Hi all. I'm looking for a little guidance on starting out in self-studying analysis. The reason is two-fold. Firstly, I am very interested in it and want to have a deeper understanding of Calculus rather than the computational plug and chugs I learned in Calculus I last semester. Second, I've noticed that many graduate schools for physics suggest having an understanding of analysis. See http://pma.caltech.edu/information-for-applicants-2 as one example.

I have read the self study guide posted by micromass, but sadly it appears he's no longer a member so I can't ask directly! https://www.physicsforums.com/insights/self-study-analysis-part-intro-analysis/

My question is, in my first link, it states to have an understanding at the level of T.M Apostol's Mathematical Analysis. Micromass suggests using Numbers and Real Analysis by Bloch.
How do these two compare? I am unequipped to make the necessary comparisons, as right now I'm just starting out on the basics like logic and set theory in How to Prove It by Velleman.

Any tips?
I suggest that, if possible, go to your school/local library and browse through the books in this section to see which feels right for you. As a rule of thumb I exclude books without clear descriptions of notation used or poor indexes, lack of solved problems ( at least a few in each chapter). These last traits to me reflect whether the author has taken care to write a clear book.

#### opus

Gold Member
I suggest that, if possible, go to your school/local library and browse through the books in this section to see which feels right for you. As a rule of thumb I exclude books without clear descriptions of notation used or poor indexes, lack of solved problems ( at least a few in each chapter). These last traits to me reflect whether the author has taken care to write a clear book.
I completely agree. The issued texts for my Calculus classes have a severe amount of typos and notation inconsistencies. There's nothing worse!

#### MidgetDwarf

I would not suggest starting to learn Analysis at this stage. If you have only finished calculus 1, then maybe you are not ready. Unless you're calculus was taught from a book such as Spivak, Apostol, or Courant Calculus. Most people, which is typical of US undergrads, do have a rigorous introductory calculus class.

I you are familiar with proofs. Ie, what a direct statement is, a converse statement, proof by contrapositive, induction, proof by contradiction, existence and uniqueness proofs etc., then you are ready to learn Analysis. What a set is and facts regarding sets... If you are not sure what this is, then maybe look online for resources that have this content.

I do not really like proof books, but there is a nice free one you can download legally from the author, or buy cheaply from Amazon.
Hammock:Book of Proof.

If this you're first time trying to learn rigorous mathematics, then maybe start with Linear Algebra. I believe, Linear Algebra, is the easiest math branch to introduce students to proof based mathematics. Since it is not too abstract, same ideas are used over and over (ie., linear independence and spanning), and a very useful mathematics branch. In the beginning sections of an introductory book you will see examples of the following proofs:

Matrix multiplication is associative: induction is used for this proof
Direct proof: Let A be nxn matrix. If A can be reduced to an identity matrix by a sequence of elementary row operations, then A can be written as a product of elementary matrices.
If A and B are invertible matrices, then their product AB is invertible. Now do the converse for this. Note that it also holds. But in general, converses are not always true.

a really neat example of a proof method: if AX=0 has only the trivial solution, then A can be reduced to the Identity matrix by a sequence of elementary row operations.
I won't say which one it is, but there are some really neat and easy to digest examples of proofing techniques in linear algebra that will carry you through for further mathematics.

#### opus

Gold Member
I've heard from a few people that they don't like proof books, but can be a necessary evil. Right now I am going through How to Prove It by Velleman to get an understanding of the types of arguments you have described.
I didn't know that about Linear Algebra. I'll be taking it in the summer, so I'll definitely keep that in mind. Thanks!

#### CCofADoa

The easiest way to self-study analysis is perhaps by going through R.P. Burn's Numbers and Functions: Steps into Analysis. It's a problem book and it has all of the essentials.

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