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2

There is no right or wrong answer on your decision to not pursue something. Will you look back later in life and regret that decision? I don't know so maybe you will or maybe you wont. If you know analysis is not what you want to do in math, that is perfectly fine since mathematicians around the world specialize in many different areas of mathematics, and ...


2

"I always thought that it was beyond me and that there's no way for me to do well in it." This is your first problem, its called a limiting belief, you need to change that to the belief that you can master analysis. "I spend so many hours and put in so much effort into studying" This is the second problem, more time studying does not produce better ...


1

For the math subject test, you shouldn't just focus on Calculus, but if you do, Spivak would be my suggestion since it provides more rigor. However, you need to know analysis (real and complex), linear algebra, topology, probability, abstract algebra, combinatorics, and some other fields at a decent level. Here is the link to practice book supplied by ETS. ...


0

I suggest Thomas's Calculus as well as Hille's Calculus (as a secondary source).


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$\bullet$ Patience! $\bullet\bullet$ Persistence. $\bullet\bullet$ Work hard. $\bullet\bullet\bullet$ Learn things very well. (in detail) $\bullet\bullet\bullet\bullet$ Ask yourself lots of questions, even stupid ones! (when does this lemma work? when it doesn't? is there a generalization of it? is there a similar lemma about ...) ...


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$\color{green}{\text{Howie's Galois theory (Easy)}}$ $\color{orange}{\text{Stewart's Galois Theory (Intermediate)}}$ $\color{red}{\text{Cox's Galois Theory (Hard)}}$


2

For $p\leq 0 $ we have that $f(x)=|x|^{-p}$ is continuous and thus in the compact $[a,b]$ it has a maximum value let's say $M>0$. This means that $I\leq M(b-a)$. Now if you mean $p>1$ , in order for $I<\infty$ we must have that $0 \not \in [a,b]$. Because if $0\in [a,b]$ then $I\geq \int_{0}^{b}|x|^{-p} dx=\int_{0}^{b}x^{-p} dx=\infty$. Think if ...


0

If the pure mathematics you are referring to requires 'more' theorems and proofs than the applied mathematics you have done, or more rigorous proofs, which is also the case in other universities, then it is no surprise if it comes out harder. If all the theorems you used in applied mathematics were to be proven with no hand-waving, it will suddenly become as ...


0

I can recommend the publicly available courses on the MIT website: http://ocw.mit.edu/courses/mathematics/ Start with Single Variable Calculus, Linear Algebra, Introduction to Analysis.. If I remember right, they all have video lectures. Most important thing when learning maths, do ALL the exercises (don't look at solution until you solved them all with ...


0

I would like to add 2 more points to the excellent answers already provided. In studying pure mathematics: (1) definitions and counter examples are quite important in getting a mastery over the subject. Try to understand each and every word in the definition of a concept. What happens if you take away a particular word in the definition or if you substitute ...


3

My answer to should you just stick to applied math: The courses you mention as applied math, Calculus 1-3 and ODEs, I view as basic set for math majors in general. I can't think of any math curriculum where Calculus 1-3 aren't mandatory but ODEs is a different story. ODEs generally isn't a mandatory course but most, if not all, majors probably take this ...


1

An appropriate answer, I think, must address exactly what you plan on doing with your newly acquired mathematical knowledge. Are you planning on using it as applied to computer science, for a better pure math foundation, just for fun, etc.? If you are trying to beef up on your mathematical knowledge as applies to CS, then I don't think there is probably ...


0

I would recommend the book Basic Mathematics by Serge Lang for basic algebra and geometry. The following books by the Russian mathematician Gelfand are intended for good high school students studying on their own: Algebra The Method of Coordinates Functions and Graphs Trigonometry Gelfand's books are described here: http://www.egcpm.com/books/ You ...


0

I read this book from Lara Alcock and I think it answers your question: "How to Study as a Mathematics Major". I warmly suggest it! http://ukcatalogue.oup.com/product/9780199661312.do


1

Muhammad, I would start with high school textbooks such as algebra 1, algebra 2, plane and solid geometry, trigonometry, calculus...all in that particular order. Of course, you can always overlap at your own desire. I would also throw in high school physics, chemistry, and biology to get an idea about how mathematics is universally useful. Oh! How could I ...


2

The discipline of mathematics requires the following from you in order to excel in it: 1) Your love of the subject. 2) Your desire to do well in it. 3) Intellectual curiosity. 4) Willingness to work very hard. To excel in ANY discipline, you must work very hard. 5) Mental aptitude. This is the "luck component" of life. You have what you're born with as far ...


1

A slightly different approach would be Bogachev's two-volume 'Measure Theory', especially if you want to focus more on measures rather than general analysis. It covers all the topics you listed in depth except Hilbert spaces. It does, however, have a chapter on $L^p$-spaces. The good thing about this book is that it has plenty of exercises and supplemental ...


0

I do almost all problem solving this way: I work things out fairly completely and or develop the ideas and approach and then I write it up using Latex. The reason I write it up carefully is so that I can view my work and reasoning and explanation from the point of view of another person and critique my work. Generally my arguments become cleaner and clearer. ...


3

In my very limited experience, it seems that the ability to do well in PM takes a ton of patience and a ton of practice. It takes weeks, months, and years; most of all, though, it takes passion. You must have a burning desire to truly understand the topics with which you are dealing with in PM. You can memorize things and get by with it for the most part in ...


5

Maybe I'm off-base; but by the sound of your post, it seems like you are trying a rote memorization approach in your pure math classes. This works in certain applied classes at the freshman and sophomore level because it's largely number crunching. A lot of the numerical analysis classes also largely deal with algorithms and methods, and comparing them. ...


2

Gerald Folland's Real Analysis book is a superb choice for learning measure theory and elementary functional analysis. It covers all the topics you listed. It is a bit more difficult and abstract than most other introductory textbook (such as Royden), but it is well worth it (and yes, it is actually suitable for a first-time learner, because I was). If you ...


1

I would recommend you to start with Terence Tao's An Introduction to Measure Theory.It will cover the basic areas of measure theory .Moreover it is very well written.Not sure that it will cover every topic of yours but will cover most of them.Also the exercises are mind blowing.Try it, you will surely feel the difference from other books


2

I would start with Real Analysis by Royden. I find it very well-written and the problems are really good. It covers most of the areas you mention.


3

Some skills you need to be familiar with: 1) Techniques of proofs: How should you think to prove a statement (induction, contrapositive, minimum counterexample etc), you can start with some proofs in number theory (a most excellent book): 2) Analysis: helps you make sense of numbers and continuity, you can then use sequences to decompose mathematical ...


3

Unquestionably, it will help a lot. This is for two main reasons: 1) One needs exposure to proofs and practice solving problems using proofs at a high level. It's much easier to get this at first in dealing with relatively concrete objects (numbers, numerical functions, finite groups, etc.) than with very abstract objects (such as structures in a ...


2

Yes, of course. To master mathematical logic you need to know mathematics, for at least two reasons: -- methods of mathematical logic are mathematical, and you need to understand how they works from within a core mathematics area of knowledge, tackling pure mathematics issues; -- mathematical logic developed mostly thanks to mathematicians, if you do not ...


3

Let me tell you how I got interested in logic during my undergrad. In our measure theory class, after constructing Lebesgue measure, Vitali sets were given as examples of Lebesgue non measurable sets. I wondered if one could extend the Lebesgue measure to a measure defined for all sets of reals. My instructor (an expert in applied probability and statistics) ...


4

Yes, and no. It helps because you will see proofs, you will see careful statements and you will learn, even if not directly more examples that can be used later on to study mathematical logic better. But on the other hand, rarely anyone mentions formal logic in a course about analysis. You don't think about the inference rules, or what sort of statement ...


2

Well, mathematical logic is different from what we normally mean when we say 'logic'. To understand where it comes from and some of the motivation, it is generally good to take other math classes, but they will not teach you (much) mathematical logic. Your department should offer classes in logic, and you should take them, but other math classes will show ...


1

I think it is profoundly important to distinguish (at least) three different types of mathematics writing: textbook writing (especially lower-division undergrad, but also introductory graduate level), "research papers" (in traditional refereed journals... the necessary professional purpose most often being personal advancement more than enlightening any ...


0

Consider the following links: http://www.math.harvard.edu/quals/index.html http://math.berkeley.edu/programs/graduate/prelim-exams http://www.math.cornell.edu/m/Graduate/prog_content.html http://www.math.columbia.edu/programs-math/graduate-program/what-graduate-students-are-assumed-to-know/


0

Check out Stephen Siklos: Problems in Core Mathematics, which exists in two different editions. He used to be a STEP examiner. Also check out the MathsHelper website, which has links to these publications. Also Warwick University created a STEP course for prospective candidates, devised by one of the STEP examiners from MEI. Also check out Meikleriggs, the ...


-1

i am trying to do some studying with help of an personal wiki see https://en.wikipedia.org/wiki/Personal_wiki not sure how usefull this is (yet) but maybe an idea for you. the Idea is to cut everything up in litte notes (zettels) see http://christiantietze.de/posts/2013/06/zettelkasten-improves-thinking-writing/ and http://takingnotenow.blogspot.co.uk/ so ...


1

I was recommended to work through the book "How to think like a mathematician" before starting my undergrad. It focuses on the transition from high school math to university math, in particular how to write proofs. It might be helpful for you to go though it. Otherwise than that, just practise, practise, practise. EDIT: One more thing: if there are other ...


1

tl;dr: just find another area of math, there's too many to sweat this. But also consider taking number theory to really whet your interest in algebra before you dismiss it. Firstly, I just really think you might be taking the wrong approach on what "math" means. In some circles math means the purest math possible with the most abstract rigorous proofs and ...


0

There are several questions here. The first is whether you can start with Rudin's analysis book now. I would say that for most people this would be difficult after only Lang's calculus book, although Lang is far better than most calculus books in use these days. I don't recommend reading Spivak or Apostol (Vol. 1) now. It will take too long, as they're ...


1

I disagree with your friend saying it is more rewarding to do some applied than pure. I am in applied math and engineering so I am not a pure mathematician trying to sway you back to pure. What will be rewarding to you is simply what you love to do not necessarily something that is practical. Also, Physicist use algebra and topology in some of their branches ...



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