# Tag Info

1

Let me offer a few points, though I'm probably forgetting many things. First, you typically need a reason for motivation. It could just be you find a subject beautiful and want to learn more, or it could be because you want to use the theory for something else. My approaches to these situations are somewhat different, but hopefully it's not merely because ...

2

I will propose an answer just so there in one! For me it helps to ignore all literature and go for a walk. I ask myself what bothers me, what do I want to know and what are my interests. Then I go back to the library and look for an answer; without trying to understand how or why. I merely try to connect two dots, even if the method is completely stupid. ...

0

I started writing a proper answer to your question, with the general thesis that the answers to all three of your questions depend on how you are willing to answer them, but then I digressed. Simply because I was in your situation once and I realize that this question regarding the "mathematician's block" (which is $\cong$ "writer's block", but not quite the ...

0

While not comprehensive, I would strongly suggest that you read this, an excerpt from the Princeton Companion to Mathematics. In the link I provided, you will find advice from 5 prominent mathematicians aimed explicitly at young mathematicians. In it, you will find advice concerning what stance you should take on things such as research and the like. I ...

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 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 ...

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. ...

0

I believe that if you understand the nature and behavior of real numbers then you should not have any issue with most of the proofs in real analysis. Once you know reals, you will understand what goes underneath the complicated $\epsilon-\delta$ arguments. To understand the reals, it is necessary to have some understanding of infinity. Thus we can start ...

1

In my experience, proofs can often be summarized in one line usually describing the trick/s of the proof, by which I mean the parts of the proof which present new techniques or something. Consider outlining proofs perhaps if they are in parts like with Lebesgue integrals. So you may have something like Theorem 1 Use Bolzano-Weierstrass Prop 2 measure of ...

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

-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 ...

0

this is quite good method to study, when I studied the analysis, I did the exactly same thing as you are doing. I proved every theorems in the book, and read the whole book word by word many times. At last I found all my efforts worth it. Usually the methods to prove analysis problems are related to the following topics: supremum and infimum principle ...

0

From what you have said your problem doesn't seem to be in the learning of the material, rather in the retaining of it. The solution to your situation is "Anki". Anki is a spaced repetition system application (SRS for short). Within this application you can create flashcards, to cut a long story short once you've entered these flashcards into Anki, you will ...

0

Most of the above are helpful but when I took the course I had a different route. I am apparently a visual person and I visualized (and in fact wrote down on large artist's pads) every "if" condition in succession as overlapping Venn diagrams delineating truth on the inside of shapes and false on the outside. I presumed that the then part would be in the ...

1

How I discover a proof of a statement in analysis? I put definitions of all the terms in the statement before my eyes on a big sheet of paper, not simply relying on my own memory. Try to discover some natural consequences of the "If part". Also, try to write the "Then part" in primitive terms. First, try to progress some steps from the end side. If you ...

3

A last addition, do study a bit of propositional logic. Learn the difference between reductio ad absurdum (something is proven because its negation is not possible) direct proof (something follows out of the givens) contrapositive proof by cases and more that kind of constructions. Play a bit with it and see how these proof methods reappear in your ...

16

One thing that works for me when learning a theorem is to go through all the conditions and find corresponding counter-examples, as well as seeing exactly where the proof fails. Take for instance Rolle's theorem: If a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, ...

4

I hope you already got a lots of good advises...one little advice from me is that some time try to understand the geometry behind those definitions like closed set connected set continuous function and all...because you already know there is a beautiful geometric structure over R as well as $R^n$ so if you manage to relate those abstract definitions with ...

2

Sometimes when working on an exercise that requires me to reference a theorem whose proof I do not remember, instead of simply citing the theorem I like to reprove the theorem using the notation in the exercise, possibly simplifying the proof when some of its parts become unnecessary.

13

If you just remember individual proofs, and arrange them in a circle on your notepad, you will notice that it's $2\pi r$ to go around and remember all the proofs, but if you start at the centre (the basics) and learn the methods it's only ever $1\cdot r$ to any proof. It's much easier if you learn the basics and the 'how to step', rather than individual ...

46

Don't try to memorise the proofs: try to memorise the methods that are used in most analysis proofs. That way you only have to memorise a handful of methods instead of 30-50 proofs, and you can adapt them to prove things you have never seen before as well.

16

Do not memorize proofs. Just become comfortable with how to think about certain proofs and the basic framework of proving certain ideas. That is, you do not need to memorize how to prove that $x^3$ is continuous at $a=-1$ but you do need to be familiar with how to prove continuity as a global property and prove continuity at a point. This way of thinking ...

1

It is often extremely difficult to immediately recognize a pattern, though the first numbers happen to be somewhat important. $\{3,6,15\}$ happen to be three of the first 5 triangular numbers (numbers of the form $\sum\limits_{i=1}^n i$). Further checking reveals that the rest of the numbers on the list are also triangular numbers, but with some missing. ...

2

Each division multiplies the number of small triangles by $4$, and you start with one triangle at $0$ divisions, so the number of triangles after $n$ divisions is $4^n$. The number of distinct horizontal sides of small triangles after $n$ divisions is $$\sum_{k=1}^{2^n}k=\frac{2^n(2^n+1)}2=2^{n-1}(2^n+1)\;.$$ There are the same number of distinct sides of ...

2

The notation $$\sum_{k=1}^n 3^k$$ is a sort of a shorthand for $$3^1 + 3^2 + \cdots + 3^n$$ except that instead of requiring that you figure out what the $\cdots$ part should mean, it makes this explicit: You write the “$3^k$” part $n$ times, once with $k$ replaced by each integer from $1$ to $n$ inclusive, and then you add up the resulting $n$ ...

0

$P$ is a function, $n$ is the variable. I suppose the 3 is quite self-explanatory. $\sum_{k=1}^{n}{3^n}$ just means the sum of all values of $3^n$ as $k$ goes from 1 to $n$. So for example for $n=3$, the sum would be $3^3+3^3+3^3=81$, since there are three distinct values for $k$ as it goes from 1 to 3 (namely 1, 2, 3). Therefore$P(3)=3+81=84$.

2

This is a typo. It should probably be $P(n)=3+\sum_{k=1}^n 3^k$. $P$ is just some function, a way of defining various numbers, one for each number $n$. The parentheses just indicate that $n$ is what we're associating to $P$. So for instance the number $P(2)$ is $3+\sum_{k=1}^2 3^k=3+3+3^2=15$. But this explanation might not be clear: if you don't know what ...

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