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2
votes
3answers
76 views

What's “the most right” symbol to use for “defined to be equal to”?

What's the most used symbol for "defined to be equal to", at least in your experience (and I'm sure there are a lot of experienced people here)? Also, which one do you think is the "the most right" of ...
0
votes
2answers
102 views

Uncommon question for everyone that is related to mathematics [duplicate]

I'm still a student and something funny that I've seen is that many mathematicians evade the question, "What is mathematics?" And when they don't evade it, every person gives a different definition. ...
6
votes
1answer
96 views

Is the actual learning of a course done in second reading?

I am now a graduate student. When I was an undergraduate, I would mostly study on my own directly from the books, and not concentrate much on lectures. Also, since I was a pretty good student, I would ...
0
votes
0answers
15 views

Rotation of curves

It is a pretty simple question, but I can only find specialized answers. How can I rotate any curve by any angle in a graph and find the equation that describes it? Is there a methodology that I have ...
4
votes
2answers
278 views

I can do the math but not the problems, help? [closed]

So whenever my instructor / teacher is going over the notes and teaching the new lesson to the class, I listen to what he says, take notes, and do the practice problems along with him. Often times I ...
74
votes
9answers
4k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
0
votes
0answers
64 views

Hard-to-put-together but easy-to-prove results

What are the most important examples of theorems and definitions which are post factum obvious, i.e., hard to put together but easy to understand and use (and prove, in the case of theorems) once you ...
23
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
1
vote
1answer
30 views

Tips on “short” math lecture presentation for compsci students

I just got interview to teach college (of applied science) compsci student math. This is the first time I get this type of interview. The type of interview is called a "hearing" where a dozen people ...
7
votes
2answers
93 views

Are these proposed rules for the canonical factorization of algebraic integers complete?

In $\mathbb{Z}$, the rules are fairly well established, a few minor quibbles notwithstanding. But in, say, $\mathbb{Z}[\sqrt{7}]$, there are, as far as I can tell, no established rules. What I've seen ...
2
votes
0answers
54 views

pictures or graphs for real analysis.

Most of my previous math courses have had some graphs or pictures to help explain the ideas and concepts. I have tried looking for visual representation of the concepts of Real analysis; however, I ...
5
votes
2answers
112 views

Construct numbers using digits $123456789$ and the operations $+,-,×,÷$

From an old book I found the following question. Use the digits $1,2,3,4,5,6,7,8,9$ and the operations $"+,-,×,÷"$ with $( )$ for construct the result $100.$ During the computations the order of ...
10
votes
2answers
109 views

Is the “product rule” for the boundary of a Cartesian product of closed sets an accident?

Given two closed sets $A$ and $B$ living in topological spaces $X$ and $Y$, the boundary of $A\times B$ in the product topology, denoted (suggestively) by $\partial(A\times B)$, is given by ...
4
votes
0answers
66 views

Do Natural transformations make 'God given' precise?

Often in mathematics, one encounters certain structures in which there are natural choices to be made. For example, there could be a 'god given' map relating two mathematical objects, in the sense ...
-1
votes
4answers
55 views

Intuitive way of solving this inequality: $x^2>y^2 \wedge x>0 \Rightarrow x>y$

I have to prove that $x^2>y^2 \wedge x>0 \Rightarrow x>y$. I decided to do this: $x^2>y^2 \Leftrightarrow \sqrt{x^2}>\sqrt{y^2}\Leftrightarrow |x|>|y| \Leftrightarrow x>|y| \vee ...
3
votes
0answers
90 views

What is set theory about? [closed]

What is the subject set theory about? What knowledge is required? I am thinking about what subjects I will choose, but I am not sure if I should take this one. That is the course content: Brief ...
0
votes
2answers
67 views

How do you represent $\Im$ and $\Re$ on paper?

Do you draw $\Im$ and $\Re$ just like they are or you write $\mathtt {Im}$ and $\mathtt{Re}$?
0
votes
4answers
48 views

Explaining what is Pathwise-connectedness.

I'm an average guy but interested in explaining myself maths through illustrations and intuition(which at times fails!!!).I'm preparing myself for my calculus of several variables exam. I'm studying ...
1
vote
0answers
38 views

No simple closed form for Bell numbers

The Bell number $B_n$ is the number of partitions of $[n]$. Unlike other basic combinatorial quantities, $B_n$ has no simple finite closed form. This seems surprising to me. Can anyone explain why ...
6
votes
8answers
210 views

Calculate $\pi$ By Hand?

All over the internet the only hand equation i found was $$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$ But this takes something like a thousand iterations to get to four digits, is there a ...
3
votes
0answers
34 views

My attempt to gather mathematical spaces in a drawing

There are many spaces (Sobolev, inner product spaces, etc.) that I am not using on a daily basis, but have to recall their definition from time to time. So I decided to draw a synthetic scheme of the ...
1
vote
1answer
84 views

Abstract algebra book suggestion [duplicate]

I have been suggested Artin or Herstein's books. What do you think is more rigorous but at the same time clear and good to read?
2
votes
2answers
53 views

Reverse Polish notation in (abstract) algebra

If I have something like $\phi\circ \psi(x)$ this means first apply $\psi$ and then $\phi$. Going right to left is pretty contrary to my intuition. In computer science some programming languages (and ...
2
votes
2answers
56 views

Does the word 'ten' have a base?

My friends and I had a debate: "Does the word 'ten' have a base?" My Argument: 'ten' is only 10 in base 10 so if i have 10 objects, counting in base 10, when I get to the end of the list, I will ...
2
votes
2answers
47 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
0
votes
1answer
52 views

Fun mathematics exercises [closed]

I'm looking for fun mathematical exercises to do in my spare time, when I say fun I mean exercises which requires original manipulations and also requires some work, it would also be nice if they gave ...
6
votes
0answers
62 views

How to master general topology for analysis?

I started learning topology long ago. I first exposed myself to metric topology in Baby Rudin and Munkres Topology 2nd ed. Part I. Munkres is my most revisited book ever since. The first big ...
1
vote
0answers
26 views

Continuous maps vs. open maps [duplicate]

When we study topology, we typically study topological spaces and continuous maps between them. From a categorical perspective, this is "wrong," because continuous maps are not the structure ...
2
votes
1answer
62 views

How to conceptualize unintuitive topology?

I found Project Origami: Activities for Exploring Mathematics in my university's library the other day and quickly FUBAR'd (folded-up beyond all recognition) the couple sheets of paper I had with me ...
1
vote
1answer
34 views

Differential Structure of Two Dimensional Manifold

I am a beginner in Differential Topology, and have learnt that Theorem One dimensional differential manifold without boundary is diffeomorphic to $\mathbb R$ or $\mathbb S$. So I am curious ...
2
votes
0answers
47 views

Can I use theorems that are beyond the scope of my class in my CBSE exams? [closed]

I am a student of class $9^{th}$ of a CBSE board school.Tommorow I have my half yearly maths exam.My question is that can I use theorems(such as manelaus theorem,stewart's theorem,ceva's ...
4
votes
1answer
90 views

How important is the “prestige” of your university for undergraduate mathematics?

I go to a university in Canada. It's not really considered a top university (it is one of the best universities in the province but is ranked really low nationally). I study mathematics here and ...
4
votes
2answers
52 views

Squaring the Circle

Is squaring the circle possible in any sort of metric at all? It's known that within the Euclidean metric it is impossible, but does there exist some world or space where it is possible? Much like how ...
7
votes
3answers
147 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
2
votes
1answer
46 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
1
vote
4answers
184 views

Why is the expression $\underbrace{n\cdot n\cdot \ldots n}_{k \text{ times}}$ bad?

For writing a (german) article about the power with natural degree I have the following question: In school one defines the power with natural degree via $$n^k = \underbrace{n\cdot n\cdot \ldots ...
6
votes
3answers
381 views

Undergraduate mathematical magazines to improve mathematical knowledge

I'm sorry my ignorance, I don't know very much about mathematical magazines. I'm finishing my master degree in pure mathematics and I'm looking for mathematical magazines which could improve my ...
0
votes
1answer
19 views

Soft: Interpretation of a periodic event on circle group

Recently I've been exploring probability measures on topological groups, derived from the (essentially) unique Haar measure defined thereon. I had begun to focus on the example of the circle group ...
1
vote
1answer
61 views

Book to self-learn probability

I am reading some lecture notes (completed with exercises and competition-like problems) provided by my college professor, but I would like to study probability from a proper book. Can you suggest one ...
2
votes
0answers
19 views

Differential Equation for brownian bridge?

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is ...
4
votes
1answer
254 views

What is meant by “folkloric result” in category theory?

I often will see the word "folklore" used in papers on category theory, e.g. in Barr's paper on Isbell Duality, he states a result, which he proves, is "folkloric", on page 512: The following ...
1
vote
0answers
42 views

Question regarding advanced calculus textbooks

I'd like to start off by mentioning that I'm gonna begin studying Computer Science at the local polytechnic university. As I found out, the study of mathematics stops after two years. Recently I've ...
0
votes
0answers
33 views

Soft question regarding real analysis

I have just started studying ' Principles of mathematical analysis' by Rudin. And I have heard that it is hard to complete . I seriously want to study it and I need some suggestions about how should I ...
0
votes
1answer
26 views

Applications of random walks

I am searching for a clear and interesting exposition of an application of random walks to some physics topics accessible to advanced high school students.
8
votes
4answers
295 views

What is linearity?

Once someone asked me the question "What is linearity?" in a proficiency exam. I went hot and cold all over. Although, I heard and even used the term linearity many many times, I had not really ...
1
vote
4answers
41 views

SAT Math problem

In a volleyball league with 4 teams, each team plays exactly 2 games with each other 3 teams in the league. What is the total number of games played in this league? the book says the answer is 12, i ...
3
votes
1answer
72 views

Why Riemannian metrics have to be smooth?

Why do Riemannian metrics have to be smooth? Can you give an example of a smooth curve with a none smooth metric and show me what possibly will go wrong if our metric is not smooth?
0
votes
2answers
68 views

What is global differential geometry?

What is the difference between local and global differential geometry? I cannot find their (exact) definitions. There are some other terms in geometry like "rigid" (e.g. that structure is more rigid ...
3
votes
0answers
148 views

How is $\sqrt{3} * \sqrt{2} = \sqrt{6}$ mathematically possible? [duplicate]

Well, this question is basically a method to point a loophole in Maths which I'm not sure how it is possible... ...
4
votes
4answers
160 views

Why is differential geometry called differential geometry?

Why is differential geometry called differential geometry? Why it is not called differential and integral geometry? Isn't integration and finding areas as important as differentiation? Is it the case ...