1
vote
1answer
57 views

Length of a curve by integration: why won't flat segments do?

Maybe my question is a duplicate, but I guess I don't know the right terminology to find it elsewhere. I would be happy to delete it if someone can point out a duplicate. From elementary calculus, ...
0
votes
1answer
99 views

Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
1
vote
0answers
24 views

basic analytical set of tools — integration, differentiation, convergence and handling of applied maths equations

I am more into mathematical logic, algebra, etc. I am asking for a scrib sheet or short and precise collection of a set of tools which somehow demonstrates the everyday set of tools for integration, ...
43
votes
5answers
2k views

The Intuition behind l'Hopitals Rule

I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to ...
0
votes
2answers
125 views

Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
2
votes
2answers
104 views

Convergence of alternating nested radicals

Last evening, after reading a couple of questions about nested radicals, I started to wonder about problems involving what I will term "alternating nested radicals;" below is an example, which I found ...
65
votes
12answers
5k views

Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
7
votes
2answers
182 views

Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega $$ if ...
7
votes
2answers
378 views

Convolution intuition: clarifying Terence Tao's “blurring”/“fuzz” interpretation

On this math.MO post, "What is convolution intuitively?", Terence Tao's answer (in the case where one function is a bump function) involves "blurring" and "fuzz." Could someone clarify his ...
4
votes
1answer
97 views

Geometric intuition behind the Uniform Boundedness Principle

Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
3
votes
1answer
170 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
3
votes
1answer
52 views

conditional convergence

This is an practice question from "Advanced Calculus, Folland" Chapter 6.3, Q.2 (not HW) I am not sure how to go about this question :: suppose $\sum { { a }_{ n } } $ is conditionally convergent. ...
2
votes
3answers
255 views

Limit Question - Explanation

The limit of $f(x) = x$, as $x$ tends to zero is zero. What's the limit of the function $\dfrac{x^2}{x}$ as $x$ tends to zero? and What's the limit of the function f(x) = (modulus of x)/x ? I am ...
4
votes
1answer
363 views

proof of Poisson formula by T. Tao

I do not understand one thing in an article on the blog of Terence Tao: For instance, restricting a function $f: G \rightarrow \mathbb{C}$ to a subgroup $H$ causes the Fourier transform $\hat f$ ...
4
votes
0answers
66 views

Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
1
vote
1answer
318 views

Intuitive explanation of Residue theorem in Complex Analysis

The residue theorem that states that if a) $U$ is a simply connected and open subset of the complex plane, b) $a_1,\dots,a_n$ are finitely many points of $U$, c) and $f$ is a function which is ...
6
votes
2answers
247 views

How to 'analyze' problems in analysis; Computing $\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$

If $a, b \in \mathbb{R}$ with $a > b > 0$, compute this ungodly thing; $$\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$$ I'm really not a fan of complex analysis... I can't visualize ...
0
votes
1answer
94 views

How does a myopic interpret Wiener's Tauberian?

I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...
5
votes
2answers
470 views

Are Legendre transforms of non-convex functions useful?

Do Legendre transforms have any applications that do not appeal to convexity? What is the intuitive interpretation of the Legendre transform of a non-convex function?
2
votes
0answers
115 views

Intuition behind the proof for Wiener's theorem?

I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
11
votes
8answers
6k views

Continuous versus differentiable

A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same ...
2
votes
0answers
68 views

Intuition about moment function derivation [OR] derivatives involving a time varying integration domain

$$ m_{{pq}}(t)=\iint\limits_{R(t)}h(x,y) dx dy $$ where $ R(t)$ the domain of integration is time varying (In fact it is the only one which is time varying). And $$ h(x,y) = x^p y^q f(x,y) dx dy ...
5
votes
0answers
175 views

Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
7
votes
2answers
801 views

The determinant is the integral of algebra. The integral is the determinant of analysis

This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The determinant in algebra has a lot in common with the ...
2
votes
1answer
109 views

Efficient Sampling

I'm trying to sample a lot of points efficiently. I'm wondering if the following method is possible. I sample points of a function (evaluate the function) mod $n$. I.e. I calculate f(element one), ...