Articles of analysis

General solution for the Eikonal equation $| \nabla u|^2=1$

Does there exist a formula for the general solution of the Eikonal equation? $| \nabla u|^2=1$. I’m looking for something similar to “the general solution of $\dfrac{\partial u}{\partial x}(x,y)=0$ is $u=\varphi(y)$, for an arbitrary function $\varphi$”. That is, the formula should include one arbitrary function. Thank you

$\zeta(2)=\frac{\pi^2}{6}$ proof improvement.

Recently in one of my calculus exercise I have made out a (quite novel to me) proof for $\zeta(2)=\frac{\pi^2}{6}$ via the famous infinite product below: $$\sin(x)=x\prod_{i=1}^{\infty}(1-\frac{x^2}{i^2\pi^2})\tag{0}$$ I feel that the proof which I am to give below should count as one proper way to compute $\zeta(2)$ apart from the others (usually very complicated and requiring […]

Why do differential forms and integrands have different transformation behaviours under diffeomorphisms?

Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold. A differential $n$-form (or rather its coefficient function which is obtained by using the canonical one-chart atlas on $\mathbb R^n$) then transforms essentially by multiplication with $\mathrm{det}(Df)$, while integrals […]

“Proof” that $g(t) = (t,t,t,…)$ is not continuous with uniform topology

Let $g : \mathbb R \to \mathbb R^{\omega}$ be the function $$ g(t) := (t, t, t, \ldots). $$ If $\mathbb R^{\omega}$ is equipped with the uniform topology, and $\mathbb R$ with the standard topology, then is $g$ continuous? According to this post it is. I have a different proof which yields another conclusion, but […]

tangent space of manifold and Kernel

I want to understand the proof of this Theorem : Theorem: Let $ S $ be a submanifold of $ E $ and let $ a \in S $. Assume that there exists a submersion $f : E \supset U \rightarrow F$ of class $ \textit{C}\ ^{1} $ and $ b\in f(U) $ such that […]

Tietze extension theorem for complex valued functions

Why is this theorem always only stated for real valued functions, and not for complex valued functions? Thanks.

Approximation by smooth function while preserving the zero set

Let $\mathbb{T}$ denote the unit circle. Given $f \in \mathcal{C}(\mathbb{T})$, can we approximate $f$ by smooth functions having the same zero set ? i.e. for $\varepsilon >0$, can we find $g \in \mathcal{C}^\infty(\mathbb{T})$ such that f(x) = 0 if and only if $g(x) = 0$; $\|f-g\|_\infty < \varepsilon$. Both tasks can easily be performed separately. […]

Solving a challenging differential equation

How would one go about finding a closed form analytic solution to the following differential equation? $$\frac{d^2y}{dx^2} +(x^4 +x^2+x+c)y(x) =0 $$ where $c\in\mathbb{R}$

A limit about $a_1=1,a_{n+1}=a_n+$

Let the sequence $\{a_n\}$ satisfy $$a_1=1,a_{n+1}=a_n+[\sqrt{a_n}]\quad(n\geq1),$$ where $[x]$ is the integer part of $x$. Find the limit $$\lim\limits_{n\to\infty}\frac{a_n}{n^2}$$. Add: By the Stolz formula, we have \begin{align*} &\mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{n^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{a_{n + 1}} – {a_n}}}{{2n + 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{\left[ […]

Maximum volume change for two sets with small Hausdorff metric in bounded part of $\mathbb{R}^n$

Given two subsets $S_1$, $S_2$ of a bounded part of $\mathbb{R}^n$, say $[-M,M]^n$. Is there a way to relate the difference in volume $vol(S_2)-vol(S_1)$ to the Hausdorff metric distance between the sets $S_1$ and $S_2$ given that $S_1 \subset S_2$? Intuitively I can see that the the fact that $S_1$ and $S_2$ lie in $[-M,M]^n$ […]