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

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 […]

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 […]

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 […]

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 […]

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

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. […]

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 $c\in\mathbb{R}$

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[ […]

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 $[-M,M]^n$ […]

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