Denseness of $C^{\infty}(M)$ within $C^k(M)$
Let $ M $ be a compact smooth manifold. For $ k \geq 1 $, let $ C^k(M) $ denote the space of real-valued functions on $ M $ of class $ C^k $, equipped with the uniform $ C^k $ norm—that is, the sum of...
View ArticleFinding $\lim \frac{(2n^{\frac 1n}-1)^n}{n^2}$.
I want to find limit of$\displaystyle\quad a_n = \frac{\left(2n^{1/n} - 1\right)^{n}}{n^{2}}\quad\mbox{as}\ n \to \infty$.$\displaystyle a_{n} =\frac{\left( 2n^{\frac{1}{n}} -1\right)^{n}}{n^{2}}...
View Article$A= \{(x,y,z) \in \mathbb{R}^3:x^2+2y^2+z^2 < 4z\}$ limit: $\lim_{n \to...
$A= \{(x,y,z) \in \mathbb{R}^3:x^2+2y^2+z^2<4z\}$. Calculate the limit:$$\lim_{n \to \infty}\frac{1}{n} \int_A \frac{y^2z}{ln(x^2+2y^2+n) - ln(n)} \ d\lambda_3$$Solution:First, I can parametrize the...
View ArticleTriangle Inequality and bounds
I need help answering the following question:Use triangle Inequality and reverse triangle inequality to find upper bound for $|(x^2-3)/(x-2)|$ if $x$ ranges over $|x-1| \lt \frac23$I'm having trouble...
View ArticleNewton approximation in Tao Analysis 1
I'm reading Tao's Analysis 1 and I'm confused about this. Help!Definitions that are relevant:Differentiability at a point:Let $X$ be a subset of $\mathbb{R}$, and let $x_0 \in X$ be an element of $X$...
View ArticleProve that $\lim_{t\to 0}\left(\sum_{i=1}^n\alpha_ix_i^t\right)^\frac 1t...
Let $x=(x_1,\cdots,x_n)$, $\alpha=(\alpha_1,\cdots,\alpha_n)$$\sum_{i=1}^n\alpha_i=1$, we have $\alpha>0,x_i\ge 0$ for all $i$. Define $$M_t(x,\alpha)=\left(\sum_{i=1}^n\alpha_ix_i^t\right)^\frac...
View ArticleProving existence of certain step functions implies integrability
In problem 26(b) of Chapter 13 of Spivak's Calculus, the following problem is givenSuppose that for all $\epsilon>0$, there are step functions $s_1\le f$ and $s_2\ge f$ such that $\int_a^bs_2...
View ArticleOn the geometric meaning of restricting $f$ along a curve
Forgive me in advance for this probably stupid question. I was thinking about when we restrict a given function $f(x, y)$ to a curve, in order to study certain properties in an easier way. For example...
View ArticleNonstandard Analysis research project ideas
Before I ask my question, this is my first post on MSE, so I apologise if I have not met the standard etiquette of a post.I have finished my first year at a UK univeristy for a maths degree. In our...
View ArticleIf $\forall n,\sum_ka_{n,k}^2
Let $\ell^2$ denote the metric space of all the square-summable sequences of real numbers. Let $p_n = \left( a_{n1}, a_{n2}, a_{n3}, \ldots \right)$ for $n = 1, 2, 3, \ldots$ be a sequence of points in...
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Urysohn's Lemma from RCA Rudin
I found out the proof of Urysohn's Lemma from Rudin's book but I have couple questions which I am not able to answer.1) Why Rudin wrote that "in terms of characteristic functions, the conclusion...
View ArticleWill the following Method of engineering analysis work?
Analytical Engineering Analysis of 3D ShapesUsing volume integral($\iiint_{}^{}{f(t)}dx dy dz$) to do a AnalyticalEngineering Analysis of 3D Shapes without using mesh based FEA. Likeintegration...
View ArticleInformations about a sequence from tail behaviour
Suppose $\{c_n\}_n$ is a sequence of non negative reals. We have the following three informations about it.(a) $\sum_{k \ge n}c_k \sim \frac{1}{2n}$(b) $\sum_{k=2}^n \frac{kc_k}{\log n} \to...
View ArticleThere is at least one point of every non-empty open subset of the $\ell^2$...
Here we take$$\mathbb{N} := \{ 1, 2, 3, \ldots \}.$$Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\lvert...
View Articlepasting lemma for smooth functions example
Can someone give me the example of two smooth functions $f(x)$ and $g(x)$ defined on closed set $A$ and $B$ and smooth on that domain such that h(x) defined as follows $$h(x)=\begin{cases}f(x), &...
View ArticleConditions for function to be periodic
I am investigating the following type of functions\begin{equation}I(\alpha) = \int_{0}^{\pi}f(t)\cos(\alpha t)\,\mathrm{d}t\,.\end{equation}where $f(t)$ is a real-valued function with non-negative...
View ArticleA corollary of Rolle's theorem
Let $a,b \in \mathbb{R}$ such that $a<b$. Let $f$ differentiable on $[a,b]$ such that $f'(a) < 0 < f'(b)$.Show that : $$\exists c\in ]a,b[, \hspace{1mm} f'(c)=0$$My attempt :Since the...
View ArticleIntegral of Thomae's function
Define the following function $f : \mathbb{R} \to \mathbb{R}$ as follows:$f(x) = 0$ if $x$ is irrational$f(x) = 1/n$ if $x$ is rational and $x = \frac{m}{n}$ in lowest terms.Some googling comes up that...
View ArticleUnderstanding Rudin's PMA Theorem 9.17
$9.17$ Theorem Suppose f maps an open set $E\subset R^n$ into $R^m,$ and f is differentiable at a point $\mathbf{x}\in E.$ Then the partial derivatives $D_jf_i(\mathbf{x})$ exist, and...
View Article$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$.Calculate $\int_A...
$A = \{ x^2 + y^2 + z^2 < 2x + 2y \} \subset \mathbb{R}^3$Calculate:$$\int_A xyz \ d \lambda_3$$Solution:We know that: $x^2 + y^2 + z^2 > 0$ and therefore $2x + 2y > 0 \iff x + y > 0$We can...
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