-1
votes
0answers
20 views

Max value of sine wave using any 3 points

Measuring a circular part height with 3 points 120 deg. apart. How do you calculate the maximum height of the part.
1
vote
0answers
39 views

Proving $\sigma$-additivity and interchanging order of summation/integration just because positive

Prove $\sigma$-additivity in the ff: Let $\Omega = {\omega_1, \omega_2, ...}$ be some countable set. Let $\mathfrak{F} = 2^{\Omega}$. Consider a sequence {$p_n$} in [0,1] s.t. $\sum_{n=1}^{\infty} ...
0
votes
1answer
43 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
1
vote
0answers
26 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
1
vote
0answers
17 views

Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
2
votes
0answers
57 views

Strategies for swapping the order of integration with dependent bounds

What are the general strategies for swapping the order of integration given dependent bounds? Specifically, in $\mathbb{R}^2$, Fubini's theorem allows us the following $$ \int_{a}^b\int_{c}^d ...
3
votes
1answer
59 views

Show that B is measurable

Let $\mathbf{g}:\Delta \subset \mathbb{R}^n \rightarrow D\subset \mathbb{R}^n $ be univalent, $C^{(1)}$ and with the Jacobian different from zero, $\forall t \in \Delta$. Let $B=\mathbf{g}^{-1}(A)$, ...
0
votes
1answer
40 views

Integral converges in $E^n$

How do I prove that $\displaystyle \int_{E^n}|\mathbf{x}|^{-|\mathbf{x}|}d\lambda(\mathbf{x})$, where $\lambda$ is lebesgue measure, converges? I was thinking of finding an upper bound function for ...
0
votes
1answer
109 views

If a function is differentiable almost everywhere, can it be written as an integral?

Consider a function $f:\mathbb{R}^n \to \mathbb{R}$. If $f$ is differentiable with Lebesgue integrable derivative, we may write $$ f(x+y) - f(x) = \sum_{i=1}^p \int_0^1 y_i \nabla f_i(x+ty)dt $$ by ...
3
votes
1answer
157 views

Problem 3-26 in Spivak´s Calculus on Manifolds

Let $ f: [a,b] \to \mathbf{R}$ be integrable and non-negative and let $$ A_f = \{ (x,y) : a \leq x \leq b \mbox{ and } 0 \leq y \leq f(x)\}$$ Show that $A_f$ is Jordanmeasurable and has area $ ...
0
votes
3answers
54 views

Why and when $\lim_{r\to0}\int_{\partial B(x,r)}u(y)\;dS(y)=u(x)$?

Let $U\subset\mathbb{R}^n$ be an open set, $x\in U$ and $u\in C^2(U)$ a harmonic function. I would like know what is the theorem that is used to conclude that $$\lim_{r\to0}\int_{\partial ...
2
votes
1answer
92 views

Derive $\frac1n \|x\|_p^p \leq \|x\| \leq n^{p/2}\|x\|_p^p$ from Holder's inequality?

Given a vector $x = (x_1, \dotsc, x_n)\in \mathbb{C}^n$, I wanted to compare $|x_1|^p + \dotsb + |x_n|^p$ to $\|x\|^p$. I discovered that if $m=\max_i|x_i|$, we have $$m^p \leq \|x\|^p \leq ...
3
votes
1answer
57 views

Integral of the gradient of a semilinear function.

Let $u:\mathbb{R}^n \rightarrow \mathbb{R}$ be a semilinear map so that for any $k\in \mathbb{R}$ the surface $\partial C_k^+=\{x:k=u(x)\}$ is contained in the union of finitely many hyperplanes. ...
1
vote
1answer
59 views

Ham sandwich for measuers implies the classical one

Ham sandwich theorem for measures: Let $\mu_1,\mu_2,\mu_3 $ be finite Borel measures on $\mathbb{R^3}$ such that every hyperplane has measure $0$ for each of $\mu_i$. Then there exists a ...
1
vote
0answers
243 views

Euler's theorem for homogeneous functions

Let $\textbf{R}_{+}$ be the set of positive real numbers. The following is a well-known theorem due to Euler: A differentiable function $f:\textbf{R}^n_{+} \rightarrow \textbf{R}_{+}$ is positively ...
3
votes
1answer
118 views

Show $g(\mathbf{x}) \leq h(\mathbf{x})$ implies $\int g(\mathbf{x})\mathrm{d}\mathbf{x} \leq \int h(\mathbf{x})\mathrm{d}\mathbf{x}$

Suppose I have $g$ and $h$ from $\mathbb{R}^p\to\mathbb{R}$ such that for all $\mathbf{x}$, $g(\mathbf{x}) \leq h(\mathbf{x})$. I want to prove that the integral over all $\mathbb{R}^p$ of $g$ is less ...
3
votes
0answers
91 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
2
votes
1answer
204 views

Is there a measure

Is there a measure $\nu$ on $[0,\infty)$ such that $$ \ln x=\int_{0}^{\infty}d\nu\left(y\right)/\left(x+y-1\right)? $$ Thanks for any helpful answers!
2
votes
0answers
131 views

Showing S is Jordan Measurable and Calculating the Volume

If S is the solid obtained by intersecting the ball $x^2+y^2+z^2\le4$ and $x^2+z^2\le1$ 1) How do I show that S is Jordan measurable? Can I simply say the following: "Clearly S is bounded, and the ...
2
votes
2answers
139 views

Integral variable substitution using Hausdorff measure

Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral: $$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$ where $B \subset ...
31
votes
2answers
1k views

Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject(single-variable calculus): the ...
3
votes
0answers
86 views

Notation for a certain kind of discrete measure

Suppose $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ is smooth, $Z=\{x: \phi(x)=0\}$ and $D\phi\neq0$ on $Z$. Is anyone familiar with use of the notation $dZ$ for the measure $$\sum_{x \in Z} ...