All Questions
24
questions
3
votes
1
answer
109
views
Stokes theorem for currents on manifolds with corners
Let $M\subset\mathbb R^N$ be a compact oriented $n$-(sub)manifold with corners and $\omega$ be an $(n-1)$-form on it. The usual statement of Stokes theorem
$$\int_M d\omega=\int_{\partial M}\omega$$
...
- 1,851
2
votes
0
answers
85
views
Integral inequalities with total variation measure
I know and I can prove that, given $f:\Omega \rightarrow \mathbb{R}$ in $L^1_{|\mu|}(\Omega)$, then
$$\left|\int_{\Omega}f\,d\mu\right|\leq \int_{\Omega}|f|d|\mu|$$
where $|\mu|$ is the total ...
- 45
5
votes
1
answer
342
views
Confusion on integration by parts on a Riemannian manifold
For two vector fields $X$ and $Y$ on a Riemannian manifold $M$ with metric $g$, we define
$$\langle X, Y \rangle_{L^2} = \int_M g_{ij}X^iY^j dV.$$
I have been unable to find a similar expression for ...
- 6,277
0
votes
1
answer
128
views
Calculate the part of the sphere that lies inside the cylinder.
Hey I have problems with this problem.
Let $\mathbb{R}^3$ be a sphere, described by $x^2 +y^2 +z^2 ≤ R^2$ , and a cylinder, described by $(x − R/2 )^2 + y^2 ≤ ( R/2 )^2$ .
a) Calculate the part of the ...
- 900
0
votes
1
answer
66
views
Function with harmonic properties
Let $g(z)$ be a continuous function on $\mathbb R^n\setminus \{0\}$.
$$ \int_{B_R(0)} |g(z)| dz \leq C_1 $$ for some constant $C_1$, and with $B_R (0)$ being the ball of radius $R$ centered at the ...
- 1,065
1
vote
1
answer
210
views
Application of Fubini theorem to a proof of the coarea formula, or why the product of $\mathcal H^{n-m}$ with $\mathcal L^m$ equals $\mathcal L^n$
While reading the proof of the coarea formula in Evans and Gariepy's book, Measure Theory and Fine Properties of Functions, I stumbled upon the following affirmation
Let $A\subseteq\mathbb R^n$ be $\...
- 2,743
2
votes
1
answer
599
views
Can I write the integral of a function in terms of its level sets?
I have a function like this $f:\mathbb{R}^2\to\mathbb{R}_+$
$$f(x, y) = e^{-\frac{x^2+y^2}{2}}$$
Its level sets $f(x, y) = c$ are simply circles centered at the origin
$$
x^2 + y^2 = \log\left(\frac{1}...
user318854
0
votes
1
answer
163
views
If $R\subseteq\Bbb{R}^n$ is a rectangle then $m(R)=0\Leftrightarrow v(R)=0\Leftrightarrow\overset{°}R=\varnothing$.
What shown below is a reference from "Analysis on manifolds" by James R. Munkres
Definition
Let $A$ a subset of $\Bbb{R}^n$. We say $A$ has measure zero in $\Bbb{R}^n$ if for every $\...
- 4,906
1
vote
0
answers
162
views
Under what conditions can we use change of variables in an integral with respect to the Hausdorff measure?
Let ${\cal H}^m$ be the $m$-dimensional Hausdorff measure, $A$ be a ${\cal H}^m$-measurable subset of $\mathbb{R}^n$ ($m<n$), and $f:A\rightarrow \mathbb{R}^n$ be an injective Lipschitz function.
...
- 31
1
vote
0
answers
159
views
Integrating and extending differential forms
I am looking for some clarification regarding integrating differential forms. Hopefully this is very basic.
Suppose that I have a compact Riemannian manifold of dimension $n$, say $M$, and a ...
- 155
6
votes
2
answers
222
views
A counter-example for integration by parts when there are "small" singularities
I am looking for a "counter-example" to integration by parts of the following type:
$\Omega \subseteq \mathbb R^n$ is an open, bounded, connected domain with smooth boundary. $u,v:\bar \Omega \to \...
- 25.3k
1
vote
1
answer
189
views
formula of the Lebesgue measure of $E$ in terms of the integral regarding the Hausdorff measure
Let $E\subset\mathbb{R}^n$ be such that for the boundary of $E$ holds $\partial E=\{(1+u(x))x \mid x\in \partial B_1(0)\}$, where $u:\partial B_1(0)\to (-1,\infty)$ is a function of class $C^1$, and $\...
4
votes
1
answer
374
views
Prove that if $E \subset \mathbb{R}^n $ has finite perimeter, then almost every vertical slice has finite perimeter too.
Before explaining my problem, I recall the definitions:
Let $E \subset \mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K \...
- 281
0
votes
1
answer
80
views
understanding an estimation of the perimeter of sets, $|P(\hat{V})-P(V)|\le P(B_r(x))$
Let $V$ be a Borel set in $\mathbb{R}^n$ such that the Lebesgue measure of $V$, $|V|$, satisfies $|V|\approx |B_1(0)|$, but $|V|\neq |B_1(0)|$ (i.e. $|V|$ is slightly greater or less than $|B_1(0)|$).
...
7
votes
1
answer
171
views
Constructing a function with constant line integral in $\mathbb{R}^n$?
Suppose, $\Omega \subset \mathbb{R}^n$ is a bounded convex set. If, there is an integrable function $f:\Omega \to \mathbb{R}$ s.t. $$\int_{\Omega \cap \ell} f = 1$$ for every line $\ell \subset \...
- 18k