Tagged Questions
2
votes
1answer
45 views
How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?
Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$?
Where $M$ is a ...
3
votes
1answer
84 views
Homeomorphism vs diffeomorphism in the definition of k-chain
In "Analysis and Algebra on Differentiable Manifolds", 1st Ed., by Gadea and Masqué, in Problem 3.2.4, the student is asked to prove that circles can not be boundaries of any 2-chain in ...
2
votes
1answer
65 views
Formal finite sum for integration on k-chains
This question is related to the integration of differential forms on chains, as exposed in the document at: http://www.math.upenn.edu/~ryblair/Math%20600/papers/Lec17.pdf . The author gives the ...
6
votes
4answers
172 views
why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?
Why the following integral means the area of surface $f(x,y)=z$?
$$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
1
vote
0answers
69 views
Integration on manifold, pullback
Define the maps $F_\pm:B^2\rightarrow S^2$, $(x,y)\rightarrow(x,y,\pm\sqrt{1-x^2-y^2})$. Prove that for any $\omega\in\Omega^2(S^2)$:
$$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega$$
...
5
votes
1answer
137 views
Closed not exact form on $\mathbb{R}^n\setminus\{0\}$
I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$
I think something like ...
0
votes
1answer
50 views
Proof of the naturality of integration
I have a bit of a problem with the following identity:
Suppose that $U, V \subset \mathbb{R}^n$, are two open sets.
Let $x^1,...,x^n$ be a system of coordinates of $U$ and $y^1,...,y^n$ one on $V$. ...
2
votes
0answers
60 views
Correct use of substitution rule for Integration on Riemannian manifolds
Let $(N,g_N)$ be a Riemannian manifold and let $\psi: M \rightarrow N$ be a
a diffeomorphism. Now I know how the Riemannian metric on $M$ defined by the pull-back of the metric on $N$ looks like (this ...
10
votes
1answer
320 views
Problem 3-38 in Spivak´s Calculus on Manifolds
This is not homework. Problem 3-38 reads:
Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
3
votes
3answers
79 views
understanding change of variable
the following is drawn from a rather rough set of lecture notes and I am not sure I understand it.
the goal is to determine for which values of $p$ we have
$$
\int_{|x|\leq 1} \frac{1}{|x|^p} \,dx ...
3
votes
2answers
201 views
Integral over a torus
I've been asked to solve the following integral:
$$\int_{\mathbb{T}^2} xyz \, dw\wedge dy$$
where $\mathbb{T}^2\subset\mathbb{R}^4$ is the 2-torus defined by: $$w^2+x^2=y^2+z^2=1$$
I've tried ...
3
votes
0answers
56 views
why we cannot integrate on a nonorientable manifold?
I feel it rather weird that there is a notion of integration when you glue a patch of paper to get a surface of cylinder while there is not a suitable notion when you glue it differently to get a ...
1
vote
1answer
37 views
Question about integrable functions on Riemannian manifolds
I was wondering about the sense of the following definition:
Let (M,g) be a Riemmanian manifold and $\varphi\in C^{\infty}(M)$ a smooth function, such that $\varphi\geq 0$ or $supp(\varphi)\subset M$ ...
7
votes
1answer
93 views
What is the average length of all integral curves of a vector field?
Considering a vector field with a source and a sink in a finite comact space, are there any bounds on the length of the integral curves?
Specifically, I am interested in the average length of ...
4
votes
3answers
2k views
Simple proof of integration in polar coordinates?
In every example I saw of integration in polar coordinates the
Jacobian determinant is used, not that i have a problem with the Jacobian,
but I wondered if there's a simpler way to show this which ...
1
vote
0answers
164 views
Gaussian Curvature
Can anyone see the connection between the Gaussian curvature of the ellipsoid $x^2+y^2+az^2-1=0$ where $a>0$ and the integral $\int_0^1 {1\over (1+(a-1)w^2)^{3\over 2}}dw$?
I am guessing ...
3
votes
1answer
298 views
integral of Laplacian of a positive function
I've encountered the following, rather elementary, problem:
$K$ is a compact subset of some 2-dimensional oriented manifold with smooth boundary, $f$ is a positive smooth function on $K$ that ...
9
votes
1answer
260 views
Are integrations on forms “different” from Riemann integrations?
I was amazed by the power of integration on forms when I learned that the Stokes' theorem can be written in a beautiful way (don't assume that I know more than this fact itself):
$$
...
