0
votes
1answer
52 views

Restricting the domain of an integral on a manifold

I would like to prove the following: Guess. Suppose M is an orientable smooth manifold without boundary and W is an open set in M. If $\omega$ is a smooth n-form on M such that $\operatorname{supp} ...
2
votes
0answers
128 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 $\omega\in\Omega^2(S^2)$: $$\int_{S^2}\omega=\int_{B^2}F_+^*\omega-\int_{B_2}F_-^*\omega$$ ...
8
votes
1answer
411 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 ...
11
votes
1answer
499 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 ...
1
vote
2answers
194 views

Do we need additional assumptions for problem 3-37 (b) in Spivak´s calculus on Manifolds?

Problem 3-37 (b) reads: Let $A_{n}=[1-1/2^{n},1-1/2^{n+1}]$. Suppose that $f:(0,1)\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for any $x\notin$ any $A_{n}$. Show that ...
3
votes
2answers
346 views

Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables

Problem 3-29 (p. 61) in the section treating Fubini´s theorem reads: Use Fubini´s theorem to derive an expression for the volume of a set of $\mathbb{R}^{3}$ obtained by revolving a Jordan-measurable ...
0
votes
1answer
148 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...