# Tagged Questions

**1**

vote

**2**answers

146 views

### How to show this integral on boundary of Lipschitz domain is finite?

Sorry for asking a basic question but this did not get answered on M.SE.
Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that
$$\int_{\partial\Omega} ...

**2**

votes

**2**answers

491 views

### Stokes theorem for manifolds without orientation?

Hello!
In textbooks Stokes theorem is usually formulated for orientable manifolds (At least I couldn't find any version not using orientability). Is Stokes theorem: ...

**11**

votes

**5**answers

2k views

### Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...

**1**

vote

**1**answer

211 views

### evaluating an integral related to the volume of Hessenberg orthogonal matrices

Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
\sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm ...