Recent content by semigroups

Let $$M$$ be a surface with Riemannian metric $$g$$. Recall that an orthonormal framing of $$M$$ is an ordered pair of vector fields $$(E_1,E_2)$$ such that $$g(E_i,E_j)=\delta_{ij}$$. Prove that an orthonormal framing exists iff $$M$$ is orientable and $$M$$ admits a nowhere vanishing vector...

Let $$f:U \to \mathbb{R}^3$$ be a surface, where $$U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$$ Consider the two closed square regions $$4F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, |u^2|\leq1\}$$ and $$F_2=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq2, |u^2|\leq2\}.$$ While the first...

Followed by Willie's hint. Write $$u^1=u^1(v^1,v^2)$$ and $$u^2=u^2(v^1,v^2)$$ to get $$\mathrm{d}u^1 \wedge \mathrm{d}u^2=\mathrm{det}(A)\mathrm{d}v^1 \wedge \mathrm{d}v^2$$where $$A:=\begin{vmatrix}\frac{\partial u^1}{\partial v^1}&\frac{\partial u^1}{\partial v^2}\\ \frac{\partial...

I got $$\mathrm{d}u^1 \wedge \mathrm{d}u^2=\mathrm{det}(\Phi) \mathrm{d}v^1\wedge\mathrm{d}v^2$$. I also know $$g_{ij}=\langle \frac{\partial}{\partial u^i} \frac{\partial}{\partial u^j} \rangle$$, moreover, for a surface $$f: U \to \mathbb{R}^3$$ the components of its first fundamental form $g$...

Let $$(M,g)$$ be an oriented Remannian surface. Then globally $$(M,g)$$ has a canonical area-2 form $$\mathrm{d}M$$ defined by $$\mathrm{d}M=\sqrt{|g|} \mathrm{d}u^1 \wedge \mathrm{d}u^2$$ with respect to a positively oriented chart $$(u_{\alpha}, M_{\alpha})$$ where $$|g|=\mathrm{det}(g_{ij})$$...

Thanks for pointing it out! My answer should be $$(\nabla_X \omega)(Y)=X^iY^j(\omega_{j,i}-\Gamma^{k}_{ij} \omega_k)$$ I'm now trying to differentiate a covariant 2-tensor (may not be a 2-form), namely $$(\nabla_{X}S)(Y,Z)=X^i \frac {\partial S(Y,Z)}{\partial...

Let $$f:U \to \mathbb{R}^3$$ be a surface with local coordinates $$f_i=\frac{\partial f}{\partial u^i}$$. Let $\omega$ be a one-form. I want to express $$\nabla \omega$$ in terms of local coordinates and Christoffel symboles. Where $$\nabla$$ is the Levi-Civita connection (thus it coincides with...

Let RP2 denote the real projective plane (it can be obtained from glueing a Mobius band and a disk whose boundary is the same as the boundary of the Mobius band). I know if one punches a hole off RP2 then the punched RP2 is homotopy equivalent to a Mobius band which is in turn deformable to a...

Thanks for reply! Please find the attachements for deatailed working (due to the size of screen capture I have to divide it into three pictures).

Please see the attachment. Under the given range of parameters the integral converges, but I can't find a closed form solution. It seems one has to integrate very special functions other than simple Beta, Gamma, Error functions etc..

Once (1-x)^n-1 is replaced by (1-a)^n-1 where a>1, the form of the integral changed drastically (I can't find a proper substitution in order to transform the new integral to Beta), does anyone know how to compute the new integral then? (I presume the result will still involve Gamma functions...