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\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ... 2answers 96 views The euclidean space \Bbb R^n is orientable as a manifold. I know that The euclidean space \Bbb R^n is orientable as a manifold. I think that it is orientable because it has a nowhere vanishing n-form. But I am not sure. Please can you explain ... 0answers 33 views Stoke's theorem application to curl theorem. I did. Please can you check it? Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept w=Pdy∧dz +Q dz∧dx + R dx∧dy \in Ω^2(M) dim(M)=2 M is the subset of \Bbb ... 0answers 50 views Real projective space is Hausdorff I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ... 1answer 53 views Real Projective Space How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me. 1answer 31 views Locally finite or not I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you 1answer 72 views What is overlop I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ... 2answers 54 views An open cover that is not locally finite I could not understand that why is not locally finite for example 13.4 can you give me explanation please. 0answers 74 views Show that the projection map is Orientation preserving iff n is even My question is that Orient the unit sphere S^n in \Bbb R^{n+1} as the boundary of the closed unit ball. Let U be the upper hemisphere U ={x∈S^n |x^{n+1} >0}. It is a coordinate chart on ... 1answer 52 views Manifolds with boundary and definition Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ... 1answer 45 views Boundary orientation for a cylinder Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :( 1answer 33 views Now I am asking that the topological and manifold boudary for real line I am grateful to explain me more clearly and instructively. Let M be the subset [0,1[ ∪  {2} of the real line. Find its topological boundary bd(M) and its manifold boundary ∂ M. I know that while I find the topological boundary, I need to show ... 1answer 68 views The open Möbius Band is not orientable Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points (0,0) and (1,0) are also e_{1},e_{2} 1answer 81 views Why is the cylinder surface on \Bbb R^3 orientable? Why is the cylinder surface on \Bbb R^3 orientable? Please can someone explain me clearly? 1answer 58 views Diffeomorphism of open intervals in \mathbb{R} with specified values I know two open intervals on \mathbb{R} are diffeomorphic to each other. My question is if I have a intervals (a-\varepsilon,b+\varepsilon) and (c-\delta,d+\delta), is there a diffeomorphism ... 1answer 55 views F(x,y) = (x^2 +y^2,xy). compute F^{∗}(u \, du+v \, dv) Let F : \Bbb R^2 → \Bbb R^2 be given by If u,v are the standard coordinates on the target \Bbb R^2, compute F^{∗}(u \, du+v \, dv).F(x,y) = (x^2 +y^2,xy).$$I am confused so much. I ... 1answer 49 views Partitions of unity and bump function I can not image this guestion in my mind.can you give me graph and help how ı can prove this question please. 0answers 56 views Prove a torus is a 2-dimension C∞-Manifold and find its tangent space (i) A torus is a doughnut-shaped surface in \mathbb{R}^3 that can be constructed as follows. Let a > b > 0 and consider the circle C of radius a in the xy-plane. By definition, the ... 1answer 59 views Find a 1-form ω on \mathbb R^2 −\{(0,0)\} such that ω(X) = 1 and ω(Y) = 0. Please ı dont know what I need to do. thus, help me to solve. 1answer 48 views Show that this is a diffeomorphism I have a function F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2 with (\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi))) and want to show that this is a smooth(meaning ... 1answer 33 views prove that supp(π^{∗} f) = (supp f)×N. Please can you check my answer? Also more explanation please. My question is that Let f \colon M \to R be a C^{\infty} function on a manifold M. If N is another manifold and π \colon M \times N \to M is the projection onto the first factor, prove ... 2answers 81 views Lie bracket in local coordinates. Find the formula c^{k} in terms of a^{i} and b^{j} This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help. 1answer 64 views How are the isometries h:(\mathbb{R}^n,||\cdot||_p)\longrightarrow(\mathbb{R}^n,||\cdot||_p)\;? An isometry of \mathbb{R}^n is a function h:\mathbb{R}^n\longrightarrow\mathbb{R}^n that preserves the distance between vectors:$$||h(x)-h(y)||_p=||x-y||_p\;\;, \;\;p\ge1$$for all x and y ... 1answer 54 views If \alpha:[a,b]\longrightarrow \mathbb{R}^n is a continuous function and injective \Longrightarrow  \text{int}(\alpha([a,b]))=\emptyset\;? If \alpha:[a,b]\longrightarrow \mathbb{R}^n a continuous function and injective, n>1 We can say that \text{int}(\alpha([a,b]))=\emptyset ? Any hints would be appreciated. 1answer 50 views  l(\alpha)=\int_a^b\|\alpha '(t)\| \;\mbox{dt} , \alpha a differentiable function whose derivative is integrable. Let \alpha:[a,b]\longrightarrow \mathbb{R}^n a differentiable function whose derivative is integrable. We can say that \displaystyle l(\alpha)=\int_a^b\|\alpha '(t)\| \;\mbox{d}t\;? , \alpha ... 2answers 92 views Implicit function theorem - how to approach? I have a question that I have been working on for a while. I was wondering how I should approach the following question: Are there any points on the graph of the equation$$x^3+3xy^2+2xy^3=1$$... 0answers 57 views State of the art of the Implicit Function Theorem What is the most general form of the Implicit Function Theorem? Quite a general form of this theorem was given by Kumagai (1980): An implicit function theorem. So I am wondering what are the weakest ... 4answers 171 views why \iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy means surface area? Why the following integral means the area of surface f(x,y)=z?$$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$1answer 48 views Every closed C^1 curve in \mathbb R^3 \setminus \{ 0 \} is the boundary of some C^1 2-surface \Sigma \subset \mathbb R^3 \setminus \{ 0 \} How can I prove it? This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof. (Proving this will help me apply Stokes' theorem to ... 0answers 116 views Baby Rudin, Chapter 10, Problem 23 (d) - Differential forms. In problems 21 and 22, Rudin defines the differential forms \eta=\dfrac{xdy-ydx}{x^2+y^2} and \zeta=\dfrac{x dy \wedge dz+ydz \wedge dx+z dx \wedge dy}{r^3} and the reader is asked to prove ... 2answers 80 views Is \omega_n exact in \mathbb R^n -\{0 \}? For n \ge 2 consider the differential form \omega_n=r^{-n} \sum_{i=1}^n(-1)^{i-1}x_idx_1 \wedge \ldots \wedge dx_{i-1} \wedge dx_{i+1} \wedge \ldots \wedge dx_n, defined on \mathbb R^n \setminus ... 1answer 58 views Is \zeta=\frac{x dy \wedge dz+y dz \wedge dx+z dx \wedge dy}{r^3} exact in the complement of every line through the origin? r=\sqrt{x^2+y^2+z^2} of course. If the line is the z axis, it is given in the book (Rudin) that \zeta=d \left( -\dfrac{z}{r} \dfrac{xdy-ydx}{x^2+y^2} \right) I've managed to figure out 2 ... 0answers 48 views Difference between distance between two points and metric if i have a line element given e.g. ds^2=\frac{dx^2+dy^2}{2y}  is it then always possible to derive a distance between two points in this metric? and how would one determine the length of a curve if ... 1answer 133 views shortest distance between two points on S^2 Length of Curve in 2D is l_{\gamma}(\mathbb{R}^2)=\int_{0}^{1}\sqrt{(dr/dt)^2+r^2(d\theta/dt)^2} Length of a curve in 3D is ... 1answer 91 views Length of a curve on S^2 1. Could any one tell me what is the shortest distance between 2 points on S^2? 2. Could any one tell me how to measure explicitly a length of a curve on the S^2 using polar co-ordinates? ... 0answers 73 views Prove \left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$\left.{\partial X\over \partial Y}\right|_Z=-\frac{\left.{\partial Z\over \partial Y}\right|_X}{ \left.{\partial Z\over \partial X}\right|_Y}$$The above is an identity frequently used in ... 1answer 60 views Rademacher theorem for Riemannian manifold Let M be an open set of \mathbb R^n  and let  ds^2  be some metric on M. Let  d  be the distance induced by  ds^2  on M. If  f  is a Lipschitz function with respect to  d , is it ... 2answers 74 views Composite of an immersion with the inverse map of another immersion is a diffeomorphism Let U\subset \mathbb{R}^k be an open set, n>k and \varphi_1,\varphi_2 : U\to \mathbb{R}^n be immersions, meaning continuously differentiable such that the differential taken in any point of ... 1answer 87 views Implicit function theorem-show that in a neighbourhood of the point -can be described by a pair of functions Let g_1(x,y_1,y_2)= x^2(y_1^2+y_2^2)-5 and g_2(x,y_1,y_2)=(x-y_2)^2+y_1^2-2. Use implicit function theorem to show that in a neighbourhood of the point x=1, y_1=-1, y_2=2, the curve of ... 1answer 120 views any two simply connected open set in the plane R^2 are diffeomorphic Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ... 1answer 70 views Equivalent definition of Tangent Spaces There are about 4 definitions of tangent spaces 1) using velocities of curve 2) via derivations 3)via cotangent spaces 4) as directional derivatives. I am not getting the intuition about what tangent ... 1answer 35 views Find tangential plane of a surface given as an equation on a given point I have a surface given by the equation f(x,y,z)=0 and need to find the Tangential plane on a given point p. 2answers 195 views Why does the condition of a function being differentiable always require an open domain? Going through Spivak's Calculus on Manifolds and in his definition of a differentiable function from a subset A of \mathbb{R}^n to \mathbb{R}^m, f is said to be differentiable if it can be ... 0answers 54 views Computing evolute o f a curve by finding the surronding of the normal rects. So, I have to compute the evolute of the curve: y^{2} = 2px But I have to do that by computing the surronding of the normal rect family. So I start taking the positive side of the function and ... 0answers 44 views what are conormal distributions? According to the first answer in this post, a conormal distribution u on a manifold X relative to a (closed, embedded) submanifold Y is an element of a Banach (or Hilbert) space H such that ... 1answer 39 views Hessian equivalence Let F: R^n \longrightarrow R be twice differentiable and x,y \in R^n with F(x)=F(y). Further let \phi [0,1] \rightarrow R^n be a nice curve with \phi(0)=x and \phi(1)=y. If we know that ... 1answer 121 views Uniform convergence in \mathbb{R}^2 a, b are 2 points in \mathbb{R}^{2},\rho_{n}(t)\,:\,[0,1]\to\mathbb{R}^{2} is a sequence of continuously differentiable constant speed curves with \|\rho_n(t)\|=L_n for all t from 0 to ... 3answers 78 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 ...
Let $U$ be an open subset of $\mathbb{R}^2$ and $f:U\to \mathbb{R}$ a differentiable function. Let $S=\{(x,y,f(x,y)): x\in U\}$ be the graph of $f$. Let $V$ be an open subset of $\mathbb{R}^2$ and ...
Let $U$ be an open set in $\mathbb{R}^2$ and $F:U\to \mathbb{R}^3$ a one-to-one differentiable function such that its inverse from $F(U)$ to $\mathbb{R}^2$ is also continuous. Is it possible that ...