# Tagged Questions

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### Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
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### A notational confusion on gradient

Given a parametrized function $f_{w}: \Bbb R ^{m} \to \Bbb R ^{k}, w \in \Bbb R^d$, I see in a book the following notation $\bigtriangledown ^ {w} f_{w}(.)$ denote its gradient w.r.t. $w$. What is the ...
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### A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
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### one problem on multivariable claculus

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes ...
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### For which values of $a$ is this set a manifold?

Let $f:\mathbb{R}^3\to\mathbb{R}, f(x,y,z)=(x-y+z-1)^2$. For which values of $a$ is $\{(x,y,z)\in\mathbb{R}^3:f(x,y,z)=a\}$ a 2-manifold? Instead of $(x-y+z-1)^2=a$ seems a better idea to write ...
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### Prove that inverse of $f$ defines a manifold

Let $f:\mathbb{R}^3\to\mathbb{R}$ be given by $f(x,y,z)=z^2$. Prove that $0$ is not a regular point but $f^{-1}(\{0\})$ is a manifold. I divided this in two parts: $(1)\; 0$ is not a regular ...
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### k+1 Differential form

Consider the k-form given by, $w = \sum_{i_{1}<i_{2}<...<i_{k}} w_{i_{1}i_{2}...i_{k}} dx^{i_{1}}\wedge dx^{i_{2}}\wedge...\wedge dx^{i_{k}}$ Define $k+1$ form $dw$ , the differential of ...
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### Proving a set of $2\times 3$ matrices is a manifold?

The way I have always been told to check if something is a manifold (I haven't had a whole lot of experience with them), is to check if the derivative of the function representing the loci of the ...
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### Directional, differential and lie derivatives on manifolds intuition?

Trying to translate elementary multivariable calculus into the language of manifolds: Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single ...
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### finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
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### Derivative of function on parametrized manifold

Given $U\subset\mathbb{R}^d$. Let $\alpha:U\mapsto \mathbb{R}^k$ be an injective function such that $\alpha(U)$ is $d-$dimensional parametrized manifold. Now define $\beta:\alpha(U)\mapsto U$ by ...
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### Derivative on parametrized manifold

Let $U\subset \mathbb{R}^m$ is open and $\alpha: U\mapsto\mathbb{R}^n$ is $m-$dimensional parametrized manifold with $m\leq n$. I have two following questions: If we are given $v\in \alpha(U)$, ...
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### Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
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### How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $(\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{pmatrix} a & b\\ c & d ...
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### Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
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### Integral curves of $X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}$ on a cylinder

Consider coordinates $(\theta, z)$ on $S^1 \times \mathbb R$, and a vector field $$X = z \dfrac{\partial}{\partial \theta} - \sin \theta \dfrac{\partial}{\partial z}.$$ Show that the integral curve of ...
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### Surface of graph of a function.

Let $G:=\{(x,y,z)\in\mathbb{R}^3:|x|<|z|^2,|y|<|z|,0<z<1\}$ and $f:G\to\mathbb{R},f(x,y,z)=2x+2y+z^3$. Calculate the surface of the graph of f. We recently got introduced to Stokes' ...
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### Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n$ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M$ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
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### Show that $\dot{n_s}=-\kappa_s t$

I found the question in a differential geometry textbook while studying. This question seems so intesting to me. So please help me solving it. Show that, if $\gamma$ is a unit-speed plane curve, ...
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### Prove equivalence of two conditions to be a smooth $k$-manifold $M^k \subseteq \mathbb{R}^n$

For the first couple classes of differential geometry, we have used the more concrete characterization of a manifold (given in #1 below). I am trying to prove that the following two conditions are ...
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### Is there any relationship between Cauchy-Riemann equations and vector fields on manifolds?

Well, suppose we have $f : \mathbb{C} \to \mathbb{C}$ analytic, then if $f = u + iv$ the functions $u,v : \mathbb{C} \to \mathbb{R}$ satisfy the Cauchy-Riemann equations: $D_1u=D_2v$ and $D_2u=-D_1v$. ...
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### local parametrization of regular surface

I am doing excercises of Do Carmo's dg of curves and surfaces Chapter 2.2 and need some help with the following excercise: Show that the set $S=\{(x,y,z)\in R^3;z=x^2-y^2\}$ a regular surface and ...
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### Basic (multivariable) calculus question

I need some help with basic calculus. I asked a question the other day and got a decent answer but there is one step in the answer I just don't understand. Why is ${\partial y_1 \over \partial x_1}$ ...
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### Polar decompostion should be a diffeomorphism, right?

I seem to have gotten stuck in the mud verifying what I thought was going to be a completely straightforward fact. I would appreciate if somebody could help dig me out. Inside the $n \times n$ ...
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### Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
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### Finding a direct basis for tangent space of piece with boundary of an oriented manifold.

I have the following definition (from Hubbard's vector calculus book) for an oriented boundary of piece with boundary of an oriented manifold: Let $M$ be a $k$ dimensional manifold oriented by ...
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### Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
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### Problem about differential of a linear map

Please can you tell how to solve this problem clearly? Please solve this explanatorily. Thank you
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### Compute the velocity vector.

Can you solve explicitly? please. I don't know how to solve. Thank you for help.
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### Is $S$ a regular submanifold?

$M=M_{n\times n}(\Bbb R)$ $S=\operatorname{SL}(n, \Bbb R) = \left \{ A \in M \mid \det(A)=1 \right \}$ $M$ is an $n^{2}$ dimensional $C^{\infty}$ manifold. Is $S$ a regular ...
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### How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
Consider this definition of the parametrization of a manifold, found in Hubbard & Hubbard: A parametrization of a $k$-dimensional manifold $M\subset\mathbb{R}^n$ is a mapping $\gamma:U\subset ... 1answer 134 views ### A “Manifold with Boundary” Question I'm given the following scenario: Letting$U$be an open subset of$\mathbb{R}^n$and$f,g:U\rightarrow \mathbb{R}$, two smooth functions such that$f(\vec x)\lt g(\vec x)$for all$\vec x\in U$, ... 1answer 514 views ### Problem 3-38 in Spivak´s Calculus on Manifolds This is not homework. Problem 3-38 reads: Let$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$for ... 2answers 484 views ### Showing something isn't a manifold So I'm following some notes that are introducing manifolds with pretty minimal prerequisites. What I want to do is show where the image of$\phi: \mathbb{R}\rightarrow \mathbb{R^2}t\mapsto ...
I'm following the proof of the local expression for the Laplacian on a compact manifold and I'm having problems understanding how the integral on a manifold translates into an integral in $R^n$, in ...