2
votes
1answer
37 views

Torsion vs Curvature of a curve

I was wondering if there is a simple explanation of the torsion and curvature in $\mathbb{R}^3$ of a curve. The curvature measure somehow the acceleration perpendicular to the tangent vector, but ...
1
vote
1answer
109 views

Curvature of curve not parametrized by arclength

If I have a curve that is not parametrized by arclength, is the curvature still $||\gamma''(t)||$? I am not so sure about this, cause then we don't know that $\gamma'' \perp \gamma'$ holds, so the ...
0
votes
1answer
35 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
1
vote
0answers
31 views

The intuition of the rank theorem

In Rudin's Principle of Math Analysis, I know the proof of the rank theorem. However, I fail to understand its content. 9.32 Theorem: Suppose $m,n,$ are nonnegative integers, $m\geq r,n\geq r$, $F$ ...
0
votes
0answers
31 views

Reparametrisation of closed not closed

I would like an example of a closed curve and a reparametrisation of the same curve that is not closed.
0
votes
1answer
94 views

Parameterise $y^2-x^2=1$ - not possible.

I'm doing stuff from a book and it has just spoke of the importance of not parameterising half a curve (with the example of a circle). However I am not sure what to do. First of all ...
0
votes
1answer
98 views

Interpretation of differential form

We know what is the interpretation of a total differential, ex.: $$df=\frac{\partial f}{\partial x} dx+ \frac{\partial f}{\partial y} dy$$ But what is the interpretation of a 1-form and its exterior ...
0
votes
1answer
43 views

Relation between a function on $N$ sphere and a function on $(N-1)$-cell.

Let $S^N$ be a unit $N$-sphere. Let $f:S^N\to\mathbb{R}$ be a function. Let $\bf{\Sigma}$ be a unit $(N-1)$-cell, consider the function $g:S^N\to\bf{\Sigma}$ such that, for any $\hat{a}\in S^N$, ...
0
votes
1answer
49 views

Metric tensor and einstein's notations .

Here , $\Omega \subset \mathbb R^n$. Can someone explain it to me what $F$ is ? I also don't understand how we can get the unit outer normal in the second para . Please kindly help me to understand ...
0
votes
0answers
26 views

Laplacian in arbitrary spherical coordinates?

assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function that just depends on the radius vector $r$, so no angular dependence(!). Can we say what $\Delta f$ is in arbitrary coordinates $r ...
0
votes
2answers
40 views

Given a measurable vector field, construct another such that together they form a basis at every point

Let $v_1:(0,1)\rightarrow \mathbb{R}^2$ a measurable function such that $v_1(x)\neq 0$ for all $x$. I wonder if it is possible to construct a measurable function $v_2:(0,1)\rightarrow \mathbb{R}^2$ ...
1
vote
1answer
28 views

Questino about integration of differential forms

Here is the theorem about the integration of two-forms in Edwards' "Advanced Calculus" The definitions of "charts" in this theorem are given here. What I do not understand about the Theorem is ...
0
votes
0answers
107 views

Why is it hold for types of operators?

We ‎let ‎the ‎state ‎space ‎be‎ ‏‎‎‎$ \mathcal{H} =‎ ‎‎H_{E}^{2}(0 , 1) \times L^2(0 , 1) $‎ equipped with the norm ‎ \begin{align} \| (f , g) \| = \int_{0}^{1} [ |f''(x)|^2 + |g(x)|^2] ‎\mathrm{d}x‎ ...
1
vote
3answers
49 views

Length of Difference Curve

Let $\varphi : [a,b] \to \mathbb R^n$ be a curve, and for some partition $\pi = \{ t_0 = a, t_1, \ldots, t_m = b \}$ of $[a,b]$ set $$ l(\pi, \varphi) = \sum_{i=1}^m \| \varphi(t_i) - ...
0
votes
1answer
43 views

Local isometric embedding

Every $n$-dimensional smooth Riemannian manifold admits a local isometric embedding of class $C^1$ into $\mathbb R^{n+1}$ by the Nash-Kuiper theorem. An example by Nadirashvili and Yuan shows that in ...
0
votes
1answer
36 views

Estimate for boundary points and exterior normal vector of bounded domain of class $C^2$

Consider a bounded open set $\Omega\subset\mathbb{R}^d$, s.t. the boundary set $\partial \Omega$ is a manifold of class $C^2$. Let $x,x_0\in\partial\Omega$ be boundary points and $\nu_x$ the exterior ...
0
votes
1answer
84 views

Integration by substitution and separation of variables.

Let's say I want to integrate over a sphere $S^2$. Take $f \in (L^1(S^2),dS)$, then we have that $$\int_{S^2} |f| dS = \int_{S^2} |f| \sin^2 (\theta) d \theta d \phi < \infty,$$ right? Now, ...
2
votes
1answer
49 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
2
votes
1answer
42 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
4
votes
1answer
121 views

Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
3
votes
1answer
52 views

Question on Inductive Proof of Implicit Function Theorem

I am struggling with an inductive proof of the implicit function theorem, concretely with the final part of construction of a function, up to this final point everything is perfectly clear to me. ...
1
vote
0answers
27 views

Proving that a function is a $C^\infty$ submanifold in $\Bbb{R}^2$ of dimension 1

We need to prove that for all $c\in\Bbb{R}$ the set $\{x\in\Bbb{R}\,\colon\, g(x)=c\, \}$ is a $C^\infty$ submanifold ($g\,\colon\,\Bbb{R}^2\rightarrow \Bbb{R};(x_1,x_2)\mapsto x_1^3-x_2^3$) in ...
3
votes
1answer
61 views

Smooth Manifold, covered by 2 Charts is orientable if the Intersection is Connected

I came across this Question: Atlas on a smooth manifold that contains 2 charts in which Professor Lee commented that this Proposition is true only if the Intersection of the two Maps is connected, so ...
9
votes
1answer
141 views

Is every scalar differential operator on $(M,g)$ that commutes with isometries a polynomial of the Laplacian?

On $(\mathbb{R}^n, g_{\text{std}})$ with $\Delta$ the Laplacian, the following holds: Fact: Every scalar differential operator $D$ that satisfies $D \circ F^* = F^* \circ D$ for all isometries $F ...
0
votes
3answers
72 views

A diffeomorphism with negative Jacobian swaps the orientation?

Let C be a simple close oriented curve $C^1$ in $\mathbb{R}^2$ and let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a diffeomorphism such that $\forall (x, y) \in C$ it holds that the determinant of the ...
0
votes
0answers
31 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
3
votes
0answers
45 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
7
votes
1answer
68 views

Translate a vector field

Imagine that you have a vector field $A = \frac{A_0}{r} e_{\theta}$ in cylindrical coordinates, where $A_0 \in \mathbb{R}$. Now you translate your coordinate system in $e_x$ direction by $x \mapsto x ...
1
vote
1answer
41 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
0
votes
1answer
20 views

Change of variables for vector valued measures

So in this question I am assuming that $f$ is of bound variation on $\mathbb{R}^{n}$ so $\nabla f$ is a vector valued measure and $|\nabla f|$ is its total variation but you can assume that $f$ is ...
1
vote
1answer
25 views

Differentiability of “positive part” function

Let the positive part function be defined as $\max(0,x)$; this function is obviously not differentiable in $x=0$. But what about the (more smooth) function $\big( \max(0,x) \big)^{2}$. I suspect the ...
2
votes
1answer
65 views

curve integral - intersection between plane and sphere

I am going to calculate the line integral $$ \int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
0
votes
2answers
86 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
2
votes
1answer
100 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
1
vote
1answer
36 views

Regularity of weakly harmonic map

Suppose $(M,g)$ is a smooth $n$-dimensional manifold with $C^k$-metric $g$ and let $U\subset M$ be an open subset. Does anyone have a reference for a statement about the regularity of a map ...
0
votes
1answer
39 views

The extension of functions

Let $f$ be a smooth function defined on $[a,b]$ and $g$ a smooth function defined on $[c,d]$. If $a<b<c<d,f'(x)>0, g'(x)>0$ and $f(b)<g(c)$, then can we find a function $h: \mathbb R ...
0
votes
1answer
23 views

Length of an globally continuous parameterized curve differentiable nowhere

Say we have a globally continuous parametrized curve $c : \mathbb{R} \longrightarrow \mathbb{R}^k$ such that $s$ is nowhere differentiable. We define the length of a curve between a point $a$ and $b$ ...
1
vote
0answers
40 views

Smooth bijection has a dense open subset in which the inverse is also smooth

Let $f:\mathbb{R}^n \longrightarrow \mathbb{R}^n$ be smooth and bijective. Prove there exists open subset $U$ and $V$ dense in $\mathbb{R}^n$ such that $f: U \longrightarrow V$ has a smooth inverse. ...
2
votes
1answer
43 views

Del on Riemannian manifolds

I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor. My ...
0
votes
0answers
264 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
0
votes
0answers
49 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
0
votes
1answer
67 views

Tangent at a singular point

I'm looking at this question If the tangent at the point $P$ with coordinates $(h, k)$ on the curve $y^2 = 2x^3$ is perpendicular to the line $4x = 3y$, find $(h, k).$ This is how I attempted it ...
2
votes
3answers
161 views

How to parametrize a curve by its arc length

I am reading on Wikipedia that ''...Any regular curve may be parametrized by the arc length (the natural parametrization) and...'' I know that if $a(t) = (x(t),y(t),z(t))$ is a curve (say, smooth) ...
2
votes
1answer
136 views

Surjectivity of an integration map

N.B.: Thanks to studiosus answer I realised I should ask for more conditions or otherwise the answer is straightforwardly wrong. I rechecked my problem and added new assumptions that I boldface. ...
1
vote
2answers
45 views

Dual space (Wikipedia)

I am struggling to understand something on Wikipedia: ''If $V$ consists of the space of geometrical vectors (arrows) in the plane, then the level curves of an element of $V^*$ form a family of ...
2
votes
1answer
58 views

Very basic questions on chain rules and product rules

I have serious gaps in maths and would like to ask some basic questions. I know there is the following chain rule for the first derivative: $$ Dh(x) = Dg(f(x))Df(x)\quad\quad (1) $$ where $h(x) = ...
0
votes
1answer
46 views

Some trivial questions about Tangent Spaces

I'm studying submanifolds $M \subset \mathbb{R}^n$ and now I've got some questions about tangent spaces. First question: Let $\gamma:I\subset\mathbb{R} \rightarrow \mathbb{R}^n$ be a path, which ...
2
votes
1answer
41 views

Partitions of Unity-Integration on Manifolds

So lets say I have a $k$-manifold $M$ in $\mathbb{R}^n$, and I cover it up with coordinate patches $\{\alpha_i\}$. I can find a set of partitions of unity $\{\phi_1,...\phi_l\}$ on $M$ which is ...
0
votes
1answer
105 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
2
votes
0answers
62 views

Is it generally difficult to memorize 'multivariable calculus' theorems?

There are many weak forms of "Fubini's theorem" with strong hypotheses in elementary calculus texts. However, these strong hypotheses are very unnatural and are thus hard to memorize. Compared to ...