2
votes
1answer
39 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
29 views

Level set of a real valued harmonic fucntion

Let $f$ be a real valued harmonic function defined on a neighborhood $U$ of origin in $\mathbb{R}^2$. And $f$ is such that its gradient vanishes at origin. Then how do i show that the set given by ...
0
votes
0answers
30 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
37 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
45 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
59 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) ...
1
vote
2answers
39 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
47 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
44 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 ...
1
vote
1answer
23 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
92 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
50 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 ...
3
votes
1answer
66 views

Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
2
votes
2answers
57 views

Solving a differential equation $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$

Let $\alpha$ be a regular curve in $\mathbb{R}^3$ such that $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$ for $w$ a constant vector. How can we determine $\alpha$ ? $\displaystyle w ...
1
vote
1answer
32 views

How do I convert OS coordinates (X and Y) to longitude and latitude coordinates?

How do I convert OS coordinates (X and Y) - Eastings and Northings to longitude and latitude coordinates? For example X and Y below X (Eastings): 347904 Y (Northings): 287484
1
vote
0answers
16 views

How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
6
votes
1answer
91 views

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{equation}\begin{pmatrix} a & b\\ c & d ...
1
vote
1answer
62 views

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location?

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location? I don't mind how many coordinates that takes. For instance: Latitude = ...
0
votes
2answers
51 views

If $ \lambda'(t) \in {\rm Span}(v)$ then $\lambda(t)$ belongs to a straight line

If $\lambda:(a,b)\longrightarrow \mathbb{R}^3$ is a path differentiable (not necessarily of class $C^1$) where $\displaystyle \lambda'(t)\in {\rm Span}(v)$. How to prove that $\lambda(t)$ belongs to a ...
2
votes
1answer
72 views

Brouwer's fixed point theorem

Theorem: If $f:D^n\rightarrow D^n$ is continuous then there is $x \in D^n$ such that $f(x)=x$. To prove the theorem we assume that $f$ is cts but has no fixed point, that is $f(x)\neq x$ for all ...
2
votes
0answers
50 views

differentiability of functions at a manifold without a zero set

I am curious to whether there exists a notion of differentiability (corresponding to $C^{\infty}(M,\mathbb{R})$) that still makes sense on a manifold after removing a Lesbesugue zero set. Does the ...
1
vote
2answers
75 views

How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
1
vote
2answers
63 views

How can I prove that the angles of the tetrahedral structure is $109.5^o$ with calculus. I could do it with geometry.

How can I prove that the dihedral angles of the regular tetrahedral structure is $109.5^o$ with calculus or any other technique. I could do it with geometry by considering a cube.
0
votes
1answer
35 views

Change variables of a Jacobian

If I have a change of variable, made by a diffeomorphism, the determinant of the Jacobian preserves the sign?(if it is positive it remains positive)
1
vote
1answer
52 views

How should I prove $\operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr$ without using spherical coordinates?

Let $B_n:=\{x\in{\Bbb R}^n:|x|\leq 1\}$ and $S^n(r):=\{x\in{\Bbb R}^{n+1}:|x|=r\}$. Then we have the following formula $$ \operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr. ...
2
votes
1answer
70 views

finite length of a spiral

consider a "spiral" $\alpha(t)=r(t)\left(\cos(t),\sin(t)\right)$, where $r$ is $\mathcal{C}^1$ and $0\le r(t) \le 1$ for all $0 \le t$ Show that if $\alpha$ has finite length on $ [0,\infty)$ and ...
0
votes
0answers
46 views

Is a manifold over $\mathbb{R}$ normal?

We have manifold $G$ over the reals with its finite atlas ($g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}, G=\bigcup U_i$). The atlas induces a topology in the normal way ($A \subseteq G$ is open iff ...
0
votes
1answer
45 views

Diffeomorphism of closure of open sets

Let $F:\overline{X} \to \overline{Y}$ be a map between the closure of two open Lipschitz domains $X$ and $Y$ in $\mathbb{R}^n$ (with boundaries). $F$ is such that it maps $X$ to $Y$ and it maps ...
3
votes
1answer
68 views

Differentiable parameterization of a curve $\Gamma$

If $\alpha:I\longrightarrow \Gamma$ and $\beta:J\longrightarrow \Gamma$ are two bijective differentiable regular parameterizations of the curve $\Gamma\subset\mathbb{R}^2$ (not necessarily of class ...
0
votes
1answer
38 views

$C^{0}$-norm of a map

I have the following question: Consider a continous map $f:M\rightarrow N$, where $M,N$ are smooth manifolds. How can one define its $C^{0}(M,N)$-norm, i.e. $||f||_{C^{0}(M,N)}$ ? Greetings, Daniel
2
votes
3answers
82 views

Geometric interpretation of deriavative of a function of more than one variable

A function $f$ is defined on an open set $D$ of $\mathbb R^{2}$ is called a differentiable at a point $x\in D$ if there is a vector $m \in \mathbb R^{2} $ such that $$\lim_{h\to 0} ...
1
vote
1answer
89 views

Differentiability of the arc length function : $\sigma (t)=l(\alpha)[a,t]$

Let $\alpha:[a,b]\longrightarrow \mathbb{R}^n$ be a rectifiable path (not necessarily continuous) The function arc length $\sigma:[a,b]\longrightarrow \mathbb{R}$ is defined ...
12
votes
2answers
130 views

Curve of length $L=1$ contained in a semicircle of diameter $2R=1$.

How prove that for any curve $\alpha(s)$ of length $L=1$ in the real plane, there is a semicircle of diameter $2R=1$ that contains it. Any hints would be appreciated.
4
votes
3answers
111 views

How to prove that $\alpha''(s)$ goes towards the inside of the curve $\alpha(s)$

Let $\alpha:[a,b]\longrightarrow \mathbb{R}^2$ be a plane curve parametrized by arc length by $\alpha(s)$. $T(s):$ unit tangent vector Note that $||T(s)||=1\Longrightarrow T'(s)\perp T(s)$ How to ...
1
vote
0answers
32 views

Newton polygon and asymptotic behavior near a singular point

As we know, Newton polygons could be used to determine the Puiseux series of algebraic curves (see, for example, Kirwan's Complex Algebraic Curves, chapter 7). Different branches correspond to ...
2
votes
0answers
59 views

Inverse Function Theorem (results using it)

Hi i'm thinking in some ideas for my bachelor thesis. I'm working in a more "general" framework than manifolds, and i found that the Inverse Function Theorem is valid in such structures. So i was ...
0
votes
2answers
60 views

Orientability of surfaces in $\mathbb{R}^3$

I have a question: I'm currently reading a few things about orientability and understood this concept as the answer to the question: Given a surface and a unit normal vector field on it: Is there a ...
4
votes
1answer
89 views

The unsolved extension problem on manifolds.

I have been struggeling for quite a while with this problem: Let $M \subset \mathbb{R}^n$ be a compact $C^k-$ submanifold and $\phi_i: B_i(0) \rightarrow M$ be the associated set of charts ...
1
vote
1answer
133 views

How do 1-d compact submanifolds look like?

I was wondering whether it is true that every 1-d compact submanifold $\subset \mathbb{R}^n$ that is $C^1$ is a closed curve that is also $C^1$, cause I cannot think of more examples. Therefore, I ...
1
vote
1answer
60 views

Partition of unity. Does this one exist?

Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an ...
2
votes
1answer
175 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
3
votes
0answers
98 views

Volume of “deformed torus”

I'm trying to find explicit form of volume of "deformed torus": Suppose we have a curve $\gamma(t)$ in $\mathbb{R}^n$, $t\in[0,1]$. The curve closed and smooth : ...
1
vote
1answer
36 views

How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
1
vote
0answers
42 views

Smooth parametrizations of continuous curves

Consider the curve $t\to (t,|t|)$ in $\mathbb R^2$. Even if it has a cusp in $(0,0)$ i can reparametrize it with a smooth function. Take for example $$ t\mapsto \begin{cases}(\mathrm e^{-1/t},\mathrm ...
2
votes
2answers
51 views

Derivative: a special tangent

I've learned in Euclidean Geometry that the tangent is a line which pass through only a point. For example, if someone ask me to find the tangent at this point $A$, I can easily say that the tangent ...
3
votes
1answer
93 views

Why is the derivative the tangent vector?

I'm trying to understand, at least intuitively why the derivative of a function at a point is the tangent vector at this point. If we see the functions of this form $f:\mathbb R\to \mathbb R$ we see ...
3
votes
1answer
50 views

Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
0
votes
2answers
55 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
1
vote
1answer
74 views

Show that the Kelvin transformation is a diffeomorphism and find the inverse

Let $\Omega:=\mathbb{R}^n\setminus\overline{B}_R(0)$ with $R>0$ and $n>1$. We call $\phi\colon\Omega\to G:=B_R(0)\setminus\left\{0\right\}$ with $$ y=\phi(x):=\frac{R^2}{\lVert ...
2
votes
1answer
55 views

Constructing submanifolds. Did I understand this right?

I just want to know whether I understand the construction of a submanifold in some $\mathbb{R}^n$ properly. Please correct everything that you think could be wrong. As far as I know so far, it is ...