0
votes
0answers
63 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
18 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
34 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
109 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 ...
2
votes
0answers
35 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
26 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
43 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
131 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
64 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
26 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
37 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
63 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
35 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
19 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
21 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
52 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
74 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
89 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
29 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
36 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
19 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
37 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
70 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
45 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
57 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
81 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
43 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
53 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
45 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
34 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
100 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
59 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
92 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
64 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
35 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
18 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
93 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
76 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
53 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
95 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
51 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
78 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
67 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
43 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
53 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
84 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
1answer
51 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
70 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
39 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