Tagged Questions
0
votes
0answers
18 views
Prolate spheroidal coordinates
If $\alpha \in (0,\infty)$, $\beta \in (0,\pi)$ and $\theta \in (0,2\pi)$.
$$\varphi (\alpha,\beta,\theta) =(
\sinh(\alpha)\sin(\beta)\cos(\theta),
\sinh(\alpha)\sin(\beta)\sin(\theta),
...
1
vote
2answers
36 views
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 ...
1
vote
1answer
43 views
explain $df(tx).x = \sum_{i=1}^n {\partial f\over \partial x_i}(tx)x_i \hspace{1cm} x\in \mathbb R^n$
the question is :
let $U$ be a Neighbourhood of the origine of $R^n$ and :
$x\in U \Rightarrow tx \in U , \forall t\in U $
let f be a numeric function defined in U , and $f(0)= 0$
if we have ...
1
vote
1answer
35 views
if the curvature is constant and positive, then it is on the circunference
I'm trying to prove that if $\alpha(t)=(x(t),y(t))$ is a $C^2$ regular curve $(\alpha'\neq0)$ with constant and positive curvature, then $\alpha$ is on the circunference and if $\alpha$ is the ...
2
votes
1answer
56 views
Incomplete vector field
Is there a way I can tell if a vector field on a manifold or just $\mathbb{R}^n$ is incomplete simply by just looking at its formula. For instance on $\mathbb{R}$, $\displaystyle X= (x^2+1) ...
0
votes
1answer
65 views
wedge product and change of variables
The question is:
Let $\phi：\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map and let $y=\phi(x)$ be the change of variables. Show that
d$y_1\wedge...\wedge ...
1
vote
0answers
40 views
Cohomologies of $\mathbb R^n$ with rational differential forms
We can consider de Rham complex $0 \to \Omega^1 \to \Omega ^ 2 \to...$ on $\mathbb R^n$, where $\Omega ^r$ are $r$-forms on $\mathbb R^n$ with rational coefficients. What are homologies of this ...
0
votes
1answer
37 views
Show that anecessary and sufficient condition for $x_{p}$ to be tangent to $S^{n}$ at $p$
Please help me! How do I solve this problem? I didnt produce any idea because I didnt understand this topic properly. Thus, please can you explain the solution explicitly? Thank you for help:)
1
vote
1answer
58 views
show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$.
If $f$ and $g$ are $C^{∞}$ functions and $X$ and $Y$ are $C^{∞}$ vector fields on a manifold $M$, show that $[fX,gY]= fg[X,Y]+f(Xg)Y−g(Yf)X$
This is a proposition in a book. But I cannot prove this:(
...
2
votes
0answers
40 views
Divergence and curl united?
In my post,
In 2D we can define
$$div(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{r} dC$$
$$curl_z(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{v} dC$$
Where $C$ is a ...
2
votes
2answers
44 views
Higher orders of divergence and curl
In the standard definition of div and curl, a limit is taken.
If one instead expands the integral out into a series(Taylor?) of the vol/area then there are higher order terms that vanish. Do these ...
1
vote
2answers
54 views
Compute the velocity vector.
Can you solve explicitly? please. I don't know how to solve. Thank you for help.
0
votes
0answers
55 views
Example of differential form usage of Stoke's theorem
There are many examples that show how Kelvin-Stokes theorem is used.
But I would like to see an example that uses differential form usage of Stoke's theorem and is hard or impossible to solve by ...
0
votes
2answers
40 views
Find $T$, $N$, and $k$ for vector
$$ x(t) = ( t , \sin(4t), \cos(4t) )$$
I am unsure on what they are referring to here. would $T$ be the tangent and therefore:
$$ x'(t) = ( 1 , 4\cos(4t) , 4\sin(4t) ) ?$$
Thanks :)
Also, ...
0
votes
1answer
32 views
What happens to the Frenet-Serret frame when $\kappa=0$?
I was considering the following question for 3D curves: Does zero curvature imply zero torsion?
I think it's reasonable, because zero curvature implies the curve is a straight line, which lies in a ...
1
vote
1answer
79 views
Proof that cone not diffeomorphic to plane
What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$?
After some comments, I realize that this question The ...
3
votes
1answer
52 views
Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$
How can I prove it?
This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof.
(Proving this will help me apply Stokes' theorem to ...
7
votes
3answers
185 views
Can the curl operator be generalized to non-3D?
In three dimensions, the curl operator $\newcommand{curl}{\operatorname{curl}}\curl = \vec\nabla\times$ fulfils the equations
$$\curl^2 = ...
0
votes
1answer
119 views
Left-Invariant Vector Field of a Lie Group
How do I tell if a vector field on a Lie Group is left-invariant? I have the technical definition. But, I want to understand given a specific vector field what should I do to test if it is ...
3
votes
1answer
61 views
The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)
In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it.
Let $\gamma: ...
1
vote
1answer
142 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 ...
4
votes
2answers
72 views
Length of curve in 3D spherical coordinate
let $r$ be the magnitude of a vector in 3D with Spherical co-ordinate $(r,\theta,\phi)$ and cartesian coordinates is $(x,y,z)$, whose angle with $z$ axis is $\phi$ and projection of the vector makes ...
1
vote
2answers
70 views
How can I calculate the Euclidian displacement of two places on a sphere (earth in this case ) and calculate the
I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a ...
2
votes
2answers
91 views
How is differential form different from ordinary calculus objects?
I am going to learn differential form soon, but after reading some introductory parts of my texts, I couldn't get why differential form is needed and how it is different from ordinary mathematics ...
2
votes
1answer
108 views
Does a 3-Dimensional coordinate transformation exist such that its scale factors are equal?
Let $\vec r=(x,y,z) $ be the position vector expressed in Cartesian coordinates. Let us define the coordinate transformation as
$\vec r(u,v,w)=(x(u,v,w),y(u,v,w),z(u,v,w)) $
The scale factors are ...
3
votes
0answers
62 views
Proof of the arc length parametrization is $1$
Let $\gamma : [a,b] \to \mathbb R^n$ be a regular curve.
Let $p:[a,b] \to [0, p(b)]$ be the map $p(t) = \int_{a}^t \|\gamma' (s) \|ds$. Then $p^{-1}: [0,p(b)] \to [a,b]$.
I tried to show that ...
2
votes
2answers
56 views
Does there exist a diffeomorphism on $\mathbb{R^2}$ that flattens out the boundary of a compact set at a point?
Given a compact subset of $\mathbb{R^2}$ with ${C^2}$ boundary $S$ and a point $x \in S$, can one find a diffeomorphism $f$ from $\mathbb{R^2}$ to $\mathbb{R^2}$ for which $f(x) = x$, the image $f(S)$ ...
1
vote
0answers
40 views
Integral over a Funnel in Fermi coordinates
Suppose we are in the Hyperbolic plane, defined as
$$
\{w = u + iv \mid v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,.
$$
I am given a funnel $F$. This object is isometric to a ...
6
votes
3answers
52 views
Exact sequence involving the nabla operator
Recently I noticed that $$0 \longrightarrow \Bbb R \overset{\text{const.}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R) \overset{\text{grad}}\longrightarrow \mathcal{C}^\infty(\Bbb R^3,\Bbb R^3) ...
2
votes
1answer
94 views
What does the definition of curvature mean?
First question, am I right in saying that curvature measure how quickly the direction of a cruve changes?
Also, we have been given the "definition":
$$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$
where ...
4
votes
4answers
136 views
Does the gradient always point outward of a level surface?
Let $f:\mathbb{R}^n\to \mathbb{R}$ be a differentiable function, with $a\in \mathbb{R}$ a regular value of $f$.
Let $M=f^{-1}((-\infty,a])$. Then $M$ is an $n$-manifold with boundary, whose boundary ...
2
votes
1answer
87 views
Implicit function theorem-show that in a neighbourhood of the point -can be described by a pair of functions
Let $g_1(x,y_1,y_2)$= $x^2(y_1^2+y_2^2)$-5 and $g_2(x,y_1,y_2)$=$(x-y_2)^2$+$y_1^2$-2. Use implicit function theorem to show that in a neighbourhood of the point x=1, $y_1$=-1, $y_2$=2, the curve of ...
2
votes
0answers
64 views
Is multivariable calculus synonymous with differential geometry?
Or are they two distinct topics? For instance, Spivak's calculus on manifolds book considered a treatise on multivariable calculus, but concludes with a differential geometry theorem - Stokes' ...
5
votes
1answer
178 views
What is the Laplace operator's representation in 3-sphere-coordinates?
The three-dimensional Laplace operator in spherical coordinates can be expressed as
$$\Delta_3 = \frac1{r^2}\partial_r(r^2\partial_r) + \frac1{r^2} L^2$$
where $L^2$ is the squared angular momentum ...
1
vote
3answers
97 views
A manifold being orientable vs oriented.
I read that an oriented manifold is a manifold with a choice of orientations for each tangent space so that for $p \in M$, there is an open set $U$ and a collection of vector fields $X_1,...,X_n$ so ...
10
votes
1answer
320 views
Problem 3-38 in Spivak´s Calculus on Manifolds
This is not homework. Problem 3-38 reads:
Let $A_{n}$ be a closed set contained in $(n,n+1)$. Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $\int_{A_{n}}f=(-1)^{n}/n$ and $f(x)=0$ for ...
3
votes
0answers
77 views
Partial derivatives using variables after a transformation
I have a transformation $$(x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))$$ and I wish to find $$\partial x'_1\over \partial x'_2$$ how might I evaluate this?
If it is difficult to find a general expression for ...
3
votes
2answers
142 views
algebraic manipulation of differential form
Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$
$(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes ...
0
votes
2answers
71 views
Identifying functions on the unit disk with functions on the upper hemisphere
I've been wondering about something, and it might be nonsense (if so I apologize!). Consider the unit disk in $\mathbb{R}^2$ and a function $f$ defined on the disk. I can compute its double integral ...
1
vote
1answer
40 views
Rank of the differential of a composition
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 ...
1
vote
1answer
33 views
Differentiable surface question
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 ...
2
votes
0answers
47 views
How to apply Gauss's theorem when the metric is unknown
Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, ...
0
votes
1answer
47 views
Practise with Smooth Functions and Manifolds
I'm trying to get an intuition for smooth manifolds, and in particular the smoothness of transition functions. I haven't done that much calculus on $\mathbb{R}^n$ before, and would like to practice ...
2
votes
1answer
390 views
How to calculate minimum distance between two arbitrary ellipses in 2D?
Arbitrary ellipses means that they can be scaled, translated and rotated in any way in 2D. Do you know some high-school method (might be slightly more advanced than that) to find the minimum distance? ...
1
vote
1answer
74 views
Parametrization of a solid
Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
2
votes
0answers
39 views
laplacian for functions problem with the integral on manifolds
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 ...
4
votes
0answers
211 views
Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?
Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem".
I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
2
votes
3answers
77 views
Calculating a Multivariable derivative.
I'm trying to work through Spivak's Calculus on Manifolds and I've arrived at Differentiation. While I can usually follow his steps, I find myself lost or stuck when I try to do something on my own. ...
2
votes
2answers
197 views
What do level curves signify?
Suppose I have a function $z=f(x,y)$, say like $z=\sqrt{x^2+y^2}$. By fixing some value for $z$ and varying all possible $x$ and $y$, we would get a level curve of $z=f(x,y)$. By changing values for ...
1
vote
2answers
166 views
Coordinate-free method to determine local maxima/minima?
If there is a function $f : M \to \mathbb R$ then the critical point is given as a point where
$$d f = 0$$
$df$ being 1-form (btw am I right here?). Is there a coordinate independent formulation of ...
