1
vote
5answers
52 views

Vector field ${\bf F}$ with $\int_S {\bf F}\cdot{\bf n}\ dS=c$

Find a vector field ${\bf F}$ on $ {\bf R}^3$ with $$\int_S {\bf F}\cdot{\bf n}\ dS=c > 0 \tag{1} $$ where $S$ is any closed surface containing $0$ and ${\bf n}$ is normal Here there is a ...
0
votes
0answers
46 views

Derivative with respect to a vector and tensor on a manifold

I am reading through a paper and have come across a statement which I do not fully understand. I paraphrase below. Consider a scalar function $f = ...
0
votes
1answer
50 views

Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
2
votes
0answers
31 views

Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
5
votes
4answers
60 views

Proof: Force always perpendicular and motion in a plane implies that the trajectory is a circle

I am looking for a proof for a physics problem. Consider a particle which is subject to a force $\vec{F}(t)$ with $|\vec{F}(t)| = \text{const}$ which is always perpendicular to the velocity ...
5
votes
1answer
76 views

Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
2
votes
0answers
49 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
2
votes
1answer
44 views

How can I prove the tangential acceleration equation?

This is my assignment for this weekend. $$a=a_{TT}+a_{NN} = \frac{d^2 s}{dt^2}\vec{T} + \kappa \frac{ds}{dt} 2\vec{N}$$ Actually, I want to why $a_N=\kappa (ds/dt)$ hold Please help me.
6
votes
0answers
70 views

What is the generalization of Gauss's Theorem to a manifold?

In a (pseudo-)Riemannian manifold with constant basis vectors, one certainly has that the integral of the divergence of a tensor field $T$ over a submanifold $\Omega$ is equal to the integral over the ...
3
votes
2answers
166 views

Should diffeomorphisms preserving arc length be affine?

Problem Suppose $\varphi\colon V=\mathbb R^n\to V$ be a differmorphism and $d\varphi$ is its tangent mapping. $\langle\circ,\circ\rangle$ is a nondegenerate (symmetric or symplectic) bilinear form on ...
2
votes
1answer
80 views

Mathematically explaining a trapped surface?

I'm currently writing my thesis, and the general area is that of minimal surfaces. I have a deep interest in cosmology so have directed it towards space-time, ie applying minimal surfaces in space, ...
3
votes
1answer
76 views

Non commutative tensor products / simple example?

In O'Neil's "Semi-Riemannian Geometry" it is stated that if $A$ is a $(r,0)$-tensor field and $B$ is $(0,s)$-tensor field then it is ""from the definition'' that $A \otimes B = B \otimes A$. I still ...
2
votes
0answers
444 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
1
vote
1answer
93 views

How can I know the time difference $(\Delta t)$ between two cities aren't in the same latitude?

I'm trying to measure the time difference $(\Delta t)$ between two cities (London, Moscow) (they aren't at the same latitude) but I'm facing problem because the speed of earth rotation ($\nu$) depends ...
2
votes
2answers
191 views

How can I know the time difference between two cities almost at the same latitude?

Well I know that's the earth rotation speed is: $v=1669.756481\frac{km}{h}$ I have two cities New York, Madrid almost at the same latitude and the distance between them is: $d=5774.39$ $km$ ...
1
vote
1answer
124 views

How can I know the time difference between two cities by knowing the distance between them and earth speed?

Well I know that's the earth speed is: $v=1669.756481\frac{km}{h}$ and I have two cities Moscow and NewYork the distance between them is: $d=7518.92$ $km$ Actually I know that's : ...
4
votes
1answer
1k views

Riemann, Ricci curvature tensor and Ricci scalar of the n dimensional sphere

I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the n sphere $x_0^2 + x_1^2 + ....+x_n^2=R^2$, whose metric is $$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
2
votes
2answers
167 views

Generally covariant Klein-Gordon equation

Consider a 4-dimensional smooth manifold $M$ on which there is a Lorentzian metric $g_{ab}$ and a function $\phi$ satisfying the following two equations (in abstract index notation): \begin{equation} ...
1
vote
1answer
166 views

Bullseye-shaped interference pattern in seminar-room chair

During a break n a seminar today, I noticed that the chairs in front of me all had slightly transparent black mesh fabric. The backs of the chairs were in the shape of a hyperbolic paraboloid. The ...
2
votes
1answer
208 views

Physics related question on Divergence Theorem for “general” smooth manifolds

I'm a physics major so bear with me here on the math. This is related to a problem from the textbook General Relativity - Wald. In classical electromagnetism say we have a vector field $V$ defined on ...
4
votes
3answers
555 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for ...
4
votes
1answer
303 views

Is a twisted de Rham cohomology always the same as the untwisted one?

I am a physicist studying now some supersymmetric sigma models. My question can, however, be reformulated in a purely mathematical language: A twisted de Rham complex involves $d_W = d + dW \wedge $ ...
5
votes
4answers
545 views

What does the symbol $\operatorname{Tr}$ in the Yang-Mills action mean?

I find that many authors write the Yang-Mills action as follows: $$\mathcal{J}= \int \operatorname{Tr}(F \wedge \star F).$$ I have yet to find a formal description of the symbol $\operatorname{Tr}$ ...
3
votes
1answer
381 views

Covariant Derivatives and the Cross Product

I've recently read a paper that used a covariant derivative product rule for cross products. It was something like $\nabla_v (A \times B) = (\nabla_v A) \times B + A \times (\nabla_v B)$. Here, $A, ...
1
vote
0answers
115 views

Help on Einstein Summation

I am not sure how to interpret the following expression with regard to the Einstein summation convention \begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) \end{equation} ...
2
votes
0answers
75 views

A lower positive bound on the number of closed orbits with given energy for a mechanical system

Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the ...
3
votes
2answers
256 views

boundary defining functions

Suppose that $M$ is the interior of a compact manifold with boundary $\partial M$. I'm often faced with so called boundary defining functions - or just 'defining functions'. That are by definition ...
15
votes
1answer
647 views

Stochastic interpretation of Einstein Equations

Einsteins theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a ...