"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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looking for help with a trace/norm inequality

I'm trying to understand a derivation that seems to claim that $\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where $\rho$ is Hermitian and has ...
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0answers
23 views

Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are $3$ teams observing the movement of the birds. Team $1$ observes that on their first flight the birds move a short distance across a ...
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1answer
17 views

uniform angular distribution-change of origin

Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning ...
2
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0answers
9 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$. I'd like a ...
4
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0answers
25 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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1answer
26 views

Problem with second distributional derivative

I have the following function: $ f(x) = \begin{cases} \sqrt{x}, & \text{if $x>0$} \\ \sqrt[3]{|x|}, & \text{if $x<0$ } \end{cases} $. I have to find $f'(x)$, $f''(x)$ as ...
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1answer
31 views

Problem on string vibration

Given the standard wave equation for small amplitudes, we have been asked to find the position of string $y(x,t)$, given: $y(x,0)=\sin x$, and, $y'(x,0)=\cos x$, where $y'$ depicts partial ...
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0answers
25 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
3
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1answer
52 views

Changing the form of this equation

In quantum mechanics, a particle is described by its wavefunction, $y(x)$, which is related to the probability of finding the particle at position $x$ (roughly speaking). This wavefunction satisfies ...
2
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2answers
55 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...
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0answers
12 views

Algebraic determination of asymmetric unit (aka irreducible wedge) in Brillouin zone of lattice

In Solid State physics the reciprocal space is of utmost importance to predict the band structure and thus most of the electrical transport parameters like effective mass, etc. The First Brillouin ...
4
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1answer
100 views
+100

Nonlinear Partial DE

In my work I have faced with following partial differential equation $$\left(\frac{\partial u}{\partial x}\right)^2-\left(\frac{\partial u}{\partial y}\right)^2+f(x,y)\frac{\partial u}{\partial ...
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2answers
107 views

In what sense does $\sum_{k=0}^{\infty} 2^{2k} = - {1 \over 3}$?

In The Road to Reality Penrose remarks on an identity written down by Euler which is "obviously wrong" and yet correct "on some deeper level". He makes reference to the series again when discussing ...
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0answers
16 views

mathematical symbol in android app(offline) [closed]

i want to add mathematical equations to my offline android app. assume a multiple choice math test that includes text(utf -8)and equations , number and ... explain in simple please help me
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2answers
48 views

Time-independent Schrodinger equation

the equation : (-h/2m)y'' + U(x)y = (E)y How do you put the time-independent Schrodinger equation in the form of: y'' + G(x)y' + P(x)y = 0
3
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1answer
48 views

Calculus of variations question with two variables

If $u(x)$ and $v(x)$ satisfy $u(0)=1$, $v(0)=-1$, $u(\pi/2) =0$, $v(π/2) =0$ on extremals of functional $$ \int_0^{\pi/2}\left[\big({\frac{du}{dx}\big)^2 +\big(\frac{dv}{dx}\big)^2 +2 \,u v ...
3
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1answer
38 views

Center of mass calculation

Calculate the center of mass for : The area bounded by parabola $y = x^2/b$ and the line $y = b$. I got the following integral I just need verification that my work is correct. First I got ...
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0answers
7 views

How can I find the domain of this diffeomorphism (coordinate transformation)?

I have been struggling with this coordinate transformation in $R^2$. $Q:\begin{bmatrix}\rho\\\phi\end{bmatrix}\to \begin{bmatrix}\cosh(\rho)cos(\phi)\\sinh(\rho)sin(\phi)\end{bmatrix}$ I am ...
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4answers
789 views

What is the motivation for analytic solutions in Mathematical Physics?

I am trying to understand why one cares about solving PDE's with an analytic/theoretical solution when one can use numerical methods? If you tell me, "only mathematicians try to find theoretical ...
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1answer
24 views

Quick formula rearranging

I'm having problems rearranging this formula to solve for c, could someone lend a hand please. It's a physics formula for projectile motion. ...
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1answer
43 views

Speed as a function

we were studing the rate of the function $\frac{f{x_1}-f{x_2}}{x_1-x_2}$ if it is positive so the fonction is growing if it is negative so the function is ascending . in this moment our teacher ...
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2answers
27 views

How can I show that the function is smooth?

I got an assignment which I just can't find the right way to solve. It goes like this: Let $\Omega \in R^n$ be a domain and $b_1,...,b_n:\Omega \to R$ smooth mappings (or functions, don't know the ...
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0answers
6 views

Finding the approximate solution of ODE by perturbation method ( perihalion precession in the general spherically symmetric space time).

I am trying to find the approximate solution of the differential equation obtained for the the motion of massive particle around Riessner Nordstrom-Ads solution, differential equation to be solved ...
3
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1answer
66 views

Trouble with a Statement in Arnold's “Mathematical Methods of Classical Mechanics”

On Pg 6 of Arnold's Mathematical Methods of Classical Mechanics (2nd Edition), there is a line which reads One can speak of two events occuring simultaneously in different places, but the ...
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0answers
9 views

How can I prove that this is a diffeomorphism?

$G={{(x,y)\in R^2|x>0,y>0}}$ $ ~~~~~~$$\Omega={(u,v)\in R^2|v>0}$ Q:$G\to \Omega$, $\begin{bmatrix}x\\ y \end{bmatrix}$ $\to$$\begin{bmatrix}ln(\sqrt{\frac{x}{y}})\\\sqrt{\frac{x}{y}} ...
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0answers
21 views

Need help finding an equation of a boat.

The task is as follows: The flow rate of a river of width 2d is 0 at the river banks and linearly increases as you reach the center of the river to a maximum value of u. A boat crosses the river ...
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0answers
17 views

Forms and conservative forces

According to Tongs notes on Classical Mechanics; a force is called conservative when $F=-\nabla V$ And iff $\nabla \times F = 0$. This is in $R^3$. Also the potential $V=\int_{x_o}^{x^1} F(x)$ $dx$ ...
2
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1answer
40 views

Spinors and Möbius strips

Consider a Möbius strip; draw on one side of it an arrow aligned vertically; now take it for a trip by around the strip; then when it comes back to the same position it has flipped direction; another ...
2
votes
0answers
39 views

Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
2
votes
1answer
60 views

Do we deduce that the physical law isn't unit-free?

A small sphere with radius $1$ and density $p$ moves downwards with constant velocity $v$, under the influence of the gravity $g$, at a liquid of density $p_l$ and viscosity coefficient $\mu$. (The ...
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1answer
56 views

Finding potential of a given vector field

I am trying to solve the following problem: Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for ...
2
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1answer
20 views

According to Buckingham Theorem the rank of $A$ should be $2$

A physical system is described by a law of the form $f(E,P,A)=0$, where $E,P,A$ represent, respectively, energy, pressure and surface area. Find an equivalent physical law that relates suitable ...
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0answers
23 views

determining the fermi velocity via density of states

The problem is to determine the Fermi velocity for a fermion gas at absolute zero. the problem using integrating a function that looks like $$ v = \frac{4\pi V}{h^{3}} m^{3} \int_{0}^{\infty}{ ...
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0answers
25 views

How does the Schrodinger's potential transformer if the metric conformally transformers?

Given Schrodinger's equation $$ (-\nabla^2+V)\psi=E\psi $$ and the conformal transformation $\tilde{g}_{mn}=e^{2\phi}g_{mn}$, how does the Schrodinger's potential $V$ transformer if the metric ...
2
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0answers
23 views

How to prove the following determinant identity?

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
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0answers
30 views

Integrating Associated Legendre Polynomials

As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, but I am not sure how to proceed: ...
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1answer
22 views

What really is a path-ordered exponential?

In some texts about gauge theories in Physics I've found one object called a path-ordered exponential which I'm not sure what it means. As I understood, the idea is as follows: let $G$ be a Lie group ...
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0answers
66 views

Area enclosed by an equipotential curve for an electric dipole on the plane

I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following ...
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0answers
150 views
+100

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a sense, unorganized (see Open problems in ...
4
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0answers
46 views

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$?

How do you prove $\delta (ds^2) = 2 ds \delta(ds)$ ? To give context, this comes from: Dirac's Theory of General Relativity p19: http://imgur.com/mrkT5C7 I'm not comfortable with proofs regarding ...
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1answer
60 views

Why are nodes and nodal sets called this way?

Nodes of standing waves are points where they are zero. Generally, nodal sets of Laplacian eigenfunctions are the sets of points where they are zero. Why is this the name for them (that is, why is ...
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1answer
35 views

Newtons Law of Cooling Differential Equations

We have two differential equations, $$\begin{cases} {dT\over dt} = -\alpha(T-B)\\ {dB\over dt} = -\beta(B-T)\end{cases}$$ If $T(0) = 7$ and $B(0) =3$, determine the equilibrium temperature of the ...
0
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1answer
45 views

How to simplify the summation of log

I have a summation that involve log. I don't know how to solve this summation. I want to find an expression (even a good approximation is enough) for this summation. $\sum_{k=0}^{n}{log(a_k)}$ ...
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2answers
54 views

How can $ \Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $ possibly be the Heaviside Step Function?

How can $$\Theta(x) = \int_{-\infty}^\infty \frac{-i}{2 \pi k} e^{ikx} \, dk $$ possibly be the Heaviside Step Function? What I'm looking for is a direct visualization or maybe an approximate ...
1
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1answer
75 views

Proving that $\int \delta \dot{x} dt = \delta x$

Everytime I've seen this I've assumed it was true. It seems plausible. But I would like to rigorously prove it. I think this is correct, but I would like another opinion because my mathematics is very ...
0
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1answer
20 views

Pendulum tension force

I realize this is physics related, although the problem is really about math so I thought it would be a good fit for this site. My illustration is supposed to depict a pendulum and the forces ...
2
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0answers
34 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
0
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2answers
29 views

Solving Bernoulli equation transformation

I'm trying to solve the Bernoulli's equation via perturbation method but I need some help understanding how its done: We start off with $y'=-y+\epsilon y^2$ with $y(0)=1$. Then how is it possible ...
3
votes
1answer
62 views

What is a good reference for rigorous Electromagnetism and Electrodynamics?

Is there any good book on Electromagnetism from a more mathematical point of view? By this I mean a book which makes use of differential forms and maybe De Rham cohomology. I was also searching for ...
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Find out rotation and angle given a square

Consider I have a camera and a square, the projection of the square on the camera would give me a "distorted" square. Can I possibly find out the angle of the camera relative to the square and the ...