"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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Zeta function and heat kernel

It is easy to prove that zeta function $$\zeta_{\Lambda}(s)=\sum \frac{1}{\lambda_{n}^{s}}$$ and trace of heat kernel $$K_{\Lambda}(t)=\sum e^{-\lambda_{n}t}$$ satisfy the relashion ...
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Boundary conditions for the time-independent Schrödinger equation on the sphere

if you have a free Schrödinger operator on a sphere $-\Delta \psi(\theta,\phi) = E\psi(\theta,\phi),$ then the substitution $\psi(\theta,\phi) = f(\theta)e^{i n \phi}$ leaves you with the ...
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How to calculate Hill's discriminant?

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
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Rotation about an axis by matrix multiplication

Suppose I have three axis of rotation vectors $\vec{v_1},\vec{v_2},\vec{v_3}$ and angle of rotation as vectors $\theta_1,\theta_2,\theta_3$. Take a vector $P$ then apply rotation around $\vec{v_1}$ ...
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how to calculate the phase shift in the formula that has sin in both side?

Given formula $asin ( x ) = b sin( x + \phi)$ where $a$ and $b$ are constants. I want to calculate $\phi$.
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physical meaning of heat equation

consider the heat equation $u_t=a(t)u_{xx}+f(x,t)$, $0<x<L$, $0<t<T$ subject to the initial condition $u(x,0)=g(x)$ and boundary conditions $u(1,t)=0,$ $u_x(0,t)+hu(0,t)=0$ where ...
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What happens when this turns to $dx$?

I have this equation: $$ ds^2=c^2dt^2-dx^2-dy^2-dz^2. $$ And I've also been given $$ x=x'\cos(\Omega t)-y'\sin(\Omega t), $$ which I need to substitute into the first equation. I've squared $x$ to get ...
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Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
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k-space tensor integral in statistical mechanics [duplicate]

k is the modulus of the vector k. Please help me to integrate the above tensor expression in the infinite domain of the vector k. I have tried to let u in the direction of kz and then transform the ...
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40 views

$k$-space tensor integral in statistical physics

$$Q=\int_{\text{all space}} \frac{\hbar \nu_g \mathbf{k}\mathbf{k}}{\exp[(\hbar \nu_g |\mathbf{k}|-\mathbf{k}\cdot\mathbf{u})/k_B T]-1}d\mathbf{k} $$ Please help me to integrate the above tensor ...
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85 views

Damped simple harmonic oscillator, phase space

I want to calculate and draw the phase space trajectory of this damped harmonic oscillator: $$\ddot{x}+\gamma\,\dot{x}+\omega^2x=0$$ for the two cases $\gamma=2\omega$ and $\gamma=\omega$. I'm ...
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39 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
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1answer
207 views

Newton's Law of Cooling, age of Earth, weak math skills

I'm curious about a problem concerning the age of the earth, but I don't have the math skills to think properly about it. I've found the solution to Newton's Law of Cooling, and I can handle that ...
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176 views

question about monoidal structure of a 2-category

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the ...
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Momentum Representation vs Position Representation

I have a question involving the representation of operators in momentum representation and position representation. The question is a little long, so I'll do my best to explain it. We are given an ...
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297 views

Convert coordinates to a different coordinate axis

Sorry for any forum rules I have broken, I needed a quick answer. I want to create a plane including 3 nonlinear points on a 3d coordinate system, one being the origin. I also need to create a ...
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261 views

Learning Roadmap to Mathematical Physics

Currently, I am a graduate student specializing in algebraic geometry. On the other hand, I have also become extremely interested in the mathematical physics. However, I am not sure what steps I ...
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52 views

Calculating the electric field of a disk

I'm having trouble regarding how to calculate the electric field of a disk. Here's the scheme: The exercise states that the disk is uniformely charged. This is what I did: Density charge : ...
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1answer
44 views

Inverse laplace transform in a physics problem.

This came up during a physics problem, where we need to find the inverse laplace transform of $$X(s) = \left( 1+ \frac{k}{ms^{3/2}}\right)^{-1} \left( \frac{c_1}{s^2} + \frac{c_2}{s} \right)$$ to ...
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Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
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1answer
304 views

Guide to mathematical physics?

I am currently a math phd student specializing in algebraic geometry aspiring to work at the boundaries of the the fields of mathematics and physics and so, was looking into the field of mathematical ...
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102 views

Linear algebra too early.

I have started college few days ago. At the first exposition of physics, professor has been reminding us what is vector and what is definition of a vector. But he has been using linear algebra to ...
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Mathematical expression for map from $[0,1]$ to $S^2$

A topological space is called arcwise connected if, for any points $x,y\in X$, there exists a continuous map $f: [0,1]\rightarrow X$ such that $f(0)=x$ and $f(1)=y$. Although it is intuitively ...
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Connectedness of $O(3)$ group manifold

A topological space is said to be connected if it cannot be written as $X=X_1\cup X_2$, where $X_1,X_2$ are both open and $X_1\cap X_2=\emptyset$. Otherwise, X is called disconnected. Is it wrong to ...
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83 views

how to solve the system of differential equations for this particle?

I'm trying to solve this problem A particle of mass m moves under the action of gravity on the inner surface of a paraboloid of revolution $x^2+y^2=az$ which assumed frictionless. Obtain the ...
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Explain why we describe the flow of fluid passing through volume with this equation

I am new to this Q&A site. Recently I came across this expression while watching a video http://www.youtube.com/watch?v=GveJWPr9UOk/ about the mass of fluid flowing in through a cube. You can find ...
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1answer
257 views

Guidance regarding research in Mathematical Physics

I am currently a Master's student in Mathematics. The main focus of my undergraduate programme was on Mathematics. However as a part of the course, I have done some Theoretical Physics courses. In ...
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37 views

How do I determine both width and angles to cut on my deck

I am trying to figure out how to calculate the length of the board and both cut #1 and #2 angles. The board will go from bottom left side of deck to top right side. The far right line is my house. ...
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1answer
21 views

Regularity of the surface of a crystal

If I want to model the surface of any random crystal, is it safe to assume that it is the graph of a Lipschitz function. Is there a precise result from physicists? How wrong would it be if I assume ...
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1answer
67 views

Force field and work

How can I solve the following? Let $F_1=(-y,x,z)$ and $F_2=(y,x,z)$. Calculate for each force field the work done in moving a particle around the circle in the $(x,y)$ plane. Which of the two ...
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1answer
36 views

Simultaneous Suvat help [closed]

I learnt this in college but can't for the life of me remember how to do it. I've searched stack exchange and the internet for answers but it isn't clicking. It doesn't help my teacher has decided to ...
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1answer
60 views

Mathematical Description for Steam Rising from a Cup

I was staring at a cup of coffee I have on the desk just now. The light shines through the water vapor as they rise from the cup. The shape of the steam is not completely random, as it drift from ...
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How far down will the ball travel and what is the magnitude of the ball's initial vector?

Confused a little with $V_x$ and $V_y$ components and how to find the displacement of X. A football is kicked with an initial velocity of $V_x = 30 \text{ ft/sec}$, and $V_y=80 \text{ ft/sec}$ 1) ...
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Conceptual Understanding of Kernels

In the previous thread (Difference between kernel and function?) the question of the difference between a kernel and a function came to, in my mind, an unclear conclusion. Am I right in thinking that ...
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1answer
58 views

Integral of voltage, $\int_{-a}^a \frac{dy}{\sqrt{x^2 + y^2}}$

This is (probably) a very easy integral to solve, but for some reason the answer just isn't coming to me (or at least the one my professor got isn't). He gave us a formula for voltage along the x-axis ...
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1answer
159 views

Electrostatic Potential Energy integral in spherical coordinates

I'm having trouble with evaluating an integral that arises from attempting to find the total energy of an electrostatic system consisting of two point charges, which involves an integral over all ...
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1answer
56 views

Show that requiring Electrostatic potential to be a stationary point of Electrostatic potential energy is equivalent to Laplace's equation.

Suppose we want to find the electrostatic potential $\phi$(r) inside of some volume $V$ with known boundary conditions. The physical field configuration should minimize the electrostatic potential ...
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Diagonalization of total angular momentum over creation operators for an isotropic harmonic oscillator?

You have an isotropic three dimensional quantum harmonic oscillator so the Hamiltonian is $$ H=\frac{p^2}{2}+\frac{r^2}2 $$ If you do the creation-annihilation operator-algebra trick and define ...
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1answer
98 views

Lagrangian equivalence up to total time derivative: dependence on higher derivatives

I recently encountered the problem Show that the Euler-Lagrange equations of motion for $L_1$ and $L_2$ are the same when $$L_2(\ddot{q},\dot{q},q,t) = L_1(\dot{q},q,t) + \frac{d}{dt} ...
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What are the non-linear representations of $SO(3,1)$?

The classification of the representations of the Lorentz group $SO(3,1)$ is well known, but the representations are usually expressed in linear form. My question is whether there is a framework to ...
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1answer
73 views

Books for Tensor Algebra used in Physics?

I'm taking a dual Math,Physics undergraduate course.I want to study GR and a few parts of relativistic Quantum Mechanics.I've a decent amount of knowledge in linear algebra. Though we have tensor ...
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24 views

Heinrich Hertz on Mathematical Equations

What is the quote from Heinrich Hertz on how he could never exhaust the meaning behind a mathematical equation? (It's not mentioned in the Hertz quotations here.)
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1answer
48 views

Does Lagrangian always exist for any equation?

In mathematical physics, a lot of equations can be interpreted as a solution of least action principle for some Lagrangian. I wonder if for every equation there is a Lagrangian so that one achieves ...
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1answer
26 views

one-dimension motion

A rock is thrown vertically up at 80 ft/s. Find it's maximum height, time of flight, and final velocity as it passes the starting point. So for my parameters I have: initial y=0 final y=? ...
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49 views

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates.

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates. I have been attempting to do this using general curvilinear dot products ...
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1answer
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Visual understanding of convergence of domains in the sense of Fisher

In these lecture notes by Ueltschi here, I found in Definition 2.3 a peculiar type of convergence. Especially the second property is hard for me to visualize what it means, could anybody try to ...
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1answer
41 views

Find the average acceleration

Find the average acceleration of the tip of the 2.4-cm long hour hand in the interval noon to 6pm. I found the average velocity is -2.2x10^6 but I'm not sure how to go about finding acceleration. If ...
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91 views

help setting up a velocity question

Standing on the ground from 3.0 m from a building, you want to throw a package from your 1.5- shoulder level to someone in a window 4.2 m above the ground. At what speed and angle should you throw ...
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2answers
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For the following voltage waveform, determine the frequency f, peak voltage amplitude Vp and phase [closed]

For the following voltage waveform, determine the frequency f, peak voltage amplitude Vp and phase $ v(t)=6sin(2\pi 10000 t+30^0) Volts $ the solution is f=10KHZ Vp=6 volts $ phase =30^0 or, \pi/6 ...
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1answer
58 views

List of well-known submodular function in physics, statistics, math?

Can you please share a list of well-known submodular functions (have the diminishing return property) that you know? In physics, stats, math, etc? I am searching for a submodular function for my ...