# Tagged Questions

"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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### Toppling of a road cone that has an axis at an angle $\alpha$ to the horizontal.

A road cone consists of a $45cm$ x $45cm$ square base of height $10cm$, and a conical shell of radius $15cm$ and height $75cm$. The base has a circular hole through it , of radius $15cm$, to aid ...
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### $\nabla \sqrt{\rho} \in L^2(\mathbb{R}^3) \implies \rho \in L^3(\mathbb{R}^3)$

I found this in the INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL XXIV, 250 (1983) inside the paper of Elliot H. Lieb with the title Density Functionals for ...
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### n-dimensional wave equation proving the compactness of the support of the solution

The question is the following. Let $u\in C^2(\mathbb{R}^n\times[0,+\infty))$ be a solution of the problem \begin{cases} u_{tt}-\Delta u = 0\\ u(x,0) = \phi(x)\\ u_t(x,0)=\psi(x) \end{cases} where the ...
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### Prove a particular property of Laplacian operator

I can't prove that Laplacian $\Delta(u(x))=\Delta(u(x_1,\ldots, x_n))=0$ also implies $$\Delta\left(|x|^{2-n}u\left(\frac{x}{|x|^2}\right)\right)=0$$ for $\frac{x}{|x|^2}$ in the domain of ...
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### Finding the measured average angular velocities

Supposed I have a couple rows of data with recorded measured ratios $\omega_f/\omega_i$ and they ask me for the "Average Measured $\omega_f/\omega_i$ " This may seem like a really trivial solution but ...
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### Finding the limit of N approaching infinity: $N(x^\frac{1}{N}-1)\approx\ln(x)+\frac{1}{2N}\ln(x)^2+…$

I am having trouble understanding the linked exercise, final paragraph (not parts a or b) Entropy Calc Problem I understand this is a physics related exercise, however, my trouble comes in at the ...
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### Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a ...
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### What exactly is the diagonal subgroup of a group?

In specific consider the example of $SU(2)_a \times SU(2)_b$. What is the definition of the diagonal subgroup and how can one construct it from the generators of the group (or its algebra)? This ...
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### Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
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### How to infer $f$ from this operator: $g(k\nabla f - \nabla,.)$ where $g$ is the Euclidean metric, $k > 0.$

I have the following operator: $g(k\nabla f - \nabla,.)$ where $g$ is the Euclidean metric, $k > 0$, and $f$ is unknown. It acts on vectors in $\mathbb{R}^n$. What kind of informations can I obtain ...
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### Is it correct to think of the Laplacian as the divergence of a gradient field?

Factoring out the notation, I see that $$\nabla^2(\phi) = \nabla \cdot \nabla(\phi) = \nabla \cdot (\nabla(\phi))$$ which looks something like the divergence of the gradient of phi. Is it ...
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### How is my proof that this vector field is identically zero?

EDIT: If my work is fine, I believe that the problem statement (an old exam question from 1992) has given one too many assumptions - namely, divF=0. I think towards the end of my proof, when I ...
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### How to use the assumption that a vector field is curl-free in a “convex” region,

I don't seem to need this assumption in one of my proofs, but the problem statement gives it, so I think I had better try to use it. Does a convex region imply that it is simply connected (but that ...
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### What does this gradient-like symbol mean?

If $\nabla \phi$ denotes the gradient of some scalar field $\phi$, then what does $\nabla^2 (\phi^2)$ mean? I don't think it means taking the gradient of a gradient (of a squared-scalar field), ...
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### Advice on Mathematical Modeling with Differential Equations

I am on my fourth year studying in a bachelor program in applied mathematics and computer science and plan to write a term paper on mathematical modeling using differential equations. This will be the ...
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### when is a function changing by an order of 1,2,3…n

say for example we have the distance traveled by a vehicle as a function of time. if the speed(change in distance) is constant then this would be a linear function of order 1. if there was ...
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### Lagrangian invariant under left and right multiplication by unitary matrices, slick way to see?

Is there a slick way to see that the Lagrangian$$\mathcal{L} = \text{Tr}(\partial^\mu G\partial_\mu G^{-1}),$$where $G$ is an $N \times N$ unitary matrix, is invariant under left and right ...
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### A bead is threaded on a friction-less vertical wire loop of radius $R$.

The question is the very last sentence at the end of this post. In this post, I'll demonstrate how I reach to a contradiction(the conditions mentioned in conjecture 1 should be satisfied by all ...
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### Nonhomgeneous Linear Differential Equation: Harmonic Oscillator

Consider frictionless harmonic oscillator (w/ m = 1) driven by an external force $f(t) = A\sin{\omega t}$, so that $$\frac{d^2 x}{dt^2} + \omega_0^2x = A\sin{\omega t}.$$ Show that the particular ...
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### 2D reaction-diffusion Schnakenberg Model normalization

I need to normalice this Schnakegnber model in 2D: $u_t=D_u(u_{xx}+u_{yy})-u+av+u^2v\\ v_t=D_u(u_{xx}+u_{yy})b-av-u^2v$ We also know that $a,b>0$. Then I know that I'll applied separation of ...
I am studying the numerical aspects of fourth-order elliptic problems now, and I came across the plate problem: Let $\Omega\subset\mathbb{R}^n$ bounded domain with Lipschitz-Boundary. Find $u$ s.t. $\... 0answers 35 views ### How can I understand instantons as sheaves? In specific, instantons are considered torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two moduli spaces (... 0answers 21 views ### Moving two links in a two-dimensional space I have two robotic links. They're basically sticks. These two linked sticks can be controlled by changing their angle with respect to the former link. In this case, the first stick simply moves like ... 1answer 89 views ### Doppler effect: an understandable explanation for a mathematician I have curiosity by Question. Can someone explain me, and to the audience too, the mathematical essence behind the so called Doppler effect? Thanks in advance. Then you have the ability to ... 1answer 47 views ### Show series representation of orthogonal polynomials wikipedia has the following series expansion for hermite polynomials, namely: $$\exp \left\{xt-\frac{t^2}{2}\right\} = \sum_{n=0}^\infty {\mathit{He}}_n(x) \frac {t^n}{n!}.$$ Does anybody see how ... 1answer 71 views ### Energy functional and Euler Lagrange equation We know that for potential energy functional, its derivative is called the Euler Lagrange equation and physically, it means that at the given point there is a force balance. Now if the energy ... 0answers 19 views ### Deducing the Equation of a Transformed Sinusoid Given a wave, which you know to be a transformed sinusoid, how can you determine its equation? I have the following, which is a wave I obtained experimentally: It is a little off what we would ... 1answer 42 views ### Integration of motion using resistance and gravity. I'm having trouble with a high school mathematics question. An object of mass$1kg$falls from rest in a medium in which the resistance to motion is given by$r=kv^2$, where$k$is a constant and$v$... 1answer 21 views ### Show a function behaves as a harmonic oscillator We have a function$V(x)$(potential energy) with$x$being some variable. This function has a minimum at a certain$x_0$. We assume that$V(x)$is an analytic real function of$x$around$x_0$. ... 1answer 50 views ### Many worlds probability of getting cancer I have first asked this question on physics.SE (where I personally believe it belongs), however it was suggested that this question better fits here, so here I am. My understanding of probabilities ... 1answer 59 views ### raising/ lowering indices Here is my understanding of tensors: There is more than one way to think about tensors. One way is be thinking about tensors as objects with components which obey some transformation laws. For ... 1answer 132 views ### Why is$|\cos\theta d\omega|$the projection of the differential solid angle$d\omega$onto the$(x,y)$-plane? Let$B\subseteq\mathbb R^3$be the ball with radius$r>0$around$0$and$S_{\partial B}$be the surface measure of the boundary$\partial B$. Given a piece of the surface$A\subseteq\partial B$, ... 0answers 92 views ### Trace of six gamma matrices I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5})$$ I know that I can express this as: $$Tr(\gamma^{\mu}\gamma^{\... 1answer 34 views ### Norm of orthogonal matrices Can someone help me with this problem. I have no idea how to solve it!! If A is a p×q matrix, U is a p×p orthogonal matrix, and Z is a q×q orthogonal matrix, prove that ||A||_2=||UAZ||_2 1answer 47 views ### Finding the volume of a real egg if the volume of an egg shape(with different dimensions) on a graph is known Equation of the egg shown above: If the volume of the egg show above is: 12.00405units^3(found using calculus) if the volume of a real egg is 55cm^3 Is there anyway of finding out the ... 1answer 73 views ### Why use a particular regularization for \int_0^\infty \mathrm{d}x\,e^{i p x}? There are many badly defined integrals in physics. I want to discuss one of them which I see very often.$$\int_0^\infty \mathrm{d}x\,e^{i p x}$$I have seen this integral in many physical problems. ... 1answer 40 views ### How can I solve \beta^2=\frac{m^2g}{h}\left(-\frac{\beta t}{m}+e^{\frac{\beta t}{m}}-1\right) for \beta? This equation arose when I tried to find out how to derive \beta in Stokes' Drag Force F=\beta v as a function of the time t it takes a mass m to hit the ground after falling from a height h:... 0answers 29 views ### Special case of the inverse Ising problem with equal correlations Let s_1,\dots,s_N\in \{-1,1\} be N binary spins. The problem of finding a symmetric interaction matrix J=(J_{i,j})_{i,j=1}^N with zero diagonal and an external magnetic field h=(h_i)_{i=1}^N ... 1answer 39 views ### How does the Pauli principle work? Let H be some Hilbert space. Now in general, in quantum mechanics, the vector space representing states of n (non-interacting) particles is H^{\otimes n}, but if I consider these particles of be ... 1answer 19 views ### Christoffel connection I am trying to determine the correct expression when expanding a contravariant derivative acting on another contravariant derivative acting on the Ricci scalar. \nabla^a \nabla^b R = \partial^a \... 1answer 80 views ### Diffraction and Fresnel Integrals Migrated from Physics SE due to mathematical content I am trying to derive the intensity variation function for a single slit diffraction. Sorry for the poor diagram... So I decided to take the ... 0answers 58 views ### The solution of Allen-Cahn equation?$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega... 1answer 62 views ### Double-commutator$[f,[f, - \Delta]] = -2 |\nabla f|^2.$This book (proof of Theorem 3.2) in chapter 3.1 claims click me that an easy computation shows that $$[f,[f, - \Delta]] = -2 |\nabla f|^2.$$ where$[.,.]\$ denotes the commutator. Unfortunately, I ...
I have the following integral $$Z = \frac{1}{2\pi i} \int dx \, \frac{1}{(x-a_1)(x-a_2)(x-a_3)}\times \frac{1}{(x+\epsilon - a_1)(x + \epsilon - a_2)(x+ \epsilon - a_3)}$$ and this integral has ...