"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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88 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 ...
3
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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 ...
2
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1answer
68 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 ...
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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 ...
0
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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 $v$ ...
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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 $x_0$. ...
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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 ...
3
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1answer
58 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 ...
2
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1answer
130 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 $A\subseteq\partial B$, ...
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79 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^{\...
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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 $||A||_2=||UAZ||_2$
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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 ...
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72 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. ...
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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 $h$:...
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26 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 $h=(h_i)_{i=1}^N$ ...
0
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1answer
38 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 ...
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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 \...
3
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1answer
78 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 ...
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55 views

The solution of Allen-Cahn equation?

$$\frac{\partial\phi(\mathbf{x},t)}{\partial t}=\varepsilon^{2}\Delta\phi-F^{'}(\phi),\ \ \ \mathbf{x}\in \Omega,t>0$$ $$\frac{\partial \phi}{\partial\mathbf{n}}=0\ \ \text{on} \ \partial\Omega$$ $$...
1
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1answer
57 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 ...
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47 views

Can I split this integral to a sum over three contours?

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 ...
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1answer
35 views

Perturbation theory, why are the assumptions of the method satisfied?

I am a undergrad student interested in math taking quantum mechanics. Yesterday I was introduced to what physicists call perturbation theory, non-degenerate case. According to authors Griffiths, ...
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1answer
66 views

Proof of Kepler's Third Law

Kepler's Third Law states that the square of the time period ($T$) of revolution of a planet about the sun is directly proportional to the cube of the semi-major axis ($a$) of its elliptical orbit. ...
0
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1answer
26 views

Energy change of two discs in both the compression and restitution phases of a collision between them.

A collision occurs between two discs $A$, of mass $0.4kg$, and $B$, of mass $0.8Kg$, moving in the same direction with speeds $6$ $m/s$ and $2$ $m/s$ respectively. Given that the coefficient of ...
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1answer
164 views

Proof of Gauss' Theorem in electrostatics using Stokes' and divergence theorems

This was a problem I encountered while solving my homework. PROBLEM:The potential $\phi(x,y,z)$ at any point $P$ due to the charges $q_i, i=1,2,..,n$ with respective position vectors $\vec r_i, i=1,...
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59 views

Find the friction constant minimizing the duration of the vertical movement of a wheel

Q) The mass of a car that acts on one wheel is $100 kg$. The elasticity (spring) constant in the suspension system of that wheel is $k = 10^{4}N/m$. Design the strut (find the friction/resistance ...
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21 views

Free Particle on Riemannian Manifold

I'm trying to understand Takhtajan's "QM for Mathematicians" and I'm struggling still with the generalized coordinates. To make things simple consider a free particle on some Riemannian manifold. ...
0
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1answer
24 views

Calculate the line integral (cyl. coords)

So I have this vector field $$ \textbf{B}=K \left( \frac{\cos \varphi}{\rho^2}\textbf{e}_{\rho}+ \left( \frac{\sin \varphi}{\rho^2}+ \frac{1}{a\rho}\textbf{e}_{\varphi} \right) \right) $$ and the ...
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23 views

Simple (?) tensor index notation; When do the indices mean inner product and in what order?

In index notation, does the term $σ_{ik}x_{j}n_{k}$ mean $\bf{σx}\cdot\bf{n}$ or $\bf{xσ}\cdot\bf{n}$? Here $σ$ is a second-order tensor and $x,n$ are vectors. On the same note, is $$\frac{\partialσ_{...
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38 views

What does symmetry imply about the solution in mathematics? (Example: Gauss' law)

Suppose you have an infinite cylinder and are considering a field $\mathbf{D}$ caused by physical elements within the cylinder such that it satisfies $\int \mathbf{D}\cdot d\mathbf{a} = Q_{free}$. ...
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26 views

Cohomology of Anti-de Sitter manifold and black hole

Anti-de Sitter manifold AdS$_n$ is a maximally symmetric pseudo-Riemannian manifold with constant negative scalar curvature. This has $\mathbb{R}^{2,n-1}$ as its embedding and is a solution to the ...
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2answers
70 views

What exactly does the Hilbert scheme of points parametrize?

The Hilbert scheme of points is defined as $$ \text{Hilb}^n(X) = \{ I \subset \mathbb{C}[x,y] \text{ such that } \text{dim}_{\mathbb{C}}/I = n \} $$ or, in words, the Hilbert scheme of points ...
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36 views

Rounding in the method of least squares for linear regression analysis?

I tend to round off the intermediate values (like that for $\Sigma x^2) $ of my calculation for slope, etc., to appropriate significant digits by considering the significance of the raw data. ...
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1answer
72 views

How to plot a qubit on the Bloch sphere?

I've been reading pages such as this one: http://comp.uark.edu/~jgeabana/blochapps/bloch.html Which talk about the Bloch sphere, but I've been unable to figure out how to plot states on the sphere ...
0
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1answer
45 views

Determine if this vector field is a conservative force field?

Does $A(x, \, y) = (3x^2 + 2y^2)i + (4xy + 6y^2)j$ represent a conservative force field? If so, determine the potential $\phi$ in $A = \text{grad} \ \phi. $ From what I understand, we need to find $\...
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52 views

Jacobi Identity for Tensors

I am trying to derive the Maxwell's equations from the electromagnetic field tensor $F_{\alpha\beta}$ by using Jacobi identity: $$\partial_\gamma F_{\alpha\beta} + \partial_\alpha F_{\beta\gamma} + \...
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0answers
24 views

Reference request for complex scalar field, propagators worked out with path integral approach? [closed]

In quantum field theory, the Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^*\phi.$$Can anyone supply me a reference to where the ...
0
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1answer
43 views

Why the equivariant volume of a non-compact space can be finite?

I am very confused with equivariance (equivariant cohomology etc). In specific when one tries to evaluate the equivariant volume of, say, $\mathbb{R}^2$ (with coordinates $x,y$) one finds that it is $...
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29 views

Calabi's theorem

I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres). Theorem Let $\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$ be a full harmonic map. ...
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1answer
30 views

Is this Fourier Transform relation correct?

Is the Fourier Transform of $$\nabla f\cdot \nabla g$$ from $\vec{x}$ to $\vec{k}$ space a convolution? I know that, for a certain definition of the FT, $$\nabla f\to \vec{k} F $$ where $F(\vec{k})$ ...
2
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0answers
19 views

Finding highest weight of a $gl(3)$ submodule of a $gl(4)$-module

We have $gl(3)\subset gl(4)$. I have a $gl(n)$ $V(3,2,1,0)$-module. I want to know two things: 1) What is the $gl(3,\Bbb C)\subset gl(4,\Bbb C)$ branching rule for the $gl(n)$ $V(3,2,1,0)$-...
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1answer
31 views

Eliminate all parameters from the differential equation $u_t-Au_x-Bu^3+Cu_{xx}=0$.

The question is to scale the equation $$u_t-Au_x-Bu^3+Cu_{xx}=0$$ to eliminate all parameters. where $D>0$ and $A,B$ are nonzero. I tried to substitute $U=U(x/L,t/t_0)$ into the equation and ...
2
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1answer
26 views

Solving differential equation and obtain expressions for unknowns?

I have the following differential equation $my'' + \beta y' + mg = 0$ , with condition $y(0)=0$. I need to solve the equation and obtain expressions for the unknowns. I have attempted to use the ...
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2answers
160 views

How does curl relate to rotation?

The operation mathematically means $$(\nabla \times \vec A)\cdot\hat n = \lim_{\Delta S\to\ 0} \frac{\oint\vec A\cdot\ d\vec l }{\left | \Delta S \right |}$$ and the proof of this is quite logical. ...
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97 views

Solving recursions by calculating determinant of an infinite matrix

In this reference (pg. 4) and few others a specific parameter in a recursion formula is solved by setting the determinant of an infinite matrix to $0$. In this precise case we have $c_{n-1} - D_n c_n ...
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28 views

What is a weight vectors weight really?

What does a weight vector with weight $(\lambda_1,\cdots,\lambda_n)$ actually mean? Let $V$ be a $gl(n)$-module and I have that $v\in V$ is a weight vector if it is an eigenvector for all elements of ...
2
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2answers
65 views

How do you swap x/y to y/x?

I saw this video on Lorentz transformation and needed to refresh my memory a bit. If $$\frac{t}{t'}= \sqrt{1-\frac{v^2}{c^2}}$$ and $$\gamma = \frac {t'}t $$ How do you make this equal ? $$\...
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45 views

$H$ self-adjoint with mass gap, $P \ge 0,\Omega \in D(P)$, $H + \lambda P$ self-adjoint $\implies$ for $\lambda$ small, $H+ \lambda P $ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
2
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1answer
96 views

What does $\operatorname{supp}(A)$ mean?

I'm looking at a paper (specifically this one). In the paper, we have a positive operator $A$, and the operator $\operatorname{supp}(A)$ is supposed to be a projection operator. Does anybody know ...
2
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0answers
46 views

Affine geometry textbook

What's a good recommendation for a book on affine geometry at the undergrad level? I ask because I skimmed through the first bit of Vladimir Arnold's Mathematical Methods of Classical Mechanics and ...