# 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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### Intuition on the necessity of the Lipschitz condition and a physical example of an ODE

The Picard-Lindelöf theorem states that the initial value problem $$y'(x) = F(x,y(x)), \ y(x_0) = y_0$$ will always have a unique solution on some closed interval containing $x_0$ assuming that the ...
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### Integrate a multivariable funtion w.r. to one variable or make triple improper integral

I am using Matlab 2010Ra and I want to do triple integration on my function below: My function (first Fock state): $$\psi(x) = \frac{1}{\pi^{\frac{1}{4}}} e^{-\frac{x^2}{2}}$$ In fact I have a ...
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### inhomogeneous heat equation with mixed boundary conditons

Solve $$U_{t}=U_{xx}+u$$ with mixed boundary conditions $$U_x(0,t)=0, U(l,t)=0$$ and initial condition $$U(x,0)=\varphi(x)$$ I know that I have to use separation of variables and I have an idea of ...
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### Levi civita symbol identity with n dimension

There is an identity $\displaystyle{\epsilon_{i_1...i_k i_{k+1}...i_n}\epsilon_{i_1...i_kj_{k+1}...j_n} =k!\epsilon_{i_{k+1}...i_n }}$ in wikipedia. https://en.wikipedia.org/wiki/Levi-Civita_symbol ...
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### Solution for the inhomogeneous 3D heat equation with initial temperature distribution

Can anyone describe the general solution for the inhomogeneous 3-dimensional heat equation: $u_t = K\nabla^2u + \frac{1}{c\rho}f$, with initial condition $u(x, 0) = g(x)$, no boundary conditions. ...
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### solve the initial value problem on the half line for the diffusion equation $U_x(t,0)=\sin t$ [on hold]

solve $U_t-U_{xx}=0$ for the half line with initial conditions: $$\quad Ux(t,0)=\sin t\\ U(0,x)=x$$
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### Action variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian who doesn't depend on the new coordinates $w_k$, but only in the momenta,...
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### Easier solution to first order non-linear differential equation?

Im am dealing with this differential equation: $$m\frac{dv}{dt}=mg-kv^2$$ where $m,g,k$ are constants. I am able to solve this by treating this as a separable differential equation, but that method ...
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### Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
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### How to derive a logarithmic potential from Newtonian?

Suppose we believe that the formula for Newtonian potential in $R^3$ is correct: $\varphi(\bar{x}) = \frac{1}{|x|} = \frac{1}{r}$, disregarding the constant. What is the justification of the fact ...
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### Trace identities for $\text{SO}(n)$

The Green-Schwarz mechanism in Type I string theory involves certain identities relating traces in the vector and adjoint representations of $\text{SO}(n)$ of dimension $n$ and $n(n - 1)/2$ ...
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### Number of states in microcanonical ensemble

for the non-physicists, all you need to know to answer my question is that I'm talking about a $6N$ dimensional space of the coordinates $\{\vec{q}_i,\vec{p}_i\}_{i=1} ^{N}$ which I call the phase ...
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### Taylor expansion of Crystal Field potentials

I am trying to work through Michael Tinkham's "Group Theory and Quantum Mechanics". In discussing crystal field theory he uses the following example: We start with an atom at the origin. We want to ...
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### Overview of Geometric analysis [closed]

Can anyone tell me what geometric analysis is about? After reading some articles I have a view that it uses PDE extensively for geometric problems. Am I right in this point? Also what kind of ...
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### Classifying continuous maps from closed 2-manifolds to various closed manifolds

I believe my question should be simple. The question is more physically oriented and originated from one of Witten's papers, "On Holomorphic Factorization of WZW and Coset Models", where he considered ...
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### How can I understand the step by step calculations for the formula from the blog below?

I am studying clustering and found a useful article on the blog post here Finding the K in K-Means. But I am having difficulty in understanding the formulas below and how I can do step by step ...
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### Cauchy horizon of a future Cauchy hypersurface

I'm studing on the book Semi-Riemannian geometry by O'Neil. I'm tryng to understand the proof of the Hawking's singularity theorem (theorem 55A in the book). What I don't understand is why if $S$ ...
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### Biorthogonality of vectors

This question is equal parts math and physics, though I chose to ask it here because I am more concerned with the mathematics behind it, rather than physical implications. Let $\hat{K}$ be a non-...
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### Future Space Opportunities for a Mathematician [closed]

I don't know if this question should be asked here or on "Mathematics Educators", however I'll post it here for the moment. I've just finished my first year of Mathematics and I do really like maths. ...
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### Integral of Bessel Functions Multiplying “polynomials”

How can I compute the following integral: $$\int_{0}^{1} (1 - x^{2})^{\nu - \mu - 1} x^{\mu + 1} J_{\mu}(\alpha_{\nu}x) dx$$ where $\nu > \mu \geq 1$ and $J_{\nu }(\alpha_{\nu}) = 0$. The $J_{\nu}$ ...
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### shifting integration variable and taking derivative seemingly giving problem

I am doing loop integral in quantum field theory, and an issue in shifting integration variable is giving me a problem. Let me illustrate with an example. I have an integral that looks approximately ...
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### Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra?

Does the Hodge dual (star) operator make the exterior algebra an involutive (*-) algebra? https://en.m.wikipedia.org/wiki/Hodge_dual https://en.m.wikipedia.org/wiki/*-algebra This would seem to be a ...
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### What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition? [closed]

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...
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### Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
Let $F(x)=x$ be a function that describes the magnitude and direction of a force that varies with distance from the origin. I understand that $m \frac {d^2} {dt^2} p(t) = F(p(t))$ is used to derive ...