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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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 ...
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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 ...
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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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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 ...
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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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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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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 ...
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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 ...
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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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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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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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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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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. ...
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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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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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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 $F(\vec{k})$ ...
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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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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 ...
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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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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. ...