# Tagged Questions

Questions related to mathematical physics which include application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories

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### Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
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### Importance of Representation Theory

Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I ...
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### What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.) What is the geometric, physical, or other meaning of the third derivative of a function at a point? If you have interesting things to say about the meaning of the ...
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### Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$

If $T_f$ is a distribution, i.e. a linear functional, continuous according to the convergence defined here, defined on the space $K$ of the functions of class $C^\infty$ that are null outside a ...
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### Teenager solves Newton dynamics problem - where is the paper?

From Ottawa Citizen (and all over, really): An Indian-born teenager has won a research award for solving a mathematical problem first posed by Sir Isaac Newton more than 300 years ago that has ...
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### Multivariable Calculus Book Reference

I am looking for a multivariable calculus book that is really physics oriented. Anyone know of any? EDIT: My wife is looking to brush up on multivariable at the same time she needs to brush up on ...
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### Dirac Delta Function of a Function

I'm trying to show that $$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$ Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by ...
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### Physicists, not mathematicians, can multiply both sides with $dx$ - why?

The following question is asked without malicious intentions - it's not intended as a flamebait! In my physics textbooks (Young & Freedman in particular) I have often seen derivations of ...
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### Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
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### Property of Dirac delta function in $\mathbb{R}^n$

How does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$ where $S$ is the surface inside $V$ where ...
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### Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
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### Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real ...
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### The vertices of an equilateral triangle are shrinking towards each other

For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
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### What is the optimum angle of projection when throwing a stone off a cliff?

You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the ...
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### Is there a mathematical theory of physical knots?

From the point of view of people tying real knots (canonically, sailors) mathematical knot theory ignores much of what makes the problem of knot-tying interesting. Some matters that come up in the ...
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### Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
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### What Mathematics questions can be better solved with concepts from Physics?

Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics ...
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### Can this gravitational field differential equation be solved, or does it not show what I intended?

This is the equation I'm having trouble with: $$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$$ That's the non-vector form of the universal law of gravitation on the left and Newton's second law of ...
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### Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
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### Residue of $z^2 e^{1/\sin z}$ at $z=\pi$

A while back I was working through many problems in Mathews and Walker's Mathematical Methods of Physics. In the appendix is this problem: A-6. Find the residue of the function $z^2 e^{1/\sin z}$ ...
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### Energy norm. Why is it called that way?

Let $\Omega$ be an open subset of $\mathbb{R}^n$. The following $$\lVert u \rVert_{1, 2}^2=\int_{\Omega} \lvert u(x)\rvert^2\, dx + \int_{\Omega} \lvert \nabla u(x)\rvert^2\, dx$$ defines a norm on ...
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### Functional Derivative ${\delta q_a(t)}/{\delta q_b(t')}$

$\newcommand{\fdv}[2]{\frac{\delta #1}{\delta #2}}$ $\newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d}#2}}$ $\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}$ I'm from a physics background and ...
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### Should isometries be linear?

Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a ...
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### Could this be called Renormalization?

Quoted from   Space-Time Approach to Quantum Electrodynamics   by R. P. Feynman, Phys. Rev. 76, 769 1949 : We desire to make a modification of quantum electrodynamics analogous to the ...
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### Free-fall according to Newton's gravitation law

Most analysis of free-fall assume that bodies fall with constant acceleration. If however one analyses free-fall according to Newton's gravitation law, one is lead to a differential equation which I ...
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### Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
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### Can somebody help and explain me with angular and linear Velocity textproblems?

So we got this as homework today, and I just don't understand it; how to start or how to do it. Explanations and/or setups would be great :) A wheel with ...
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### Good Physical Demonstrations of Abstract Mathematics

I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to ...
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### Very *mathematical* general physics book

I am searching for a book to study physics. So far, I've been suggested Resnick, Halliday, Krane, Physics, but it doesn't seem to be very suited for a math major. Can you suggest some more ...
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### Intuitive reasoning behind $\pi$'s appearance in bouncing balls.

This video is about an interesting math/physics problem that when cranked out churns out digits of $\pi$. Is there an intuitive reason that $\pi$ is showing up instead of some other funky number ...
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### reference for multidimensional gaussian integral

I was reading on Wikipedia in this article about the n-dimensional and functional generalization of the Gaussian integral. In particular, I would like to understand how the following equations are ...
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### Does Gödel's Incompleteness Theorem really say anything about the limitations of theoretical physics?

Stephen Hawking believes that Gödel's Incompleteness Theorem makes the search for a 'Theory of Everything' impossible. He reasons that because there exist mathematical results that cannot be proven, ...
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### In what ways has physics spurred the invention of new mathematical tools?

I came across this comment: Mathematical rigor is not a criterion that physicists have for evaluating their theories. From a mathematical perspective, the non-rigorous theories are far more ...
So called pseudovectors pop up in physics when discussing quantities defined by cross products, such as angular momentum $\mathbf L=\mathbf r\times\mathbf p$. Under the active transformation \$\mathbf ...