Questions tagged [mathematics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com The mathematics tag covers non-applied pure mathematical disciplines that are traditionally not part of the mathematical physics curriculum, such as, e.g., number theory, category theory, algebraic geometry, general topology, algebraic topology, etc.

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Is $g_0(u + ct)$ in d'Alembert's formula evaluated at $u(t)$ or $(u(0))$?

I'm a complete noob to d'Alembert's general solution the one-dimensional wave equation. Very slowly, I'm starting to understand it, although this has been difficult because the usual example ...
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Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} (\dot{u}(x,t))^2 - ca(u'(x,t)^2\tag{1}$$ with respect to $u$, ($\delta L / \delta u$). The ...
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Can a singular function plus a Dirac delta have be non-infinite?

In QFT we sometimes encounter functions of the form: $$K(x-y) = \delta(x-y) + \frac{k}{(x-y)^n} $$ Where $x$ and $y$ are $d$ dimensional vectors and $k$ is a (possibly imaginary) constant. These can ...
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Does physicists have a pre-conceived notion of continuity?

In many physics lecture on GR/ mathematical physics, one of the first things discussed is topology. I have seen many times that the reason for topology being discussed is that it's the weakest ...
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on ...
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Calculating Fourier Coefficients for this 50% bipolar square wave [closed]

I'm getting stuck trying to derive an equation for the Fourier coefficients of this waveform. Following along from this example: https://lpsa.swarthmore.edu/Fourier/Series/ExFS.html It is clear that ...
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Are there physics theorems that can prove maths theorems? Eg Pythagoras' Theorem

There's this recent post on maths overflow Which theorems have Pythagoras' Theorem as a special case? that has an answer by dxiv that appears to use a physics theorem to a prove a maths theorem, ...
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Potential of a Point charge in a spherical cavity

I'm solving some problems in Mathematical methods for physics. The section is about Legandre polynomials and I have problem because the problem is not mathematics. It's about electromagnetism and I ...
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Are there ways to find representations of matrices given an algebra?

Given an equation (or a set of equations) involving matrices, is there an algorithm to find possible representations of these matrices? For example, we can consider a matrix $A$ such that $A^2=\begin{...
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Radial position operator

While trying to find the expectation value of the radial distance $r$ of an electron in hydrogen atom in ground state the expression is: $$\begin{aligned}\langle r\rangle &=\langle n \ell m|r| n \...
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Spherical harmonics integral

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi $$ I've tried to use the definition of the ...
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What is the idea behind 2-spinor calculus?

In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
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Do operators always give a number after operating?

I am having some doubts regarding operators. In QM, when operators work on a wave function, will it always give a number times the wave function? Suppose I applied it on any normal function of x. Will ...
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7 votes
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Basis Vectors as Partial Derivatives Issues

I have been introduced a number of times to people defining vectors as derivatives of a curve, with basis vectors as partial derivatives, but I have several issues with this that make this formalism ...
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1 answer
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Finding eigenstates of $S_x$ using Stern Gerlach experiment [closed]

Quantum mechanics, McIntyre, pg 62 For above spin $1$ Stern Gerlach experiment a set of results is $$ \begin{array}{c} \mathcal{P}_{1 x}=\left.\left.\right|_{x}\langle 1 \mid 1\rangle\right|^{2}=\...
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Orthogonality relations for matrix elements of irreducible representations

I am reading Howard Georgi's "Lie Algebras in Particle Physics" and have a question concerning the presented orthogonality relations for matrix elements of irreducible representations. To ...
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Why can I write $\frac{d}{dt}=\frac{d}{dt'}\frac{dt'}{dt}+\frac{d}{dx'}\frac{dx'}{dt}$?

I’m dealing with a Lorentz invariance problem, and in one of the solutions I’ve seen to prove the wave equation the term above was used. However I don’t really understand why it can be written that ...
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Path integral - operator formalism and continuum limit

Correlator of the position and momentum operators in quantum mechanics $$\langle x_f, t_f|[\hat{x}(t), \hat{p}(t)]| x_i, t_i \rangle = i\hbar \langle x_f, t_f| x_i, t_i \rangle$$ since $[\hat{x}(t), \...
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Mathematics of the star-mesh transformation

I'm trying to understand the star-mesh transform from a mathematical perspective. This transformation removes a node and by definition, the topology of the network changes, but the resulting network ...
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How Do You Calculate Effort Force Required to Move a Load on a Gaffe Lever?

First, I'd like to apologize for the misuse of any terms and my general ignorance- this is not my normal field. The question revolves around mechanism of two levers with two pivots. There are two rods....
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Unique solutions to divergence equation?

A very common problem in physics is to search for a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $$ \nabla \cdot f = g $$ for some given source density $g: \mathbb{R}^n \rightarrow \...
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Solving Young Sheldon's 100th episode vanity card

Please define the terms. Below is Chuck Lorre's 700th vanity card which congratulates Young Sheldon on reaching 100 episodes. Part 1. Find $x$ in $J_0(x)=0$. --> I guess this refers to Bessel of ...
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Inner product evaluation in QM

On wikipedia on the page for inner product it states that for any two $x,y$ in a vector space $V$ the inner product $(\cdot , \cdot)$ satisfies $(ax, y) = a(x,y)$ where $a\in\mathbb{C}$. The inner ...
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How to find the Lie Algebra from a matrix

how do I find the Lie algebra of this matrix group using the definition of the matrix Lie algebra: $$\begin{equation} A= \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \...
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Was asymptotic expansion also a form of symmetry?

Consider the infinitesimal expansion, which was used to describe the behavior of of the expression when taking the parameter to be small. The infinitesimal expansion was usually used to describe the ...
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2 answers
107 views

Explain this equation mathematically

$$\Bigl( \frac{\partial S}{\partial T} \Bigr)_H = \Bigl( \frac{\partial S}{\partial T} \Bigr)_M + \Bigl( \frac{\partial S}{\partial M} \Bigr)_T \Bigl( \frac{\partial M}{\partial T} \Bigr)_H$$ How can ...
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Why $\det(\mathbf{F}J^{-1/3}) = \det(\mathbf{F})J^{-1}$?

$J$ is the jacobian, and $F$ is the deformation gradient. As illustrated in the picture above, I don't understand why is the $\det(J^{-1/3}) = \det(J^{-1})$ ?
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1 answer
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Is it possible for a point-like system to behave like $x(t) = \frac{t}{2}\log(t^2)$ near $t=0$? (infinite speed) [closed]

Is it possible for a point-like system to behave like $x(t) = \frac{t}{2}\log(t^2)$ near $t=0$? (infinite speed) I know beforehand that relativity theory forbids anything with mass from travel faster ...
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Function must be discontinuous over $r=0$ due to physical impossibility [closed]

In this question it is asked what the upper-bound of the ratio of a solid object's surface area can be visible through direct, unaided observation. The accepted answer says that there is no such upper-...
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Can't understand this equation

We are in phase space of $6N$ dimensions. Each point $ \mathbf r$ in this space has $6N$ coordinates. Pathria writes Consider an arbitrary "volume" $\omega$ in the relevant region of the ...
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Constructing irreducible Lie algebra representations

In physics, a (irreducible) representation $\rho:\mathfrak{g}\rightarrow\mathrm{End}(V)$ of a Lie algebra $\mathfrak{g}$ is often "constructed" by finding the weights of the representation, ...
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Will it be $\vec{AB}+\vec{AC}=\vec{AD}$ or $\vec{AB}-\vec{AC}=\vec{AD}$? [closed]

The resultant of $\vec{AB}$ and $\vec{AC}$ is $\vec{AD}$. Now, which of the following is correct? $$\vec{AB}+\vec{AC}=\vec{AD}\tag{1}$$ $$\vec{AB}-\vec{AC}=\vec{AD}\tag{2}$$ I think $(1)$ is correct.
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Proof of linear independence of coordinate basis vector [closed]

Given a chart $(O,\psi)$ on an $n$-dimensional differentiable manifold $M$, let $$X_\mu(f)=\frac{\partial f\cdot\psi^{-1}(x_1)}{\partial x_\mu}|_{\psi(p)},$$ where $p\in O$. How do you prove that $\...
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Numerical solution of the 2D linear partial differential equation of first order

It is an actual geophysical problem where we study the liquid flow. We measure at each 2D grid point and each time interval three components of the liquid velocity and we want to compute how the ...
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Deriving an expression for extrema separation $\Delta E$ as a function of extrema order $n$ in the Franck-Hertz curve

Background. In the Franck-Hertz experiment on a mercury tube, the $\mathrm{Hg}$ ${}^3P_1$ excitation energy may be estimated by performing a linear fit to the accelerating potential at minima/maxima ...
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Expressing radiation flux density through a surface with Dirac delta

What is the general way in which radiation density for a given surface is expressed using Dirac deltas? Consider this surface expressed in cylindrical coordinates (for any $\phi$ and $r_0$ an ...
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Relation between equations of the form "Derivative" $f=0$

I'm currently taking an introductory course in QFT, and I've noticed that lots of equations in physics take the form of "Derivative" of a funcition equal 0. Some examples being the wave ...
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2 votes
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Why do we describe gauge fields by connections?

Let $\pi:P\rightarrow M$ denote a principal $G$-bundle, where $M$ is thought of as some spacetime and $G$ is an appropriate group (such as $\mathrm{U}(1)$ or $\mathrm{SU}(2)$). I want to understand ...
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Convergence of an oscillatory integral to real number

I have a physical model, where the long-time behavior of the system can be described by $$C(t)=\frac{1}{2\pi}\int_{-\pi}^\pi\mathrm{d}k\,\mathrm{e}^{-t\omega(k)}$$ with $\omega(k)\geq0$ and $\omega(t)\...
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Probability in a small interval is $P. dx$

Reif says ... variable $u$ which can assume any value in the continuous range $a_{1}<u<a_{2}$. To give a probability description of such a situation, one can focus attention on any ...
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Approximation when $N$ is large in binomial distribution

Reif,pg 14. $n_1$ is the number of steps to the right in a 1D random walk. $N$ are the total number of steps When $N$ is large, the binomial probability distribution $W\left(n_{1}\right)$ , $W\left(...
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What is $\int_{x_0 - \frac{\Delta x}{2}}^{x_0 + \frac{\Delta x}{2}}\delta^2(x - x_0) dx $ [duplicate]

I'm going through basic quantum mechanics and I've got the below expression. $$\int_{x_0 - \frac{\Delta x}{2}}^{x_0 + \frac{\Delta x}{2}}\delta^2(x - x_0) dx$$ How can I evaluate it?
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Are one-dimensional tensors of arbitrary rank just scalars?

Consider a tensor of arbitrary rank (2 for this case) $A_{ij}$, and dimension one. Granted there are two indices to specify a component, but since each index can only take one value, there is only one ...
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3 votes
1 answer
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Confusion between implicit and explicit dependence, and its interpretation of the Lagrangian

I just want someone to confirm if whatever I'm writing down is correct or wrong. If the Lagrangian is given by $L(x,v,t)$, then we say that it explicitly depends on time. Here, $x$ and $v$ are ...
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1 vote
1 answer
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Plasma Dispersion relation simplification

I am trying to solve the following equation D(w,k) = $1-\frac{w_{p,e}^2}{2}[\frac{1}{(w-kv_0)^2}+\frac{1}{(w+kv_0)^2}]=0$ by rearranging it as $w^2 = D(k)$ (that is, write the it as $w^2$ = ... , ...
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Why are all elements of the Lie group generated by infinitesimal elements?

I am wondering why all elements could be generated by infinitesimal elements. --------------------------------------------------------------------- Here is the explanation I found from ...
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Meaning and differences between adding & multiplying two different functions in Physics

We all know that acc. to Newtonian mechanics , $F = ma$ and acc. to Lagrangian-Hamiltonian mechanics , $H = T + V$. I want to ask what makes the Hamiltonian, $H = T + V$ and not $H = T × V$? Similarly,...
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5 votes
1 answer
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Examples of quantum systems modelled with Type II von Neumann algebras

What are the examples of quantum systems that should be modelled with a Type $II_1$ or $II_\infty$ von Neumann algebra? I am pretty much a novice at von Neumann algebra, so I have hard time finding ...
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About derivation of Navier Stokes Equation

Following the steps of derivation, everything is clear just for one small argument which is: Why is the divergence of the transpose of gradient equal to gradient of the divergence, and why does it ...
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1 vote
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Is it possible to solve cross products using Einstein notation?

I'm considering a case where I have an equation of the form $\mathbf{x}\times\mathbf{b}+\mathbf{c}=0$; I wish to solve for $\mathbf{x}$ given that $\mathbf{b}\perp \mathbf{c}$. It was in the context ...
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