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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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Good math books for basic physics [closed]

I am good at math, but bad at applying it. Is there any book which goes over math being applied for basic physics primarily motion?
1 vote
1 answer
127 views

How do operators on kets and wavefunctions correspond?

Let $\hat{A}$ be an operator on Hilbert space vectors. How does one show that there always exists a corresponding operator $\hat{a}$ on wave functions? i.e. $\exists \hat{a}:L^2\rightarrow L^2$ s.t. $$...
Y G's user avatar
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0 answers
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Finding out the number of minima for a fourier expansion [migrated]

Suppose I have a Fourier series f(x) = $\sum_{n=1}^N t_n cos(nx)$ defined in the domain $(-\pi,\pi]$. we need to prove that mathematically we can ' at most' have N minima points excluding the boundary ...
ANIMESH GHOSH's user avatar
0 votes
1 answer
51 views

What is the mathematical support for the formula $f_n = n f_1$, used to calculate the frequency of a standing wave? [closed]

could someone explain to me the mathematical support for the formula $f_n = n f_1$. This formula refers to the fact that the frequency of a standing wave is equal to the number of antinodes times the ...
Santiago Celis's user avatar
7 votes
3 answers
400 views

Negative kinetic energy on a step potential

I'm doing an introductory course on quantum mechanics. I'm having trouble with the explanation of the kinetic energy on the classically forbbiden region on a step potential ($V=0$ for $x<0$, $V=V_0$...
Vito P.'s user avatar
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9 votes
4 answers
7k views

Imagining Graham's number in your head collapses your head to a black hole

My question is how does the above phenomenon, as mentioned in Numberphile here, occur in a semi-quantitative way through physical laws. (i.e saying statistical mechanics implies that is not ...
Mahammad Yusifov's user avatar
-1 votes
1 answer
85 views

Why is the tensor product not a multilinear application? [closed]

I was studying multilinear algebra and I wanted to have a good understanding for relativity. I started studying multilinear applications, which are kind of like a natural extension of linear ...
JL14's user avatar
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-3 votes
1 answer
56 views

Why aren't all objects and their images same in size?

Suppose there is an object in front of a convex lens and we know that the light rays from each point on the surface of object will converge at a different point and form an image. So that means that ...
Virender Bhardwaj's user avatar
5 votes
1 answer
317 views

Is physics limited to smooth sets? [duplicate]

I have come across lectures about Higher Topos Theory in mathematical physics and I am wondering about the explicit restriction to the category of smooth sets. Why should the potentially possible laws ...
Pan Mrož's user avatar
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-1 votes
0 answers
19 views

How to state that a function has a certain andament in a limit? [migrated]

Assuming we have a function $f(r)$ that has the following limit $$ \lim_{r\to0} f(r) = \frac{5}{3 r^2} \,.$$ What is the correct symbol to express that the denominator goes like $r^2$? Is the ...
Aleph12345's user avatar
-1 votes
1 answer
74 views

From any element of $\mathrm{SO}(8)$, can we always find one corresponding $\mathrm{SU}(3)$ element?

I first recap the relation between $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ and then raise my question concerning $\mathrm{SU}(3)$ and $\mathrm{SO}(8)$. Given any traceless hermitian matrix $H$, we can ...
narip's user avatar
  • 307
3 votes
1 answer
112 views

Confusions on The Gravitational Energy of a Point P in a Cube

I have been working, quite tirelessly, to try and find an answer to a question that has been bothering me for some time now. I have been working over some proofs, in the Newtonian Mechanics world, to ...
Statico's user avatar
  • 152
3 votes
5 answers
219 views

What does the $N$ in $SU(N)$ mean?

So I know this is a very basic question, but I can't really wrap my head around it. I was told $N$ is the number of dimensions in the rotations of the group theory that we are considering, so I ...
minime's user avatar
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0 votes
1 answer
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Polarization Derivation

Can how I can find the phi and theta unit vectors as described (Eq. 4a and 4b) in the following paper on antenna cross polarization? I thought the thetha and phi unit vectors were the same as those of ...
DTBolt's user avatar
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1 vote
1 answer
95 views

General Solution to Maxwell's Equations with Duhamel's Principle

In one dimension, it is easy to prove that if two solutions $\{u_1, u_2\}$ are known to $\mathcal{L}u(t) = 0$ where $\mathcal{L} \equiv{a(t)\partial_t^2+b(t)\partial_t+c(t)}$, the general solution to ...
Cody Payne's user avatar
0 votes
2 answers
87 views

Why do I get two different expression for $dV$ by different methods?

So, I was taught that if we have to find the component for a very small change in volume say $dV$ then it is equal to the product of total surface of the object say $s$ and the small thickness say $dr$...
Madly_Maths's user avatar
1 vote
0 answers
70 views

A preposterous abuse of notation involving Helmholtz decomposition theorem

Take what I am about to present with a light heart, since the mathmetically inclined may find it too out-of-the-world and devastating. The above diagram (this is drawn by me, but the original is very ...
Jonathan Huang's user avatar
0 votes
0 answers
31 views

Structure factor correct calculation

I have a set of 2D points and wish to test it for hyperuniformity. As I've learned from papers, the good idea is to calculate structure factor $S(\mathbf {q})$ and test it for $$\lim _{\mathbf {q} \to ...
lesobrod's user avatar
  • 163
1 vote
2 answers
128 views

Would it be valid to say $y \propto 2x$ instead of $y \propto x$?

Would it be acceptable to write $y \propto 2x$ or would it be wrong to add the $2$? I just want to describe a relationship in more detail.
Yifan YIN's user avatar
1 vote
2 answers
152 views

Does the exponential representation of Dirac delta function depend on choice of Fourier convention?

Is it always true that $$\delta(\omega) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i \omega t} dt , $$ regardless of your Fourier convention? For example, if I choose to use the Fourier convention ...
photonica's user avatar
  • 101
1 vote
0 answers
31 views

Wigner-Eckart for Finite groups

We know the Wigner-Eckart theorem generalizes to say $\mathrm{SU}(3)$ (see for example this answer). In a different direction, I am curious if/how it generalizes to finite groups of $\mathrm{SU}(2)$. ...
Eric Kubischta's user avatar
-1 votes
1 answer
77 views

What does the notation $d𝜏'$ mean?

$\text{I was studying helmotz theorem and saw this notation, what does it mean? How is d}\tau' \, \text{ different from d}\tau \text{?}$ From :- David J. Griffiths-Introduction to Electrodynamics-...
DocAi's user avatar
  • 33
1 vote
1 answer
86 views

Tensor identity in L&L book 2

How does the identity $$\epsilon^{prst}A_{ip}A_{kr}A_{ls}A_{mt}=-A\epsilon_{iklm}$$ with the Levi Civita symbol $\epsilon$ and the determinant A of the matrix $A_{ik}$ follow from the equation $$\...
Takitoli's user avatar
1 vote
2 answers
76 views

Static solution to an implicitly dynamic problem - heat equation

Heat equation This is the heat equation: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} $ ...
Megidd's user avatar
  • 123
-2 votes
1 answer
156 views

Exact definition of divergence. Is it really the dot product of nabla operator with a vector? [closed]

I was trying to understand the derivation for divergence in cylindrical and spherical coordinate system, and I am a bit confused here. https://www.gradplus.pro/deriving-divergence-in-cylindrical-and-...
DocAi's user avatar
  • 33
0 votes
0 answers
54 views

When can I commute the 4-gradient and the "space-time" integral?

Let's say I have the following situation $$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$ Would I be able to commute the integral and the partial derivative? If so, why is ...
clebbf's user avatar
  • 1
1 vote
1 answer
64 views

Material to Study the Definition, Algebra, and Use of Infinitesimals in Physics [closed]

This is going to be a rather general question about suggestions on best supplementary material to properly explain the use of infinitesimals (or differentials?) for the purposes of integration or ...
0 votes
1 answer
31 views

Differentiation of a product of functions

If I have three (vector)functions, all dependent on different (complex)variables: \begin{equation} a = X^{\mu_1}(z_1, \bar{z}_1), b = X^{\mu_2}(z_2, \bar{z}_2), c= X^{\mu_3}(z_3, \bar{z}_3) \end{...
j_stoney's user avatar
3 votes
0 answers
41 views

Proof Majorana spinors exists if maximal commutant of Clifford algebra is $\mathbb{R}$

I am searching for a proof of the claim made in this post. It states that Majorana spinors (I refer to both complex pinor and spinor representations which are restricted to the real Spin group and ...
anonymous250's user avatar
1 vote
0 answers
65 views

What is the Taylor series of the expectation value of an observable in quantum mechanics? [closed]

I recently came across a form of the expectation value of an observable, but a Taylor series (I think?) was taken up to second-order: \begin{equation} \langle O\rangle = \langle O\rangle_{C=0} + \...
NikNack's user avatar
  • 19
1 vote
0 answers
76 views

Motivation behind reflection positivity

I have taken a look at this physicsSE question, Wikipedia, and this paper by Jaffe which go over reflection positivity. While they all nicely explain the definition behind reflection positivity and ...
CBBAM's user avatar
  • 3,370
2 votes
2 answers
201 views

Can the composition law of a group be defined only when considering a representation or realisation of the Group?

When we talk about, lets say, the Lorentz group, we define the action of the Lorentz transformation $\varLambda$ on \begin{alignat}{1} x^{\mu} & \in\mathbb{R}^{1,3},\\ x^{\mu} & \rightarrow x'^...
HypnoticZebra's user avatar
3 votes
2 answers
83 views

In what sense is $\int (u \cdot \nabla) u \cdot u dx$ an energy flux?

Due to the nature of this question I have have cross-listed it on mathSE. Let $u$ be either a solution to either the Euler equations or Navier-Stokes equations over a domain $\Omega$. In fluid ...
CBBAM's user avatar
  • 3,370
0 votes
1 answer
107 views

Mathematical meaning of a position eigenbra $\langle x_0 |$

Let $|x_0\rangle$ be an position eigenket. The physical picture I have for $|x_0\rangle$ is a particle located at $x_0$. Thus it should be represented by a delta function $\delta(x-x_0)$. For $f\in L^...
CBBAM's user avatar
  • 3,370
1 vote
1 answer
69 views

Is there an error bound for perturbation theory?

It is usually said that perturbation theory is valid when the perturbation is much smaller than the spacing between energy levels. However, I was thinking whether there exists an rigorous error bound. ...
FusRoDah's user avatar
  • 689
2 votes
0 answers
28 views

Implications of quantized space (a la LQG) on defining "realistic" number systems [duplicate]

Disclaimer: not a professional physicist or mathematician, so (deserved) tomato-throwing is welcome. I've been pondering the "naturalness" of real numbers for some time now, in the sense of ...
RuslanD's user avatar
  • 121
0 votes
1 answer
58 views

Stokes' theorem and vector continuity equations

I have been working with homogeneous continuity equations of the general form: $$\frac{\partial \rho}{\partial t}+\vec{\nabla}\cdot \vec{J}=0$$ This has me wondering whether we can formulate other ...
Lagrangiano's user avatar
  • 1,616
1 vote
2 answers
126 views

Why does a singularity imply the need for a distribution?

I am following Section 11 of Prof. Etingof's MIT OpenCourseWare notes on "Geometry And Quantum Field Theory" in which he says: ...for $d = 1$, the Green's function $G(x)$ is continuous at $...
CBBAM's user avatar
  • 3,370
1 vote
1 answer
51 views

On complex impedance representation and Riemann surfaces

We know that a complex number, $z=re^{i\phi}$, can be represented with infinitely many phases, $\phi+2\pi n$, for integer $n$, as can be easily seen from the equivalent picture of a vector on the ...
user135626's user avatar
4 votes
2 answers
714 views

Wrong solution for Green function of one-dimensional Poisson equation

An old electrodynamics exam question asks: "Find the Green function (for the one-dimensional Poisson equation) that solves the equation $$ \frac{d^2}{dx^2}G(x,x') = -\delta(x-x'). $$ Choose the ...
F L's user avatar
  • 151
-5 votes
1 answer
137 views

Manipulation of functions inside a Dirac Delta function [closed]

It is not clear to me how this derivation proceeds through the steps. Could someone help me understand how to arrive at this result or point me towards a resource that explains these algebraic ...
Jasper amirante's user avatar
0 votes
2 answers
81 views

Solving a PDE using $x-vt$ as a variable

So I was reading this Landau and Lifshitz paper: https://doi.org/10.1016/B978-0-08-036364-6.50008-9 The article can also be found without a paywall by just searching its title, "On the Theory of ...
Andreas Christophilopoulos's user avatar
1 vote
1 answer
40 views

Electric field at a point created by a charged object (derivation/integration process)

I was hoping someone can help me understand the math behind the electric field (electrostatics). I have gaps in my knowledge about integrals and derivatives (university moves very quickly and it has ...
1899DVX's user avatar
  • 19
-8 votes
3 answers
217 views

How Taylor series is compatible with special relativity?

In mathematics, an analytic function is defined by its possession of a Taylor series with a positive radius of convergence ($R_c​>0$). Notably, certain analytic functions—such as holomorphic ...
Omid's user avatar
  • 342
0 votes
0 answers
46 views

Arc length between configurations in the "mass distance"

In classical Lagrangian mechanics, the mass $M$ is a Riemannian metric on the configuration space $Q$. Does the "arc length" of a path $\gamma : [0, 1] \to Q$, $$ \int_0^1 {\lVert{\gamma'(t)}...
Ram's user avatar
  • 1
0 votes
1 answer
89 views

Dirichlet’s Theorem and Solutions to Laplace Equation in Cartesian Coordinates

I have been reading Introduction to Electrodynamics - Griffiths about solving Laplace equation in cartesian coordinates, and in that book, I saw this statement: The functions $\sin(n\pi y/a)$ are ...
Sanjay's user avatar
  • 97
-1 votes
1 answer
51 views

How to make a parametric that matches a vector field?

So I have a vector field defined as $(X(x,y),Y(x,y))$ and I’m trying to make a parametric $(t,t)$ who’s derivative at a point is equal to the vector field at that point. for example the vector field $(...
GIORGI GOGIBERIDZE's user avatar
8 votes
3 answers
921 views

Property of the Hamiltonian's discrete spectrum

I have found a statement online saying that there must be an eigenvalue of the Hamiltonian inside the range $(E-\Delta H,E+\Delta H)$. Where the mean value and variance are defined for a random (...
user20046481's user avatar
0 votes
2 answers
54 views

Is $dJ(V,V)=0$? where $J$ is a 1-form?

So is this always 0?( Where $dJ$ is the exterior derivative and $V$ a vectorial field) \begin{align} dJ(V,V)=\partial_jJ_i(dx^j\wedge dx^i)(V,V)=\\ \partial_j J_i (v^kdx^j(\partial_k)v^ldx^i(\...
Guillermo Fuentes Morales's user avatar
1 vote
1 answer
95 views

Is wedge product a tensor or a pseudo tensor?

I'm doing an exercise where $J$ is a 1-form on a manifold $M$ of dimension $N$. The exercise ask me to calculate $J∧(*J)$ with $J=dx^0+2dx^1$ in a minkowski space with metric =(-1,1,1,1) where $*J$ is ...
Guillermo Fuentes Morales's user avatar

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