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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1answer
26 views

Correlation among the terms “convergence”, “accumulation” and “explosion”

I am developing an essay on music perception by approaching some mathematical and physical notions. As such, the term I am addressing and interested in is the "convergence", which pairs in mathematics ...
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0answers
43 views

What is the most important thing to master in string theory? [closed]

What is the most important material to MASTER in string theory? I am asking this question because there are thousands of people working on string theory and it is really hard to get a job in this ...
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0answers
25 views

How to calculate the total gravitational potential energy of a vertical object (do we use integration?)

Hello I was reading another question asked by zach466920, and when he was trying to calculate the total GPE of a water 'tower', he used this explanation: He basically used integration to calculate ...
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15 views

What is the syntax in scilab to derivative a function like $x/log(x)$, not a polynomial? [closed]

In scilab to do a derivative I used the syntax derivative as: Function y=f(x) y=x/log(x) endfunction disp(derivative ("f","x",4)) And my output was: Undefined variable: derivative Where was I ...
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1answer
29 views

Does the concept of infinity have any relevancy or application in Physics and applied Physics? [duplicate]

Does the concept of infinity have any relevancy or application in Physics and applied Physics? I must admit that I am not particularly knowledgeable in the area of Physics, but I have never seen the ...
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1answer
48 views

What paths are allowed in the Fourier form of the Dirac Delta distribution?

In this form of the Dirac Delta distribution $$\delta(x) = \frac{1}{2 \pi i}\int_{- i \infty}^{i \infty}e^{-\omega x} d\omega$$ can $\omega(t)$ be evaluated over any path (that starts at $\omega(-\...
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40 views

Does higher-than-infinite speed make sense? What about other higher-than-infinite quantities in physics? [closed]

This is somehow a survey question aimed to find applications where quantities higher than infinite would make sense. As an example, I would give speed. Logically, an object has infinite speed if it ...
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0answers
5 views

Equilibrium density matrix for a homonuclear spin system in NMR

I'm currently reading this paper. In the paragraph under fig. 3 on page 2, they write: The equilibrium density matrix for a homonuclear spin system is a sum of $n=5$ terms: $IIIIZ+IIIZI+IIZII+IZIII+...
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1answer
29 views

Is magnitude of cross product commutative?

By definition of magnitude of cross product: $\| \mathbf{F} \times \mathbf{r} \|= \| \mathbf{F} \|\ \| \mathbf{r} \| \sin (\mathbf{F},\mathbf{r}) \tag1$ $\| \mathbf{r} \times \mathbf{F} \|= \| \...
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1answer
28 views

How is the root-mean-square error related to the confidence interval?

If the car's speed is calculated as V = 100 km/s, and the RMSE of V is 20 km/s. Does that mean V = 100 +/- 10 km/s, or V = 100 +/- 20 km/s?
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1answer
99 views

How does nature calculate instantaneously? [closed]

Physicists can make calculations about the world around us. However, it takes time for humans or even computers to perform the computations. How does nature do these calculations instantly?
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2answers
199 views

Trouble Solving Partial Differential Equation [closed]

I'm solving the velocity profile of a fluid flow for a circular channel with an oscillating pressure gradient $\frac{dp}{dx}=\frac{\Delta p}{\rho L}e^{-i\omega t}$. I plugged in to the Navier Stokes ...
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0answers
57 views

Understanding Multivariable Calculus without Hand-Waving [duplicate]

I am trying to properly understand multivariable calculus as a physicist and would like to know of any recommendations for textbooks to work through. Ultimately I want to be able to properly ...
2
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2answers
100 views

When can a function $f(x_0 - x)$ be approximated as $f(x_0 - x) = f(x_0) - f'(x_0) x$?

When can a function $f(x_0 - x)$ be approximated as $f(x_0 - x) = f(x_0) - f'(x_0) x$? In Reif's statistical mechanics it is said that when $x$ is much smaller than $x_0$ then the approximation can be ...
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1answer
34 views

Cross product of partial derivative of vector field

In reading Altland and Simons book (Condensed matter field theory p. 508). I came across the following problem. The authors claim that the term $$ \partial_1 n \ \times \ \partial_2 n$$ (Where $n(x)$...
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2answers
76 views

Completeness of Norm in Hilbert Space

I am not sure what it really means for the norm to be complete in a Hilbert Space. Can you provide me a proper definition? I am aware of the formula $||\Psi|| = <\Psi|\Psi>^{1/2}$. What are ...
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1answer
27 views

How to get the structure constants from a Dynkin diagram?

I'm currently learning how to work out Lie algebras. I've learnt how to read the basics of a Dynkin diagram. So I worked out some simple examples, but I'm stuck at the $[E_\alpha, E_\beta] =N_{a,b}E_{\...
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1answer
26 views

Solution of DE regarding SHM, kinda

Firstly, let me explain the situation So this year in my core module, modern physics, all of our practicals are simulations of general stuff programmed in SageMath/Jupyter Notebook. We are still in ...
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4answers
1k views

Are there undecidable statements in classical mechanics?

Do Gödel's incompleteness theorems have any significance or application to axiomatic theories of classical mechanics like Newton's for example?
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3answers
2k views

Which mathematical property allows us to combine proportional relationships?

Coulumb's law states that $$F \propto q_1 \cdot q_2 \tag{1} $$ and $$ F \propto \frac{1}{r^2} \tag{2} $$ Why can we combine these two proportions into $$ F \propto \frac{q_1.q_2}{r^2}?$$ What ...
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0answers
26 views

Does Weyl's tile argument defeat the discrete spacetime?

Weyl shows that in a discrete spacetime Pythagoras's theorem fails to arise. Of course it may be that although Pythagoras's theorem arises naturally but actually does not model the real world. So Does ...
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0answers
125 views

Classification of higher Symmetry Protected Topological (SPT) phases

Suppose that we have a $d$ dimensional bosonic SPT phase, protected by some $p$-form symmetry, $G^{[p]}$. Suppose also that it is classified within group cohomology, so that we don't have to run into ...
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1answer
34 views

Are there cases where the use of the Grassmann variables simplifies computations in the usual bosonic analysis?

When one introduces complex numbers and complex analysis one can then use the new machinery to solve some real-analysis problems. A lamppost example is computing integrals via residues. I think I've ...
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17answers
8k views

Can a mathematical proof replace experimentation?

I know that this is very similar to How important is mathematical proof in physics? as well as Is physics rigorous in the mathematical sense? and The Role of Rigor. However, none of the answers to ...
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2answers
294 views

Are all representations of a finite group unitary?

I am reading Zee's Group theory in a nutshell for physicists and came across the following theorem (Page 96): Unitary representations The all-important unitarity theorem states that finite ...
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2answers
99 views

Is Snell’s Law valid in this case?

When light travels in a perpendicular path from one medium to another medium of different optical density, is Snell’s law valid? $\sin i$ and $\sin r$ are both 0, right? So it isn’t valid. Is this ...
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1answer
96 views

Integration and average in physics? [closed]

Many applications of physics theory involve computations of integrals. Examples are voltage, force due to liquid pressure, surfaces... In some cases, when there is linear dependence between two ...
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0answers
23 views

Why are galilean transforms affine? [duplicate]

Here is a decomposition of galilean transforms of the form $x\mapsto Ax+y.$ Why are they all of this form? $T$ galilean is distance preserving so it is also injective. Take $B_r(a)$ the closed $r-$...
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1answer
99 views

Higher dimensional version of Stoke's Theorem / Divergence theorem

I've learnt about Stokes' Theorem and the divergence theorem that relate integrals of functions over manifolds to integrals of related functions around the boundary of the manifolds but all in 3-...
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1answer
73 views

Bessel function of first kind [closed]

Can someone tell me how $$\frac{1}{T}\int_0^T e^{i(m-n)\omega t} e^{-ix\sin(\omega t)} e^{iy\sin(\omega t +\phi)}\, dt = J_{m-n}\left(\sqrt{x^2 +y^2 -2xy\cos(\phi)}\right)?$$
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4answers
190 views

Problem in $\sin\theta \approx \theta$ approximation [closed]

This approximation is used oftenly in physics: $$\sin\theta \approx \theta$$ This approximation is valid for small value of $\theta \leq10°$): But $\sin 1°=0.0174524064$ $\sin 2°=0.0348994967$ $\...
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2answers
56 views

A doubt in trigonometric approximation used in the derivation of mirror formula

The following text is from Concepts of Physics by Dr. H.C.Verma, from chapter "Geometrical Optics", page 387, topic "Relation between $u$,$v$ and $R$ for Spherical Mirrors": If the point $A$ ...
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1answer
124 views

How is a function approximated by Fourier analysis?

Quantum mechanics and QFT use extensively Fourier analysis. When trying to approximate a periodic function by Fourier series (say a rectangular wave), it is possible to increase the number of terms ...
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0answers
59 views

Fourier transform and commutation of integral and laplacian

While determining the Breit-Fermi potential, I carry out a Fourier transform by using the following identity: $$\int\frac{d^3q}{(2\pi)^3}e^{-i\vec{q}\cdot\vec{r}}\frac{1}{|\vec{q}|^2}=\frac{1}{4\pi r}$...
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1answer
48 views

Is the output of a line integral over a scalar field a vector?

In my physics book of "mathematical methods for physics", the author writes that line integral of a scalar function $\phi$ over a curve $C$ can be written as the following: $$\int_C\phi\,\text d{\...
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0answers
28 views

Advice needed on learning maths oneself [duplicate]

I am a master's student in physics trying to learn maths on my own. My classes workload is heavy and the schedule is very hectic due to which I don't get time to do mathematics. I have done Linear ...
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0answers
55 views

General relativity's prerequisites' prerequisites [closed]

I know there looks to be a duplicate: What are the prerequisites to studying general relativity? From what I read, the prerequisites are Calculus, linear algebra, differential and partial ...
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1answer
83 views

Generalizing the Stone-von-Neumann theorem

The Stone-von-Neumann theorem states (in rough terms) that a pair of operators $\left(\hat{Q}, \hat{P}\right)$ satisfying the exponentiated canonical commutation relation $e^{is\hat{Q}}e^{it\hat{P}} = ...
3
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1answer
103 views

What are real world applications of the Duistermaat–Heckman formula?

In the famous 1984 paper "The Moment Map and Equivariant Cohomology" by Atiyah and Bott, an equivariant de Rham theory was presented in relation to the Duistermaat–Heckman formula $$ \int_M e^{-itf} \...
3
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0answers
63 views

Witten's description of WZW conformal blocks

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a ...
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2answers
61 views

Application of Componendo and Dividendo Rule and Dimensional Analysis

Let us consider the following ratio: $$\frac A B=\frac C D$$ where $A$,$B$,$C$, and $D$ are of different dimensions. Can we apply the Componendo and Dividendo from Algebra as given below?: $$\frac{...
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1answer
78 views

Quantum Mechanics Spectral theorem proof [closed]

Has anyone an idea how to prove the spectral theorem $A = \sum_{i} \lambda_i P_{\lambda_{i}}$. Starting from $A|\Psi_{i}\rangle=\lambda_{i}|\Psi_{i}\rangle$ or what ever?
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0answers
52 views

Groups with cardinality larger than the Reals in physics

In what physical theories are sets with cardinality larger than $\aleph_1$ used? There are plenty of examples of finite, countable, and uncountable vector spaces in physics, but do physicists ever ...
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2answers
86 views

Does incomplete differentials $\delta Q$ or $\delta W$ have potentials? [closed]

I am very confused because my text book have following formula. $$dU = \delta Q \tag {1-1}$$ $$dU = \delta W \tag {1-1'}$$ Because these might mathematically mean "incomplete derivative = ...
3
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1answer
103 views

Which integration is right? (Integration of operators and delta functions)

Let us consider following integral $$\int \int dx dy f(x)g(y)\delta(x-y).\tag{1}$$ We can bring integral with respect to $x$ to the front or $y$ to the front and integrate out to get the same answer $$...
26
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7answers
6k views

Why use Fourier series instead of Taylor?

In dynamical systems with linear differential equations, we almost always break up the function of independent variable in sines and cosines. But suppose that my function is smooth and periodic. ...
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0answers
43 views

Why is the modified spherical Bessel function an asymptotic solution of this ODE?

I am trying to solve the radial equation with $R = u/r$ $$ \frac{d^2}{dr^2}u - \frac{l(l+1)}{r^2}u + (E-V)u = 0; \qquad V(r) = -\frac{2Z}{\alpha}\frac{e^{-r}}{1 - e^{-r}}, $$ using the shooting method....
1
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1answer
65 views

Is a curve being continuous a necessary or sufficient condition for that curve to be a physical trajectory?

In calculus classes, continuous functions are often intuitively described as functions whose graph can be drawn on a piece of paper without picking up your pencil. I'm wondering how much that ...
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3answers
95 views

Hilbert space and group theory: relationship between these two approaches to quantum mechanics, and references for a beginner?

I have read basic books on Quantum Mechanics like R. Shankar's "Introduction to Quantum Mechanics, Griffiths "Quantum Mechanics" and partly I followed Bransden "Atoms and Molecules". But none of the ...

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