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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Vector Calculus [closed]

The vector field: $$A = \boldsymbol{\mathrm{Y}}ax−\boldsymbol{\mathrm{X}}ay+\boldsymbol{\mathrm{Z}}az$$ is given in rectangular coordinates. Find the surface where $|A|$ is a constant.
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What is a Calabi-Yau space?

I had trouble understanding Calabi-Yau space while reading the theory of strings and how it works in this theory. Can someone please explain it to me?
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What is the 'free' form or 'free' version of an equation in mathematical physics? Like the Dirac equation, (among others)

From Wikipedia: In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it ...
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What is the equation for calculating magnetic field for two coils when the currents in the coils have opposite orientation?

So I was wondering what the equation is to calculate the magnetic field when the two coils are separated by a distance d, with the currents in the coils in opposite orientation? Any help would be ...
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1answer
14 views

Determining charge magnitude through comparison with force of tension

Three charges of mass $m$ and charge $q$ are tied to massless ropes of length $l_!$, and hung from a single, fixed point. At equilibrium, the charges form an equilateral triangle in a horizontal plane ...
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1answer
27 views

Properties of physical vector field vector field whose components depend only on the distance from origin

In introductory physics, we often have fields of form $ \vec{F} = \frac{k}{r^n} \hat{r}$. This leads me into the habit that all physical vector fields of radius dependent on inverse power of $r$ must ...
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2answers
87 views

Does the physical unit determine the length of the unit vector?

A vector quantity, like position, can be expressed as (using one dimension for simplicity): $$\textbf{r}=(1\ \text{m})\hat{\textbf{i}}$$ What determines how "long" the unit vector $\hat{\...
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Decoupling a system of second-order differential equations [closed]

I have a coupled system of 5, second order, differential equations. I am having difficulty making substitutions of the equations to get an equation in just one of the variables. The system may or may ...
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Help with the Hermite differential equation [migrated]

I am studying the differential hermite equation for two different books. And I noticed that they have a little difference in the expression for Hermite's EDO. I'll show you: The first author I am ...
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2answers
28 views

Computing error margin after unit conversions

Suppose I've experimentally measured some length $L$ to be $10.50 \pm 0.05 cm$. Using $2.54\;cm = 1\; in$, plugging this into a calculator, I get $L = 4.13385826772 \; in$. But $0.00000000002\;in$ is ...
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1answer
106 views

Why is the first derivative of the time-dependent Schrödinger equation continuous? Where does it come from?

I was taught in first year physics that the first derivative of the time-dependent Schrödinger equation had to be continuous. However I was never taught (or at least I don't remember) the reason why. \...
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What's the difference between differentiation and derivation? [migrated]

The question is pretty much straightforward... I just don't get the difference between those two. Is there an easy way of understanding it?
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122 views

Einstein says that this equation is true, but WolframAlpha says it's not always true. Who is right? [migrated]

Einstein says $\cos (ix) = \frac{1}{\sqrt(1-(i\tan(ix))^2)}$ but WolframAlpha says that isn't true for $x= \pm 2$ and $x= \pm 9/5$. What's happening? From page 34 of The Meaning of Relativity. $v=i\...
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56 views

Best Calculus one book [duplicate]

I’m currently in my senior year of high-school. I’m planning to major in physics. I really enjoyed basic calculus but I really want to start studying it for real. I know university courses include ...
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1answer
52 views

Function of an operator on a Hilbert space

Every linear bounded self-adjoint operator $T : {\scr H} \to {\scr H} $can be written in terms of its eigenvalues and their associated projectors (see spectral theorem): $$ T = \sum_{{\frak spec}(T) }...
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Relation between the boundedness and discreteness of conjugate operators

I have two very general questions about operators in quantum mechanics. Suppose $A$ and $B$ are self-adjoint operators associated to conjugate physical quantities (e.g. position/momentum), meaning ...
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2answers
69 views

How do I know which equations can be treated as differential equations and which can't?

I'm sometimes mystified by the use of differentials in physics. I don't understand which formulas—on which occasions—can be thought of as differential equations and which cannot. While discussing work ...
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25 views

How “smooth” is the evolution of the universe? [duplicate]

Mathematicians have developed different definitions for how "smooth" a function might be. A function (e.g. from or to $\mathbb{R}^n$ or $\mathbb{C}^n$) might be continuous, or once-...
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32 views

Alternative definition of Legendre polynomials [migrated]

I'm studying Panofsky and Philips' Classical Electricity and Magnetism. In writing the potential of a linear $2^n$-pole lying along the $x$-axis, they make use of the following definition for the ...
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25 views

Check whether the function is analytic or not [migrated]

How to check whether the functions $(3+x-iy)^7$ and $(1+x+iy)^4(7-x-iy)^3$ are analytic in the domain $|z|<2$? Is there any shortcut?
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1answer
29 views

Rotation matrices and reference frames

Lets say I have 2 3x3 rotation matrices, we will call them A and B. I am told to find frame B relative to A. how would I go about doing this basic operation? would I do it similar to position (with ...
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60 views

Will Physics become Math? [closed]

There are conflicting theories in Physics. For example, GR is only good in astronomical scale but not in quantum scale, and vice versa. Unlike in Mathematics where formulas are universal, a theory or ...
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80 views

Why does an integral turn negative when its limits are switched? [migrated]

I’m trying to find the tension at the middle of a rotating rod in a gravity-free space. To do that, I had done some math and then I integrated with the limits $L/2$ to $L$. This sounds like a math ...
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Why is 0 factorial equal to 1? Is there any pure basic mathematical proof? [migrated]

I just got a question while reading permutation. Why 0 is factorial equal to 1?
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35 views

Decoupling noises

This is an experimental physics problem: Say I have 3 random variables $P_1, P_2, P_3 $ such as : $\Delta P_i=\Delta_Q P_i +\Delta_0P_i \ \ \forall i$, where $\Delta$ is the variance, this equations ...
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1answer
54 views

Derivative of basis vector in terms of Christoffel symbols

I would like to derive the formula $$\partial_{c}\vec{e}^{\,a}=-\Gamma_{bc}^{a}\vec{e}^{\,b}$$ where $\vec{e}_{a}$ are the basis vectors on a manifold. In the lecture, we did it in the following way: $...
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On the speed of light being “algebraically infinite”

Browsing the math stackexchange recently, I found this exercise which caught my interest. https://math.stackexchange.com/q/1793358/ The exercise takes the group of real numbers $\mathbb{R} = (-\infty, ...
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Does the space of Slater determinants or bosonic permanents have any nice mathematical structure?

(This is a soft question.) In the Hartree-Fock approximation, you approximate the ground state of a many-body fermionic system by performing a variational minimization of the expected energy of a ...
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1answer
30 views

Calculus and trigonometry course

Can anyone please tell me any book or refer any kind of short term course on calculus and trigonometry required for physics.
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26 views

Finite potential whose (normalisable) wavefunction doesn't vanish at infinity

I can come up with normalisable wavefunctions which don't vanish at infinity. However, I cannot come up with a potential so that it satisfies TDSE (the examples I think of are not differentiable at ...
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32 views

What is a good book for learning quantum mechanics with mathematical derivations? [duplicate]

I know calculus (ODEs, PDEs, integration over 3 dimensions, limits, W, Zeta functions, series etc) and am working on linear algebra. What it the best book to learn quantum mechanics from with ...
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2answers
63 views

Does the SUVAT equations of motion (Kinematics) come from some differential equation?

Wikipedia says about the equations of motion that; "If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.&...
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2answers
66 views

Why quantities in physics are always talking about rates? [closed]

I get the idea that physics wishes to study changes to discover new rules. But why is everything related to rates? Acceleration,Velocity? Could we use something else apart from these? What can you ...
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1answer
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Basic Slope-Intercept Form y=mx+b [closed]

The equation for a line forms a part of my physics work. The equation is: Y = mx+b To refresh myself, I watched the following video on the formula. However, at 3:15 I went off in the wrong direction ...
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2answers
89 views

Do closed line integrals need to be evaluated in “the line's” rest frame?

I've seen it said that the definition of emf requires that the integral be carried out in the circuit's rest frame. \begin{equation*} \mathcal{E} =\oint \mathbf{f} \cdotp d\mathbf{l} \end{equation*} ...
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2answers
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An expression for inverse of inner product of two quantum states [closed]

We know: (⟨x|y⟩)†=⟨y|x⟩ and ⟨x|y⟩)*=⟨y|x⟩ Is there a similar formula for inverse of inner product of two quantum states? for example can we say??: (⟨x|y⟩)^-1=⟨y^-1|x^-1⟩ or (⟨x|y⟩)^-1=⟨y|x⟩^-1
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Like Martin Gardner's “Mathematical Games” articles in Scientific American, but for physics?

Is there good source for puzzles in the style of Martin Gardner's books (Entertaining Mathematical Puzzles, aha! Insight, etc), except the domain is physics rather than math?
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1answer
60 views

How to transform this function using Legendre transformation?

$f(x,y)=x^3y^2$ the goal is the Legendre-transformed function: $g(x,u)=uy-f(x,y)$ where $u=\frac{∂ f}{∂ x}$ and $v=u=\frac{∂ f}{∂ y}$ where g(x,u) isn't explicitly dependent on y. I derived $u=x^3y^2$....
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75 views

Analytic solution to Kepler's Problem, exegesis

From 'Solving Kepler's Problem' by Colwell, the first analytic solution to Kepler's Problem used a theorem of Lagrange, and later Burmann, to invert Kepler's equation. When you look on the internet ...
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31 views

Book recommendations to understand General Relativity [duplicate]

I just finished Multivariate calculus and I am wondering what sequence of books would allow me to work up to the mathematical knowledge needed to understand General Relativity and the corresponding ...
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1answer
18 views

Computation - can you compute the gradient, Laplacian, divergence and curl of any function?

In my physics class, we are currently studying gradient, Laplacian, divergence, and curl, and we have a problem that states to compute all four of these (I.e., (1) gradient, (2) Laplacian, (3) ...
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1answer
25 views

Expression in John R. Taylor Scattering theory book

In the book Scattering theory by John R. Taylor He has the following expression at page 138 $$ i \lim _{\epsilon \downarrow 0} \int_{0}^{\infty} d t\left\langle\mathbf{p}^{\prime}\right|V e^{i\left(E_{...
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5answers
80 views

Using Schwarz's Inequality to show an expectation value relationship of a particle

This is a question from Cambridge Tripos for 1st years Natural Science students which I just can not solve. I have spent hours on it, and I am going around in circles. A particle is located between $x$...
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59 views

Gaussian Integral with Complex Parameters — Divergence and Convergence

Although this is more of a mathematical question, I will in what follows refer to an answer of @Qmechanic that has been posted in this forum (I am sorry for creating a new post for this, but I ...
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1answer
114 views

Finding the metric tensor from a 2D line element

A 2D space has coordinates $x^1$ and $x^2$ with line element $$\mathrm{d} l^{2}=\left(\mathrm{d} x^{1}\right)^{2}+\left(x^{1}\right)^{2}\left(\mathrm{~d} x^{2}\right)^{2}.$$ I'm looking to find the ...
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1answer
63 views

Shadow method of solving differential equations

While reading this answer by Rishab Navneet here, it is shown how we can visualize the harmonic oscillator as the shadow of a body moving in a circle onto a line. How was it found that the plane curve ...
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0answers
56 views

Do We know if our mathematics enough to describe nature? [closed]

To the extent, if you make a postulate based on the observation that can not be described with classical mechanics, this is how quantum mechanics was initially put in and then we ask what's its ...
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0answers
85 views

Equipotential surfaces for a knotted charge distribution

Suppose we have a compact submanifold $K$ of $\mathbb{R}^3$ with uniformly distributed charge. Neglecting multiplicative constants, the electric potential $\Phi(\vec{x}) = \int_{K} \frac{dK}{|x - k|}$...
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1answer
58 views

Calculating time of flight given index of refraction as a function of depth

Calculating light speed given a simple index of refraction is trivial, of course ($n = \frac{c}{v}$). I'm not sure, however, how one would calculate the 'time of flight' for a light signal in a medium ...
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1answer
40 views

In what dimensions to the Navier-Stokes equations work?

I have only seen examples and references to the Navier-Stokes equations in two and three dimensions. Do we know if these equations work in one dimension or greater than three dimensions?

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