Hyperbolic Definition and 336 Threads

Alright, I've been wondering this for a while now. Say you have an infinite grid of squares in hyperbolic geometry, such that the curvature makes it so each angle of each square is 72° (5 squares at each corner). At the very 'center' of the grid, or the origin, there would be 5 straight rays...

I have, Using ##\ cosh 2x = 2 \cosh^2 x - 1## ##\cosh x = 2 \cosh^2\dfrac{x}{2} -1## Therefore, ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -1 - 1## ##\cosh x -1 = 2 \cosh^2\dfrac{x}{2} -2## ##=2\left[ \cosh^2 \dfrac{x}{2}...

With the algebra so(3) are associated the spherical harmonics. I would assume that comparably with the algebra so(2,1) are associated functions that can be addressed as hyperbolic harmonics. But I nowhere found any reference to them. Do they exist and if so, where can they be found? Thank you...

I'm using a ``downwind'' approximation for the spatial derivative: \frac{\partial v}{\partial x}\approx -\frac{3}{2h}v_{j}+\frac{2}{h}v_{j-1}-\frac{1}{2h}v_{j-2} I'm using the usual approximation for the time derivative, I get the following for a stencil...

My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...

##8 \sin^4u = 3-4\cos 2u+\cos 4u## ##8 \sinh^4u = 3-4(1+2\sinh^2 u)+ \cosh ( 2u+2u)## ##8 \sin^4u = 3-4-8\sinh^2 u+ \cosh 2u \cosh 2u + \sinh 2u \sinh 2u## ##8 \sinh^4u = 3-4+1-8\sinh^2 u+ 4\sinh^2u +4\sinh^4 u + 4\sinh^2 u + 4\sinh^4 u## ##8 \sinh^4u = -8\sinh^2 u+ 8\sinh^2u +8\sinh^4 u##...

This is a textbook question and i have no solution. My attempt: We know that ##\cosh x = \dfrac{e^x + e^{-x}}{2}## and ##\cosh u = \dfrac{{x^2 + 1}}{2x}## it therefore follows that; ##e^{2u} = x^2## ##⇒u = \dfrac {2\ln...

I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties... My approach on this; ##\tanh x = \sinh y## ...just follows from ##y=\sin^{-1}(\tan x)## ##\tan x = \sin y## Therefore...

Hello, I would like to prove that the following partial differential equation is hyperbolic. u_{tt} (x,t)+ u_{xxxx} (x,t)= 0 with x \in \left[0 , 1\right] and x \in \left[0 , T \right ] . Can anyone help me? Thank you.

Epimetheus and Janus switch places periodically, because they follow a horseshoe orbit around Saturn, which is considered a "pseudo-orbit" around each other. I'm thinking that if you look at the conic sections - taking an elliptical orbit of two moons to greater and greater extremes until they...

While deriving Lorentz transformation equations, my professor assumes the following: As ##\beta \rightarrow 1,## $$-c^2t^2 + x^2 = k$$ approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...

We have this type of very famous nicely symmetric pde in our area. However, no one knows how to handle it properly since it is a nonlinear pde. Suggestions on how it is called in general would help us further googling. I already tried keywords like "bilinear", "dual", "double", but by far could...

Can someone please tell me the book that contain integration using hyperbolic substitution for beginner? I know that hyperbolic functions is taught in Calculus book but most of them is only some identities and inverses of hyperbolic functions.

Can someone please show me an example of integration using hyperbolic substitution? Thank you

Is ##\mathrm{AdS_4}## globally hyperbolic?$$g = -\left(1+ \dfrac{r^2}{l^2} \right)dt^2 + \dfrac{dr^2}{1+ \dfrac{r^2}{l^2}}+ r^2d\Omega^2$$Letting ##r = l \tan \chi## then defining ##\tilde{g} = g \cos^2 \chi##\begin{align*} g &= \sec^2 \chi (-dt^2 + l^2 d\chi^2) + l^2 \tan^2 \chi d\Omega^2 \\...

Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$

Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...

Has anyone ever watched dragon ball Z before? In this TV show, there is something called the hyperbolic time chamber. 1 day on Earth is equivalent to 1 Year in the time chamber. In other words, if you stayed in the chamber for a full 24hours than exactly 1 year would have passed when you left...

This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...

(1) ok. (2) We start with ##\sigma(ξ) = a_{11} ξ_1^2 +2a_{12}ξ_1ξ_2 +a_{22}ξ_2^2ξ## and we replace every ##ξ_iξ_j## with ##\partial_i\partial_ju##, giving ##a_{11}\partial_x^2+2a_{12}\partial_x\partial_yu+1_{22}\partial_2^2## (3) The given equation is the following. ##\sigma(ξ) = ξ^t A ξ ##...

Ok Hi everyone! I was working on what would happen if you apply a linear increasing voltage to a series capacitor resistor. The question is : If the capacitor voltage is plotted, is the cap voltage curve hyperbolic? I've done some plots on the cap voltage and it sure looks hyperbolic but I...

The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are $$ \begin{cases} \sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x)...

Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...

In "The Geometry of Minkowski Space in Terms of Hyperbolic Angles" by Chung, L'yi, & Chung in the Journal of the Korean Physical Society, Vol. 55, No. 6, December 2009, pp. 2323-2327 , the authors define an angle ϑ between the respective inertial planes of two observers in Minkowski space with...

How to prove that every quadric surface can be translated and/or rotated so that its equation matches one of the six types of quadric surfaces namely 1) Ellipsoid 2)Hyperboloid of one sheet 3) Hyperboloid of two sheet 4)Elliptic Paraboloid 5) Elliptic Cone 6) Hyperbolic Paraboloid The...

Can anyone derive the distance formula of a hyperbola for me, please? I have not found the derivation on the internet. I can't get any clue from the picture of hyperbola.

The integral of cothx is ln|sinhx|+C. Does this mean the integral of coth2x is ln|sinh2x|+C? If not, does anyone have a link to a page on how it is achieved - I'm trying to compile a list of all common hyperbolic function derivatives and integrals. However, I can't find anything to confirm if...

Hello all, I am trying to solve a limit: \[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\] I found many suggestions online, from complex numbers to Taylor approximations. Finally I found a reasonable solution, but one move there doesn't make sense to me. I am attaching a picture: I have marked...

Homework Statement Homework Equations So the question is asking to solve an integral and to use the answer of that integral to find an additional integral. With part a, I don't have much problem, but then I don't know how to apply the answer from it to part b. I know I should subsitute all...

$$V_{HE}=\sqrt{\frac{\mu}{a}}$$ What is the rationale for this formula when we can determine the change in velocity from Earth's orbit to transfer orbit using the vis-viva equation? Likewise, what is the use of defining the radius for the sphere of influence for interplanetary transfer...

I am trying to understand why maxwell equations are correct in any reference frames? While i started to understand of his laws of physics a bit i could not imagine why he uses hyperbolic functions such as coshw instead of spherical ones in position and time relation between moving frames...

Given ##ds^2 = y^{-2}(dx^2 + dy^2)##, I am trying to prove that a demicircle centred on the x-axis, written parametrically as ##x=r\cos\theta + x_0 ## and ##y= r \sin \theta ## are geodesics. Where ##r## is constant and ##\theta \in (0,\pi)## I have already found the general form of the...

Hello, Recently, a solar power tower plant was founded next to where I work. Since it's the tallest object in the area, it's quite hard to miss it. But apart from that, every morning the reflected light is arranged in a hyperbolic- like way, as you can see in the picture. Does anyone have a...

Greg Bernhardt submitted a new PF Insights post Rindler Motion in Special Relativity: Hyperbolic Trajectories Continue reading the Original PF Insights Post.

I thought this was nice. In what follows, ##c=1##. For a particle undergoing constant proper acceleration ##\alpha## in the positive ##x##-direction, an inertial observer can use: ##x (\tau) = x_0 + \dfrac{\gamma(\tau) - \gamma_0}{\alpha}##, where the Lorentz factor is: ##\gamma (\tau) =...

Homework Statement Find a complete set of conditions on the constants a, b, c, n such that, for Cartesian coordinates (x, y, z), V = axn + byn + czn is a solution of Laplace’s equation ##∇^2V = 0##. A mass filter for charged particles consists of 4 electrodes extended along the z direction...

Today, I was at an award ceremony, where in one out of the two scientific lectures, the professor was teaching the basics of Hyperbolic Geometery. However, due to time constraints, he had to teach very fast, and there was no laser pointer, nor a chalkboard, so he couldn't explain very well...

This is a pannable simulation that will run in your browser. It shows where Pioneer 10 & 11, Voyager 1 & 2, New Horizons and ʻOumuamua are, how far they are from the Sun, and their speeds. simulation: http://orbitsimulator.com/gravitySimulatorCloud/simulations/1511746688216_hyperbolic.html...

I am filling my circular hottub, and charting the water level height. Its sides have a small, constant slope from vertical - i.e. it is a truncated, inverted cone. Imagining an ideal hottub of unlimited height*, the water level will always be increasing - but at a decreasing rate - it will...

Homework Statement I'm rather confused as to where the orbited body is placed, and where the orbiting trajectory lies in this figure. Is it right for me to say that if I placed say, the Earth on the left hand focus, a comet with a hyperbolic trajectory will travel a path defined by the...

Hello. I have this function ## v(x) = -\sum_{i=1} x^i \sqrt{2}^{i-2} \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm## which I can not seem to figure out how to simplify.I tried looking at some partial integration but repeated integration of ## \cosh ## gives polylogarithms which seemed to...

Homework Statement Homework EquationsThe Attempt at a Solution The attempt is in the picture. Is this the right method? Is there any faster method without cumbersome calculations?

Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...

I have come across an interesting simulation of hyperbolic space. I can't totally wrap my hand around it, though. But I am interested in technical part of it. As far as I see no object in the space would get curved when you are moving. The visual deformation is the result of bent light reached...

Homework Statement Two identical speakers, 10.0 m apart from each other, are stimulated by the same oscillator, with a frequency f, of 21.5 Hz, at a place where the speed of sound is 344 m/s. a) Show that a receiver at A will receive the minimum intensity of sound (Amin) due to the...

Homework Statement Recall that in hyperbolic geometry the interior angle sum for any triangle is less than 180◦. Using this fact prove that it is impossible to have a rectangle in hyperbolic geometry. Homework EquationsThe Attempt at a Solution - I wanted to use the idea that rectangles are...

Mod note: Because his caps-lock key is stuck, it's OK for this post to be in all caps. FIRSTLY, MY LAPTOP'S CAPS LOCK IS BEHAVING REALLY WEIRD AND I HAVE NO CONTROL ON IT WHATSOEVER. SO SORRY FOR POSTING IN ALL CAPS/ALL SMALL LETTERS I HAVE RECENTLY LEARNED HYPERBOLIC FUNCTIONS. HOWEVER, I AM...

Homework Statement Hy guys I am having an issue with approximating this first question, which I have shown below. Now my problem is not so much solving it but I have been thinking that if given the same question without knowing that it approximates to so for example the question I am...

Homework Statement Solve ##\displaystyle{d\sigma = \frac{d\rho}{\cosh\rho}.}## Homework Equations The Attempt at a Solution The answer is ##\displaystyle{\sigma = 2 \tan^{-1}\text{sinh}(\rho/2)}##. See equation (10.2) in page 102 of the lecture notes in...

Homework Statement Here is the equation for the general solution of an overdamped harmonic oscillator: x(t) = e-βt(C1eωt+C2e-ωt) Homework Equations β decay constant C1, C2 constants ω frequency t time The Attempt at a Solution I know (eωt+e-ωt)/2 = coshωt and (eωt-e-ωt)/2 = sinhωt but how do...