Hyperbolic Definition and 347 Threads

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...

Homework Statement A rectangular box measuring a x b x c has all its walls at temperature T1 except for the one at z=c which is held at temperature T2. When the box comes to equilibrium, the temperature function T(x,y,z) satisfies ∂T/∂t =D∇2T with the time derivative on the left equal to zero...

I am trying to illustrate how hyperbolic navigation works. In the process, I need a program that will plot two hyperbolic equations and solve their intersection. I would rather not have to change the equation into function form if possible. I know that the plots will be two hyperbolas that are...

Homework Statement $$\int \tanh=?$$ Homework Equations $$\cosh^2-\sinh^2=1$$ $$(\tanh)'={\rm sech}^2=\frac{1}{\cosh^2},~~(\coth)'=-{\rm csch}^2=-\frac{1}{\sinh^2}$$ $$({\rm sech})'=\left( \frac{1}{\cosh} \right)'=-{\rm sech}\cdot\tanh=-\frac{\sinh}{{\rm cosh}^2}$$ $$({\rm csch})'=\left(...

Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...

Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. $$2x-2yy'=0~\rightarrow~\frac{x}{y}=y'=\frac{\cosh (u)}{\sinh (u)}=\frac{e^u+e^{-u}}{e^u-e^{-u}}$$ The equation...

Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...

Hello I am trying to determine the Fourier transform of the hyperbolic tangent function. I don't have a lot of experience with Fourier transforms and after searching for a bit I've come up empty handed on this specific issue. So what I want to calculate is: ##\int\limits_{-\infty}^\infty...

Usually when gravitational lensing is discussed, the examples are those of matter bending spacetime into a positive curvature. https://commons.wikimedia.org/wiki/File:Gravitational_lens-full.jpg In these cases, distortion of light is clearly evident as images of galaxies from behind these...

I noticed the graphs of ##y=\cos^{-1}x## and ##y=\cosh^{-1}x## are similar in the sense that the real part of one is the imaginary part of the other. This is true except when ##x<-1## where the imaginary part of ##y=\cos^{-1}x## is negative but the real part of ##y=\cosh^{-1}x## is positive. I...

Consider ##y=\cos{-x}=\cos x=\cosh ix##. Thus, ##\pm x=\cos^{-1}y## and ##ix=\cosh^{-1}y##. So ##\cosh^{-1}y=\pm i\cos^{-1}y##. Renaming the variable ##y##, we have ##\cosh^{-1}x=\pm i\cos^{-1}x##. Next, we evaluate the derivative of ##\cosh^{-1}x## by converting it to ##\cos^{-1}x## using...

Homework Statement Given that the sum of interior angle measures of a triangle in hyperbolic geometry must be less than 180 degree's, what can we say about the sum of the interior angle measures of a hyperbolic n-gon? Homework EquationsThe Attempt at a Solution So in normal geometry an n-gon...

Homework Statement ∫sin(2sinh(3x)) Homework EquationsThe Attempt at a Solution okay so i did a u substitution letting u=3x so we get 1/3∫sin(2sinh(u)) but i have no idea how to get rid of the sinh, i tried writing in exponential form or maybe i have to use some identity.. I am not sure where...

Show from the definition of arctanh as the inverse function of tanh that, for $x \in (-1, 1)$ $$\tanh^{-1}{x} = \frac{1}{2}\log\left(\frac{1+x}{1-x}\right)$$ The definition of hyperbolic tangent is $\displaystyle \tanh{h} = \frac{e^x-e^{-x}}{e^{x}+e^{-x}}$ Let $\displaystyle y =...

How do you calculate the limit $\displaystyle \lim_{x \to \infty}\frac{x}{\cosh{x}}$

Homework Statement Differentiate Homework Equations Chain Rule: dg/dx = du/dx . dv/du . dg/dv The Attempt at a Solution My answer(wrong): Correct answer provided to us(not mine): I understand the correct solution that was provided to us, but what I don't understand is why my method...

Homework Statement Show that the real solution ##x## of $$tanhx=cosechx$$ can be written in the form ##x=ln(a \pm \sqrt{a})## and find an explicit value for ##a##. Homework Equations $$cosh^{2}x-sinh^{2}x=1$$ $$coshx=\frac{e^{x}+e^{-x}}{2}$$ The Attempt at a Solution I reduced the original...

then look at : the 2 curves are nearly the same while the equations are not, is there anything wrong ?

This system of coordinates: can be "translated" in terms of x and y, so: x = \sqrt{\frac{\sqrt{u^2+v^2}+u}{2}} y = \sqrt{\frac{\sqrt{u^2+v^2}-u}{2}} Exist another form more simplified of write x and y in terms of u and v? I tried rewrite these expressions using the fórmulas of half angle but...

I am familiar with both trigonometric (circular) and hyperbolic substitutions, and I have solved several integrals using both substitutions. I feel like trigonometric substitutions are a lot simpler, however. Even in cases where the substitution yields an integral of secant raised to an odd...

Hello all, Homework Statement $$x{u_{xy}} - y{u_{yy}} = 0$$ Assume $$x,y \in {\rm{Reals}}$$ Homework Equations I have been able to solve this using different methods, but my classmates and I are trying to figure out if there is a way to do this using the methods from the course's text. The...

1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...

What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.

What exactly is the integral of ##\frac{1}{\sqrt{x^2 - 1}}##? I know that the derivative of ##\cosh^{-1}{x}## is ##\frac{1}{\sqrt{x^2 - 1}}##, but ##\cosh^{-1}{x}## is only defined for ##x \geq 1##, whereas ##\frac{1}{\sqrt{x^2 - 1}}## is defined for all ##|x| \geq 1##. How do I take that into...

Osborn's rule: "The prescription that a trigonometric identity can be converted to an analogous identity for hyperbolic functions by expanding, exchanging trigonometric functions with their hyperbolic counterparts, and then flipping the sign of each term involving the product of two hyperbolic...

To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...

I don't fully understand hyperbolic space. I saw a numberphile video about it. I thought It would a cool idea to make a video game based around hyperbolic space. I was going make it in html5/css/javascript. I know I need to learn a lot of math. I was going to render the hyperbolic objects in a...

Hello, I am trying to understand the resolution of the following KdV equation. I try to demonstrate it by myself. The solitary wave solution is : At first, I created new variable as follows so I could transform the PDE into an ODE. A = A(p) p = g(x,t) g(x,t) = x - ct I succeeded to...

More than just a few problems that happen to pop up in the textbook, I mean.

I want to know Universe density according to this equation( ##k=-1##) ? ##H^2(t)-8πρG/3=-k/a^2(t)## ##ρ_U=ρ_m+p_r## ##ρ_U##=Universe density ##ρ_m##=Matter density ##p_r##=Radiation density

My calc. 2 book more or less only mentioned the hyperbolic functions to make integration easier, so, now that I have some free time, I'd like to explore the area further. Could someone recommend a good book on the subject or do I need to take more math first? A quick google search revealed...

I want to convert this system of corrdinates (see image beloow) to cartesian system. How make this? https://www.physicsforums.com/attachments/c2-png.82342/?temp_hash=1cfcfdb56cb59e415f556c06ffbe270a Tip: x = a exp(+u) cosh(v) y = b exp(-u) sinh(v)

Hey! :o I have to solve the following pde: $$u_{tt}(x, t)-u_{xt}(x, t)=0, x \in \mathbb{R}, t>0 \\ u(x, 0)=f(x), x \in \mathbb{R} \\ u_t(x, 0)=g(x), x \in \mathbb{R}$$ I have done the following: $$\Delta(x, t)=1>0$$ so it is an hyperbolic equation. We want to write the equation into...

Hi! First post on this forum, though not the first time visiting :) I am working on a model of an object falling from one layer of air density into another layer with a higher density, effectively changing the acceleration from positive to negative instantly. (Somehow I am thinking of positive...

1. The problem statement, all variables and given/known Find the length of the curve $$y=ln(x),\frac{1}{2}<=x<=2$$ Homework Equations Using hyperbolic trig isn't necessary, but it's how my text (Serge Lang's A First Course in Calculus) approaches most square roots, and as a result, it's what...

Could someone tell me what is a hyperbolic trigonometric function? What is the difference between regular trigonometry and a hyperbolic trigonometry? Also, why and how to derive and get ##\sinh x = \frac{e^x - e^{-x}}{2}## ?

Hi All, I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric. So...

I know if we set x = \cosh \theta , y = \sinh \theta and graph for all \theta 's, we get a hyperbolic curve since then x^2 - y^2 = 1. But — unlike the case of making a circle by setting x = \cos \theta , y = \sin \theta and graphing all the \theta 's — in the hyperbolic graph the angle...

The FRW metric is usually expressed as $$ds^2 = -dt^2 + a(t)^2 ( \frac{dr^2}{1-kr} + r^2 d\Omega^2))$$ where ##k=-1,0,+1## respectively for a hyperbolic, flat or spherical space. The spatial part of this metric can be derived by considering a 3-sphere embedded in a four-dimensional flat space...

I've always been having trouble with the domain and range of inverse trigonometric functions. For example, let's start with an easy one: $\sin^{-1}\left({x}\right)$ Process: First, I draw out the function of $\sin\left({x}\right)$. Then I look at its range and attempt to restrict it so that it...

Suppose we want to find: $$\int \frac{1}{\sqrt{x^2-a^2}}\,dx$$ Trig Substitution: $$=\ln \left| x+\sqrt{x^2-a^2} \right|$$ Hyperbolic Substitution: $$=\cosh^{-1}\left({\frac{x}{a}}\right)=\ln\left({x+\sqrt{x^2-a^2}}\right)$$ I know this is super minor, but how are they equivalent when one...

I read somewhere that: sqrt(a^2-x^2), you can use x = asinx, acosx sqrt(a^2+x^2), you can use x = atanx (or acotx), asinhx sqrt(x^2-a^2), you can use x = asecx (or a cscx), acoshx When would it be beneficial to use a hyperbolic trig substitution as oppose to the regular trig substitutions (sin...

Normal form of the hyperbolic equation Hey! :o I am looking at the following in my notes: $$a(x,y) u_{xx}+2 b(x,y) u_{xy}+c(x,y) u_{yy}=d(x,y,u,u_x,u_y)$$ $$A u_{\xi \xi}+ 2B u_{\xi \eta}+C u_{\eta \eta}=D$$ $$A=a \xi_x^2+2b \xi_y \xi_x+c \xi_y^2 \ \ \ (*)$$ $$B=a \xi_x \eta_x +b \eta_x...

6b) tanh^2(x) + 1/cosh^2(x) = 1 Could someone help start me off? I know that you have to sub in (e^x + e^-x)/2 for cosh and (e^x - e ^-x)/(e^x + e ^-x) for tanh. Then I'd add these together, but I'm not sure how I'd solve/simplify them arithmetically after that. Help would be appreciated! thanks.

Is known that in every rectangle triangle the following relationships are true: But, how use geometrically the function sinh, cosh, and tanh?

Hello, I'm going through Landau and Lifshitz "The Classical Theory of Fields" this summer with a friend and in section 4 I've come to a bit of a math problem. Assume you have an inertial frame K' moving at speed V relative to an inertial frame K in the x-direction. In order for invariant...

Homework Statement Attached is the problem Homework Equations My question is am i going about it the right way for question C). I have done A and B and am sure they are correct. The Attempt at a Solution Attached

Homework Statement Find the anti derivative of \int xcosh (x^2) dxHomework Equations By parts formula and Hyperbolic Identities of sinh x and cosh x as well as others The Attempt at a Solution \int xcosh (x^2) dx The problem I'm having is integrating \int cosh (x^2) dx I tried setting...

Homework Statement y=1 / (cos h x), find dy/dx Homework Equations chain rule and coshx=(e^x+e^-x)/2 The Attempt at a Solution