Change of variables Definition and 219 Threads

Homework Statement Use the change of variables to evaluate the integral (x +y ) sin(x -y )dA, where R is the region enclosed by y = x, y = x - 2, y = -x and y = -x + 1. (Hint: use u = x + y and v = x - y Homework EquationsThe Attempt at a Solution Not sure how to start it

Use change of variables to evaluate the following integral $$\int_{2}^{7}\frac{x}{\sqrt[3] {x^2 -2 }} \,dx\approx 8.577$$ $u={x}^{2}-2$ $du=2x\ dx$ $\frac{1}{2}du=x\ dx$ $u(2)=2$ $u(7)=47 $ $$\frac{1 }{2 }\int_{2 }^{47 }\frac{1 }{ \sqrt[3] {u }} \,du $$ Can't get same answer

Let's say ##f(x)=ax^2##. Then ##d^2f/dx^2=2a##. Now we can make the change of variables ##y\equiv\sqrt ax## to give ##f(y)=y^2##. Then ##d^2f/dy^2=2##. It follows that ##\frac{d^2f}{dx^2}=a\frac{d^2f}{dy^2},## but I can't replicate this with the chain rule. I would put...

I need some help to make sure my reasoning is correct. Bear with me please. I have a time distribution for a process and I want to construct a distribution for the time it takes to perform two processes. So I would define ##\tau = t + t## This would create a new distribution with is a...

## \int_{0}^{∞}\int_{0}^{∞} \frac{x^2+y^2}{1+(x^2-y^2)^2} e^{-2xy} dxdy ## ##u= x^2-y^2## ##v=2xy##I tried to find the jacobian and the area elements, I found it to be ## dA = \frac{1}{v} du dv ## I'm having problem finding the limits of u & v and getting rid of ##x^{2}+y^{2}##.

This is not really a homework or anything, just found myself hitting a wall when doodling around. If I have an integral like ##\int_{-1}^{0} x(x^2-1) dx## and I introduce a new variable: ##u = x^2## How do I calculate the limits of the new integral? In this case the upper limit is obviously 0...

Homework Statement Consider the function of two variables: u(x,y) = f(x-y) + g(x+(1/3)y) where f(s) and g(t) are each arbitrary functions of a single variable. Using the change of variables: s = x-y t = x-(1/3)y use the chain rule to determine the first and second derivatives of u with...

Homework Statement In the book it gave the example for standard change of variables as, z = x / 1 + x or equivalently x = z / 1 - z , then dx = dz / (1 - z)^2 , thus (2) 1 / (1 - z)^2 f (z / 1 - z) dz (3) This is what I am trying to accomplish but with the...

Hey I am trying to evaluate d/dx, d/dy and d/dz of a wavefunction defined on a grid. I have the wavefunction defined on equally spaced points along three axes a=x+y-z, b=x-y+z and c=-x+y+z. I can therefore construct the derivative matrices d/da, d/db and d/dc using finite differences but I...

Hello! (Smile) We have the differential equation $y''+ \frac{p}{x} y'+ \frac{q}{x^2}y=0, x>0$ and we set $z=\log x$. Then $y'=\frac{dy}{dx}=\frac{dy}{dz} \frac{dz}{dx}=\frac{1}{x} \frac{dy}{dz}$ $y''=\frac{d^2y}{dx^2}=\frac{d}{dx}\left( \frac{dy}{dx} \right)=\frac{d}{dx}\left( \frac{1}{x}...

Homework Statement Use a change of variable to show that \int_0^{\infty} \frac{dx}{1+x^2} = 2\int_0^1\frac{dx}{1+x^2} Please note: the point of this exercise is to change the bounds of the integral to be finite to allow numerical estimation, as opposed to directly solving the integral...

I'm working through an article called "Cosmic abundances of stable particles -- improved analysis" (link -- viewable only in Firefox afaik), the result of which, equation (3.8), is cited quite a lot. I'm more interested in how they arrived there. Particularly, how come momentum space measure...

Homework Statement Solve: \frac{d^{2}y}{dx^{2}} + \omega^{2}y = 0 Show that the general solution can be written in the form: y(x) = A\sin(\omega x + \alpha) Where A and alpha are arbitrary constants Homework EquationsThe Attempt at a Solution I know that I will need to change variables for...

Hi there, I'm having some difficulty in understanding how the change of variables by considering a retarded time frame can be obtained for this particular eqn I have. Say I have this original equation, \frac{\partial A}{\partial z} + \beta_1 \frac{\partial A}{\partial t}+ \frac{i...

Homework Statement Write the Laplace equation \dfrac {\partial ^{2}F} {\partial x^{2}}+\dfrac {\partial ^{2}F} {\partial y^{2}}=0 in terms of polar coordinates. Homework Equations r=\sqrt {x^{2}+y^{2}} \theta =\tan ^{-1}(\frac{y}{x}) \dfrac {\partial r} {\partial x}=\cos \theta \dfrac...

I know the formula for a change of variables in a double integral using Jacobians. $$ \iint_{S}\,dx\,dy = \iint_{S'}\left\lvert J(u,v) \right\rvert\,du\,dv $$ where ## S' ## is the preimage of ## S ## under the mapping $$ x = f(u,v),~ y = g(u,v) $$ and ## J(u,v) ## is the Jacobian of the mapping...

Hi there! The question is: if I have to prove that a function is a change of variable it is sufficient to prove that the function is a diffeomorphism? i.e. prove that the function is bijective, differentiable, and its inverse is differentiable? Thanks!

Homework Statement Use the change of variables ##u=x+y## and ##y=uv## to solve: \int_0^1\int_0^{1-x}e^{\frac{y}{x+y}}dydx Homework Equations The Attempt at a Solution So I got as far as: \int\int{}ue^vdvdu. But I just can't find the region of integration in terms of ##u## and ##v##.

Homework Statement Suggest a substitution/transformation that will simplify the following integral and find their jacobians: \int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA Homework Equations [ tex ] \int\int_{R}x\sin(6x+7y)-3y\sin(6x+7y)dA [ / tex ] The Attempt at a Solution This topic is...

Homework Statement these are on the picture Homework Equations transformation, jacobian The Attempt at a Solution I don't know how to enter the equation, so i uploaded the picture.. is it alright? I think I solved it somehow, but not quite sure if it is right... please tell me if there's a...

Homework Statement Let ##H## be the parallelogram in ##\mathbb{R}^2## whose vertices are ##(1,1), (3,2), (4,5), (2,4).## Find the affine map ##T## which sense ##(0,0)## to ##(1,1), (1,0)## to ##(3,2), (0,1)## to ##(2,4)##. Show that ##J_T=5## (the Jacobian). Use ##T## to convert the integral...

Homework Statement Integrate the following over the set E. \int_E \frac{2x+y}{x+3y} dA Bounded by the lines: y = −x/3+1 y = −x/3+2/3 y = −2x y = −2x + 1 Homework Equations None. The Attempt at a Solution I can up to the same point everytime, but always get stuck on finding the new...

Hi there, in a paper the author obtains the integral $$\int_{a}^{\infty} \frac {g(\lambda(r))}{r}\mathrm{d}r$$ which is claimed to be equivalent to $$\int_{a/A}^{1} \frac {g(\lambda(r))}{\lambda (\lambda^3-1)}\mathrm{d}\lambda$$ making use of the relationship (previously physically...

The problem is as follows. Calculate the double integral of cos ((x-y)/(x+y)) dA over R, where R is the triangle bounded by the points (0,0), (2,2), and (2 + pi, 2 - pi). I understand that you have to set U = x-y and V = x+y. However, I am having a hard time finding the bounds on the...

Homework Statement Consider the differential equation zZ'' + Z' + γ2 Z = 0, where Z = Z(z). Use the change of variables x = √(z/b) with b a constant to obtain the differential equation Z'' + (1/x)Z' + α2Z = 0, where Z = Z(x) and α= 2γ √b Homework Equations Maple commands The Attempt at a...

Homework Statement The solid region W is bounded by the ellipsoid x^2/3 + y^2/5 + z^2/7 = 1. Find the triple integral ∫∫∫cos((35x^2 + 21y^2 + 15z^2)^(3/2))dV. Homework Equations Domain: x^2/3 + y^2/5 + z^2/7 = 1 Integral: ∫∫∫cos((35x^2 + 21y^2 + 15z^2)^(3/2))dV The Attempt at a...

\[ \alpha^2T_{xx} = T_t + \beta(T - T_0) \] where \(\beta\) is a constant and \(T_0\) is the temperature of the surrounding medium. The initial temperature distribution...

Homework Statement By changing variables from (S,t,V) to (x,\tau,u) where \tau = T - t, x = \ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T-t), u=e^{r\tau}V, where r, \sigma, \tau, K are constants, show that the Black-Scholes equation \frac{\partial V}{\partial t} +...

Consider the following solution to the steady state heat diffusion problem on an infinite y domain. \[ T(x, y) = \sum_{n = 1}^{\infty}c_n\exp\left(-\frac{\pi n}{\ell} y\right) \sin\left(\frac{\pi...

Okay so I am working on this problem: Solve xu_t + uu_x = 0 with u(x, 0) = x.(Hint: Change variables x \rightarrow x^2.) However, I am not sure how to use the change of variables hint that is given or why it is needed. My thinking is that I could just use the method of characteristic as normal...

Equation found in previous question is #: x''=gsin(a)-b(v^2) 1. Rewrite the equation # as a differential equation for v as a function of x. 2. Solve the equation to find v as a function of x. Relevant equations: v=x'=dx/dt, x''=v'=a=dv/dt Attempt at 1: Using the relevant equations you can...

Hello, I have a simple question regarding changing variables in a conditional distribution. I have two independent variables r \in \mathbb{R}, r>0 \\ t \in \mathbb{I}, t>0 where r is "rate" (can be any positive real number although most likely to be around 1) and t is "time" (positive...

Homework Statement Let D=\{ (x,y)\in\mathbb{R}^2:x+y< 1;0< y< x\} calculate \int_{D} e^{-(x+y)^4}(x^2-y^2) through an appropriate change of variables Homework Equations \int_{D} f *dxdy=\int_{D} f*Jacobian*dudv The Attempt at a Solution I've tried x+y=u and x-y=v...

... ... I am curious to know why we have to multiply with e^{-j\omega t} in Fourier transform? What is the purpose of this? I have heard somewhere that the transform is merely a change of variables from one set of coordinates to another. I would like to know more about this. Can you help me?

Homework Statement Let S be the part of the cylinder of radius 9 centered about z-axis and bounded by y >= 0; z = -17; z = 17. Evaluate \iint xy^2z^2 Homework Equations The Attempt at a Solution So I use the equation x^2 + y^2 \leq 9, meaning that r goes from 0 to 3 Since y...

Homework Statement The question is: Let \phi: \mathbb{R}^n\rightarrow\mathbb{R}^n be a C^1 map and let y=\phi(x) be the change of variables. Show that dy_1\wedge...\wedge dy_n=(detD\phi(x))\cdotdx_1\wedge...\wedgedx_n. Homework Equations n/a The Attempt at a Solution Take a look at here and...

The question is: Let $\phi：\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map and let $y=\phi(x)$ be the change of variables. Show that d$y_1\wedge...\wedge $d$y_n$=(detD$\phi(x)$)$\cdot$d$x_1\wedge...\wedge$d$x_n$.Take a look at here and the answer given by Michael Albanese: differential...

Hello there, I am facing the second order ODE in the unknown function $$y(t)$$ $$ \ddot{y} = a \dot{y} y - b \dot{l} l - c\dot{l} + d$$ $$a, b, c, d$$ positive constants, such that $$ \frac{a}{b} = \frac{d}{c}$$ I would like to understand more about it before relying on numerical methods...

I have looked up a few derivations of Noether's Theorem and it seems that chain rule is applied (to get a total derivative w.r.t. q_{s} ( = q + s ) is often used. What I do not understand is why this is legitimate ? If we start with L=L(q,q^{.},t) how can we change to L=L(q_{s}...

I'm reading some course notes for a physics class that contain the following step in a derivation: \phi(\vec{x}) = \int \frac{d^3k}{(2\pi)^3}\frac{e^{i\vec{k}\cdot\vec{x}}}{\vec{x}^2 + m^2} Changing to polar coordinates, and writing \vec{k}\cdot\vec{x}=kr \cos\theta, we have: \phi(\vec{x}) =...

Homework Statement This problem has to do with a canonical transformation and Hamiltonian formalism. Below is my attempt at solving it, but I am not too sure about it. Please help! Let \theta be some parameter. And X_1=x_1\cos \theta-y_2\sin\theta\\ Y_1=y_1\cos \theta+x_2\sin\theta\\...

1. Let T = (X,Y,Z) be a Gaussian for which X,Y,Z for which X, Y, Z are standard normals, such that E[XY] = E[YZ] = E[XZ] = 1/2. A) Calculate the characteristics function Φ_T(u,v,w) of T. B) Calculate the density of T. 2. Let X and Y be N(0,1) (standard normals), not necessarily...

Given $$ u(\xi,\eta) = F(\xi) + G(\eta) $$ and $$ \xi = x+ct\qquad\qquad \eta = x -ct. $$ How do we get to $u(x,t) = F(x+ct) + G(x-ct)$? I see that we can make the sub $$ u(x+ct,x-ct) = F(x+ct) + G(x-ct) $$ but how do I then get simply $u(x,t)$?

I had a question about about the integration measure for the path integral after a unitary change of variables. First they consider a 4D spacetime lattice with volume L^4. The measure is \mathcal{D}\phi = \prod_i d\phi(x_i) They expand the field variables in a Fourier series...

Homework Statement ∫∫∫V 9z2 dxdydz, where V is the solid defined by: -1≤x+y+3z≤1, 1≤2y-z≤7, -1≤x+y≤1 Homework Equations The Attempt at a Solution I did this using the bounds, 1/3(-x-y-1)<=z<=(1/3)(-x-y+1), -x-1<=y<=1-x, -3/2<=x<=1/2 but I think the answer is wrong is there a better way to do...

Homework Statement ∫∫Se2x+3ydydx where S is the region |2x|+|3y|≤ 1 Homework Equations The Attempt at a Solution So I've done this two ways and gotten two different answers and I'm not sure which is right. I used change of variables where where u=3y+2x and v=3y-2x and I got an...

For the system of equations $$ \mathbf{x}' = \begin{pmatrix} a\cos 2t & a\sin 2t\\ a\sin 2t & -a\cos 2t \end{pmatrix}\mathbf{x}, $$ verify that under the change of variables: $$ \mathbf{x} = \begin{pmatrix} \cos 2t & \sin 2t\\ \sin 2t & -\cos 2t \end{pmatrix}\mathbf{u}, $$ the equations for $u$...

Hello everyone! I am fairly new to SDE theory, so I'm sorry if my question may be a bit naive. I have the following coupled set of SDE:s d\phi = \frac{v - v_r}{R}d t + \frac{\pi}{\sqrt{t_c}}d W d v = A\cos(n\phi - \phi_w)d t + a_v d t + \sigma_v d W. W denotes a Wiener process, and the...

Homework Statement In double integrals, the change of variables is fairly easy to understand. With u = constant and v = constant, along line KL v = constant so dv = 0. Therefore the only contributing variable to ∂x and ∂y is ∂v. The Attempt at a Solution However, in tripple...

Hi All, I've managed to confuse myself with a simple change of variables. I have an integral of the form: $$ I = \int_f^{\infty} dt \int_0^1 ds\, t\, F(t(1-s),ts), $$ where $F(a,b)$ is some well behaved function and $f$ is a positive number. I want to change variables: $$ x =...