Change of variables Definition and 208 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...

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 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...

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}) =...

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 =...

Hi, The change of variables theorem states that given a diffeomorphism g:A \rightarrow B between open sets, and a continuous function f:A \rightarrow R, then \int _A f = \int _B f \circ g |Det Dg| given that either one of the integrals exist. I was wondering if anyone here could help explain...

I have an equation that I am trying to change the variables of, it has the form; \frac{d}{dt} W = g \frac{\partial}{\partial y} U + h x \frac{\partial^2}{\partial y^2} Z Where W, U and Z are my dependent variables (This equation is just one of 3 coupled equations but have written...

Quick question about Grassman numbers and change of variables. Suppose you have the function: f=\frac{c\epsilon_{ij} }{2!} \psi_i \psi_j and integrate it: \int d\psi_2 d\psi_1 \frac{c\epsilon_{ij} }{2!} \psi_i \psi_j =c Now change variables: \psi_i=J_{ik}\psi'_k to get: \int...

Homework Statement Find the area in the positive quadrant of the x-y plane bounded by the curves {x}^{2}+2\,{y}^{2}=1, {x}^{2}+2\,{y}^{2}=4, y=2\,x, y=5\,x The Attempt at a Solution This is a graph of the region: http://img21.imageshack.us/img21/2947/59763898.jpg One thing I was...