Change of variables Definition and 219 Threads

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...

Homework Statement Evaluate \iiint_\textrm{V} |xyz|dxdydz where V = \{(x,y,z) \in ℝ^3:\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} ≤ 1\}Homework Equations Change of Variables Theorem: \int_\textrm{ψ(u)} f(x)dx = \int_\textrm{K} f(\Psi(u))|detD\Psi(u)|du Examples: 1) For a ball of...

hi, i need to know how to plot the change of variables am i just to take x, and y as different constants, and treat it as normally on the (u,v) axis? so for the first, it will be like plotting on the (x,y) axis something like y^2 = x^2 - c, for different c, how many lines would i need to plot...

Homework Statement We have the evaluation of the integral associated with the electron self-energy diagram and I am following Brian Hatfield's book "Quantum Field Theory of point particles and strings", and I am having problems with the integration limits after changing variables from z_2 to...

Homework Statement The original integral is $$\left[\int_0^{\infty} {\int_0^{\infty} {F(x + y,x - y) \cdot dx \cdot dy} } \right]$$ What should be the limits of the integrals. (position represented by '?' symbol) $$\left[\int_?^? {\int_?^? {F(u,v) \cdot (\frac{1}{2})du \cdot dv} }...

Homework Statement Let I=∫∫D (x2−y2)dxdy, where D=(x,y): {1≤xy≤2, 0≤x−y≤6, x≥0, y≥0} Show that the mapping u=xy, v=x−y maps D to the rectangle R=[1,2]χ[0,6]. (a) Compute \frac{\partial(x,y)}{\partial(u,v)} by first computing \frac{\partial(u,v)}{\partial(x,y)}. (b) Use the Change...

$x=u^{2} - v^{2}$ $y=2uv$ Show the lines in the u,v plane where x and y are constant? (what?) Is this a valid change of coordinate on the whole x,y plane? (what?) By letting u = rcost, v=rsint, show that $x=r^{2}cos(2t), y=r^{2}sin(2t)$. Hence show that the map is 2 - 1 (what?) and show that...

http://imgur.com/cbQ44 I'm just not sure how they come up with that... *note, the E0 is not ground state energy, it's a constant electric field.

Homework Statement Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0 Homework Equations I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which...

Homework Statement I am trying to solve the transport PDE using a change of variables and the chain rule, and my problem seems to be with the chain rule. The PDE is: \frac{\partial u}{\partial t}+c\frac{\partial u}{\partial x} = 0 ......(1) The change of variables (change of reference frame)...

Suppose you start with a function f(x,y,t) which satisfies some partial differential equation in the variables x,y,t. Suppose you make a change of variables x,y,t \to \xi,z,\tau, where \tau = g_\tau(x,y,t) and similarly for \xi and z. If you want to know what the differential operators...

I am working on an exam question, I have the solution but i can't figure out a step in between, I need to show that ∂P/∂r = (1/r)∂ψ/∂r - ψ/r^2 I am wondering where the - ψ/r^2 comes from. I.e. which rule is this, and sencondly how does this apply to higher order differentials. Thanks in...

Homework Statement Find the mass of the plane region R in the first quadrant of the xy plane that is bounded by the hyperbolas xy=1, xy=2, x^2-y^2 = 3, x^2-y^2 = 5 where the density at the point x,y is \rho(x,y) = x^2 + y^2. Homework Equations The Attempt at a Solution The...

Homework Statement Evaulate the integral making an appropriate change of variables. \int\int_R(x+y)e^{x^2-y^2}dA where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, 3x-y=8 . Homework Equations The Attempt at a Solution I'm not sure what change of variables I should...

I have a question regarding problem solving tips. When given an iterated integral and asked to convert it to polar coordinates, how do you select the bounds of theta - do you have to understand how the graph of r operates and therefore know where the theta bounds are based on the rectangular...

Let R denote the region inside x^2 + y^2 = 1 but outside x^2 + y^2 = 2y with x=>0 and y=>0. Let u=x^2 + y^2 and v=x^2 + y^2 -2y. Compute the integral of x*e^y over the region D in the uv-plane which corresponds to R under the specified change of coordinates. I'm having trouble with this one...

Hi there I am given xi=x/s(t) and T=h(t)F(xi,t) and I need to tranform deltaT/deltat. How do I do it? Do I use the chain rule? The answer to it is : s*(dh/dt)*F+s*h*(deltaF/deltat)-xi*(ds/dt)*h*(deltaF/deltaxi) but I don't know how to get this answer. Please help me. Thank you

The problem is: R is the parallelogram bounded by the lines x+y=2, x+y=4, 2x-y=1, and 2x-y=4. Use the transformation u=x+y and v=2x-y to find the area of R. I am not sure how to complete this problem. My first issue is that I don't know how to convert the transformation functions into...

Update: I figured out how to solve the problem. Nevermind. Homework Statement Use a suitable change of variables to evaluate the double integral: \int^{1}_{0}\int^{3-x}_{2x}(y-2x)e^{(x+y)^{3}}dydxHomework Equations \frac{\partial(x,y)}{\partial(u,v)}=( \frac{\partial x}{\partial u}...

Hi, although this may sound trivial, I stumbled upon this problem while studying decimation process in digitial signal processing. I can't find anything on the web about some definition for the change of variables in sumations (as there is one for integrations), so maybe someone here could...

I'm just curious about the proofs of Integration by Parts & the Change of Variables formula as given in this book on page 357. I think there are a lot of typo's so I've uploaded my rewrite of them but I am unsure of how correct my rewrites are. If someone could point out the errors & why I...

Homework Statement [PLAIN]http://img542.imageshack.us/img542/5600/unledsn.png Homework Equations The Attempt at a Solution The first part is fine, just struggling to find a change of variables that'll help, tried spherical due to the x^2+y^2+z^2, didn't help enormously Thanks! (from sheet...

Homework Statement xy' = yf(xy) Homework Equations The Attempt at a Solution Attempt #1: F is a function of the product of x and y so I first thought of trying v = xy so dv = xdv + vdx but that would transform the equation into xy' = 1dv - vdx = yf(v). Attempt #2: I tried vx =...

Homework Statement Use the transformation x= \sqrt{v- u}, y = u + v to evaluate the double integral of f(x, y) = \frac{x}{(x^2 + y)} over the smaller region bounded by y = x^2, y = 4 − x^2, x = 1. Homework Equations The Attempt at a Solution d:={ (x,y)| -\sqrt{2}<x<1 , x^2<y<...

Homework Statement Suppose D is the parallelogram enclosed by the lines 2x-3y = 0, 2x-3y = 2, 3x-y = 0 and 3x-y = 1. \int\int^{}_{D} [(2x-3y) e^(3x-y) dA Homework Equations The Attempt at a Solution Set u to be equal to 2x-3y -> x = (u+3y)/2 Set v to be equal to 3x-y -> v = 3/2(u+3y)-y ->...

Homework Statement Suppose D is the parallelogram in the xy-plane with vertices P(-1,5), Q(1,-5), R(5,-1), S(3,9) \int\int ^{}_{D} (6x+12y) dA HINT: Use transformation x = \frac{1}{6}(u+v) and y = \frac{1}{6} (-5u+v). Homework Equations The Attempt at a Solution Calculating the Jacobian I...

Homework Statement Find the volume of the cone bounded below by z=2root(x2+y2) and above by x2 + y2 + z2 = 1 Homework Equations The Attempt at a Solution Ok I have the solution, I just don't understand how to get it! So I know I have to change into spherical coordinates but...

Hi, I'm reading an article where an integral of the form: \int^{\infty}_{-\infty}\,\mathrm{d}\tau' \int^{\infty}_{-\infty}\,\mathrm{d}\tau''... The author then splits this into the region whereby \tau' >\tau'' , and the region \tau''>\tau' ...

Homework Statement Sorry I tried to use Latex but it didn't work out :/ Make the change of variables r = x + vt and s = x vt in the wave equation partial^2y/partialx^2-(1/v^2)(partial^2y/partialt^2)=0 Homework Equations...

Homework Statement I want to use polar coordinates to integrate 1/sqrt(x^2+y^2) dydx with limits of integration 0 < y < x and 0 < x < 3Homework Equations x=rcosO y=rsinOThe Attempt at a Solution I know that the area being integrated over is the triangle enclosed by y=x and x=3. I have my...

Hi, I have a couple of questions remaining on a differential equations example sheet that I can't seem to crack. They have a common theme -- changing variables in a PDE. Here's the first one. I'm hoping that with a gentle nudge in the right direction the rest of it should fall into place...

Homework Statement I have been given the distribution function F_X of the random variable X and I am asked to find the distribution function F_Y of Y, another random variable which is defined from X in the following way. Y={\stackrel{X^{2} if X<2;}{4 if 2\leq X < 3;}\stackrel{4(4-X) if...

Homework Statement I am confused on how exactly Jacobians variables and such. We had a problem on a test in my class that was: a double integral (i don't know how to use the notation on here) over the region R of (x-2y)e^(x+y)dxdy Where R is the parallelogram with vertices (0,0) (2,1)...

Make the indicated change of variables (do not evaluate) (Not sure how to write an iterated integral with bounds so I will try and explain by just writing the bounds) (I also tried using the symbols provided, but everything I tried just put a theta in here so I gave up) \int\int\intxyz...

I am looking through my course notes for mathematical physics, in preparation for the exam, and I've run into a concept that I can't figure out. It comes up first when talking about the modified bessel's equation (x^2)y''+(x)y'-(x^2+p^2)y=0 And supposedly this can be transformed into...

So if I'm changing from variables x,y to variables \alpha = f(x,y), \beta = g(x,y), what exactly does it mean to stay this change of variables is unitary, and how can I tell if it is or if it isn't?

1. Let X~Geometric(1/4), and let Y have probability function: pY(y)= 1/6 if y=2 1/12 if y=5 3/4 if y=9 0 otherwise Let W=X+Y. Suppose X and Y are independent. Compute pW(w) for all w in R. For this i am not sure i think its summation from K=0 to infinity (PY=w-K)(1/4)(3/4)^K...

Homework Statement Given the differential equation u_{xx}+3u_{yy}-2u_{x}+24u_{y}+5u=0 use the substitution of dependent variable u=ve^{ \alpha x + \beta y} and a scaling change of variables y'= \gamma y to reduce the differential equation to v_{xx}+v_{yy}+cv=0Homework Equations I have no...

find the image of the set S under the given transformation: S is the disk given by u^2 + v^2 <= 1 where x=au, y=bv since it is a circle i have that the boundaries are [-1,1] x [-1.1] so for my equations i got x = au, y = -b x = a, y = bv x = au, y = b x = -a, y = bv i don't know what...

I'm not sure if this is a stupid question, but I'll go ahead anyway. I understand the math aspect of it, but one thing has me confused. If you have a uv plane, and then write x=x(u,v), y=y(u,v), why is it that no matter what the function transforming the uv plane to the xy plane is, we can...

I have been trying to understand and articulate why I can't do the following. Please confirm or point out misunderstanding. There is an integral in the "hatted" system, \int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})d\bar{x}^{1}...d\bar{x}^{n} I want to express this as an integral in the...

Homework Statement Let D be the region y \leq x \leq 1 and 0 \leq y \leq 1. Find the boundaries of the integration for the change of variables u and v, if: x = u + v and y = u-v. Homework Equations NoneThe Attempt at a Solution I solved for u and for v in terms of x and y. u =...

Homework Statement show \int _1^{\infty }\frac{1}{x^2}\text{Log}[x]dx=-\int_0^1 \text{Log}[x] \, dx similarly show \int _0^{\infty }\frac{1}{x^2+1}\text{Log}[x]dx = 0 The Attempt at a Solution For the first part a substitution 1/x works. The second part I cannot do, I...

Homework Statement Theorem. The change of variables is reversible near (u0,v0) (with continuous partial derivatives for the reverse functions) if and only if the Jacobian of the transformation is nonzero at (u0,v0). 1. Consider the change of variables x=x(u,v)=uv and y=y(u,v)=u2-v2. (a)...

Homework Statement Use Green's Theorem to prove for the case f(x,y) = 1 \int\int_R dxdy = \int\int_S |\partial(x,y)/\partial(u,v)|dudv EDIT: R is the region in the xy-plane that corresponds to the region S in the uv-plane under the transformation given by x = g(u,v), y = h(u,v), and the...

Homework Statement Given the equation U(\mu) = \frac{2}{\sqrt\pi} \exp\left[ -4\mu^2 \right] [/itex] find an expression for \hat U(\hat x) given that change of variables x = \frac n2 + \sqrt n \mu, \qquad \hat x = \frac xn and \hat U is the U under this variable transformation...

Homework Statement I have to transform the following equation using variables (u,v,w(u,v))=(yz-x,xz-y, xy-z): (xy+z)\frac{\partial z}{\partial x}+(1-y^2)\frac{\partial z}{\partial y}=x+yz. Homework Equations chain rule: \frac{dw}{dx} = \frac{\partial w}{\partial u}...