Change of variables Definition and 208 Threads

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...

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 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...

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?

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}...

the problem: evaluate the following integral by making appropriate change of variables. double integral, over region R, of xy dA R is bounded by lines: 2x - y = 1 2x - y = -3 3x + y = 1 3x + y = -2 my attempt: let 2x - y = u, and let 3x + y = v then the new region in (u,v) coordinates...

suppose i want to find the following integral: 7 \int[SIZE="5"]x dx 3 now suppose for some demented reason i decided not to do it straightforward and get (49-9)/2=20 instead i use the substitution x=u2+4u+5 giving u1 \int[SIZE="5"](u2+4u+5)(2u+4)du u0 u1 \int[SIZE="5"]2u3+12u2+26u+20 du u0...

dTB/dt = -k(TB-TM). TM is held constant. (TB-TM) = Q, so: dTB/Q = -kdt. How would I change this equation so that instead of integrating wrt to TB, I can instead integrate wrt Q?

Homework Statement The problem is as follows: Let T be the triangle with vertices (0,1), (1,0), (0,0). Compute the integral \int\int\frac{sin^{2}(x+y)}{(x+y)} dxdy by making an appropriate change of variables. (Hint: check #24 Section 15.9) Homework Equations Problem 24 in 15.9 of...

Homework Statement Let B be the outside of the unit ball centered at the origin, and let c be a non-zero constant. Consider the mapping where k=1,2,3. Find the image of the set B under the mapping. (Hint: consider the norm of (y1, y2, y3)) Homework Equations The unit ball would be 2...

Let R be the elliptical regin in the first quadrant by x^2/3^2+y^2/2^2=1. x=3u, y=2v, evaluate the area of R. How do I setup the double integral? By this I mean what do I put in for f(g(u,v),h(u,v))jacobian dudv? I know the bounds of u and v and the Jacobian but I not sure what to do for g...

Homework Statement "Let f(x)=x. Define the change of variable y=5x. Then this implies g(y)=y/5. [we have g(y)=f(x(y))=f(y/5) and f(x)=g(y(x))=g(5x)] " Homework Equations N/A The Attempt at a Solution I don't understand the above statement. If we define y=5x, then WHY does it imply...

"Is f(x)=sin(x2) periodic? Answer: no." WHY? I believe "sin" is always periodic? Can someone please explain? Any help is appreciated!