Change of variables Definition and 208 Threads

Homework Statement Consider the PDE: 2dz/dx - dz/dy = 0 How can I show that if f(u) is differentiable function of one variable, then the PDE above is satisfied by z = f(x +2y)? Also, the change in variables t = x+2y, s=x reduces the above PDE to dz/dt = 0. But how can I show this? The...

Homework Statement using change of variables, find the region bounded by y = x, y = 2x, xy = 1, xy=2 Homework Equations The Attempt at a Solution i know i have to introduce the variables u, v the problem is i don't understand how to introduce them i tried read the textbook but...

[note: Ux=∂U/∂x, Uy=∂U/∂y] Example: Solve the partial differential equation 2Ux + 3Uy + U = 0 by using the change of variables V(x,y)=ln[U(x,y)] Solution: Vx = Ux/U Vy = Uy/U 2Ux + 3Uy + U = 0 Dividing both sides by U, we have 2Ux/U + 3Uy/U + 1 = 0 => 2Vx + 3Vy +1 = 0 => 2Vx + 3Vy...

Clarification of "change of variables" for multiple integration This isn't really a question about a specific math problem, but rather for the change of variables of multiple integration as a whole. When you change variables you have to multiply the new expression by the jacobian of the new...

So, I've got a problem understanding the "algorithm" for changing variables in a more-than-one-dimensional integral. For the two-dimensional case, I've got a specific problem that I'm looking at: \int^{a}_{0}\left(\int^{2a-x}_{x}\frac{y-x}{4a^2+(y+x)^2}dy\right)dx which I assume is an...

\int_{c_1}^{c_2} \int_{g_1 (x)}^{g_2 (x)} f(x,y) dy dx If f(x,y) is function such that it is not easily integrable, if we wanted to switch the bounds of integration so that h1(y) = g1(x) , same for g2(x), what would be the general way to rewrite the bounds? Would it involve inverse...

Homework Statement Let D be the region bounded by x=0, y=0, x+y=1, x+y=4. Using the change on variables x=u-uv, y=uv and the jacobian, evaluate the double integral double integral of dxdy/(x+y) Homework Equations answer is 3The Attempt at a Solution i drew the graph and found the...

Homework Statement Evaluate \int\inte^xy dA, where R is the region enclosed by the curves: y/x=1/2 , y/x=2, xy=1, and xy=2. Homework Equations None? The Attempt at a Solution I have the region graphed and I'm currently working on acquiring the change of variables functions in x and...

Homework Statement I just need to be able to change a vector field from spherical to cartesian The question is about verifying stokes theorem (curl theorem) for a given vector field within and on a given path. It says not to use spherical coordinates, but the vector field is given in...

I needed to evaluate the following integral (for constructing Chebyshev polynomials by an orthogonalisation process), but I just discovered that I'm having an issue with the change of variable technique:P The specific integral itself is unimportant as to the issue I'm having, but by means of an...

Homework Statement Let S = {x∈R^n : x_i ≥ 0 for all i, x_1 + 2x_2 + 3x_3 + ... + nx_n ≤ n}. Find the n-dimensional volume of S. Homework Equations I'm 95% sure that I'm supposed to use the change of variables theorem here. The Attempt at a Solution So far, I have calculated the values for...

Homework Statement So it's been a really long time since I've done any ode/linear algebra and would like some help with this problem. Derive the general solution of the given equation by using an appropriate change of variables 2\deltau/\deltat + 3\deltau/\deltax = 0 The thing that...

Homework Statement Use a suitable change of variable to find the area of the region R bounded by y=x^2, y=4x^2, y=\sqrt{x}, y=\frac{1}{2}\sqrt{x} 2. The attempt at a solution I am trying to first find the inverse transformations {u & v =?

Homework Statement Make the appropriate change of variables and evaluate \int\int _R\(sin(9x^2+4y^2)}\;dA Homework Equations R is bounded by the ellipse 9x^2+4y^2=1 The Attempt at a Solution I can't figure out what the substitution should be I tried u=9x^2 and v=4y^2 and that...

Homework Statement Hi all. I have the 1-D wave-equation, and I wish to make a change of variables, where a = x+ct and b = x-ct. I get: \begin{array}{l} c^2 \frac{{\partial ^2 u}}{{\partial x^2 }} = c^2 \left[ {\frac{{d^2 u}}{{da^2 }}\left( {\frac{{da}}{{dx}}} \right)^2 +...

Homework Statement Evaluate the double integral over R of cos[(y-x)/(y+x)] dA where R is the trapezoidal region with vertices (1,0) (2,0) (0,1) and (0,2). The Attempt at a Solution First I set u=y-x, v=y+x. I have 4 sides in the xy-plane that need to be transformed into the uv-plane...

I thought I grasped coordinate changes well, but now I've run into some problems. Usually I would have some function f(x,y) and transformation equations like s = a*x+b*y . I would apply chain rule and stayed left with new equations in new variables. (old ones get away through...

[SOLVED] Rudin's change of variables theorem Homework Statement Rudin's Principles of Mathematical Analysis Theorem 6.19 (for the Riemann integral case) says Suppose \phi is a strictly increasing continuous differentiable function that maps an interval [A,B] ont [a,b]. Suppose f is Riemann...

Homework Statement Hi, this is not homework exactly, I'm doing some exercises as part of my personal study. I'm analizing linear invariant systems and I'm stuck in an apparently trivial step, please, help. I have these integrals: Homework Equations integral( x(tau)*dtau, from -infinity...

Hi I just cannot understand the following transformation, where \phi(t) is the displacement of an optimal path using standard calculus of variations. All functions are defined between 0 and T. \phi equals zero at 0 and T. r is some discount rate, e it the Euler number, t is time...

Hi, everyone! I have a problem in understading the change of variables in double integrals. Here is an example \int\int x^2+y^2dx dy=\int \frac{x^3}{3}+y^2x dy=\frac{x^3y}{3}+\frac{y^3x}{3}+C_1 but if I first do a change in poral coordinates I get \int\int r^2 r...

Q1: Let S be the region in the first quadrant bounded by the curves xy=1, xy=3, x2 - y2 = 1, and x2 - y2 = 4. Compute ∫∫(x2 + y2)dA. S (Hint: Let G(x,y)=(xy, x2 - y2). What is |det DG|?) Solution: http://www.geocities.com/asdfasdf23135/advcal19.JPG I don't understand the third and...

Q1: Suppose B=[0,1]x[0,2]x[0,3]x[0,4] in R4, and that C=[0,1]x[0,1]x[0,1]x[0,1]. Given that ∫ ∫ ∫ ∫f(x)=d4x=(2pi)4 B What is the value of ∫ ∫ ∫ ∫ f(x1,2x2,3x3,4x4) d4x? C[/color] Solution: Define x=G(u)=(u1,u2/2,u3/3,u4/4) ∫ ∫ ∫ ∫ f(x1,2x2,3x3,4x4) d4x C by change of variables...

1) Find the volume of T bounded below by the cone z=sqrt(x2+y2) and above by the sphere x2+y2+z2=1. Solution: Volume = ∫∫∫ 1 dV = T b d f ∫ ∫ ∫ r (d theta)dzdr (change of variables to cylindrical coordinates) a c e where a=0 b=1/sqrt2[/color] <---I am having a lot of trouble...

[SOLVED] Integration change of variables Homework Statement An electron is confined in an infinite one dimensional well where 0 < x < L with L = 2 x 10^-10m. Use lowest order perturbation theory to determine the shift in the third level due to the perturbation: V(x) =...

Hello everybody First I'd like to thank for the work all of you are developing with this forum. I found it for casuality but I'm sure since now it will be a perpetual partner. I'm a Spanish Physics graduate and I am working about microwave guides and connectors for devices components in...

Homework Statement f(x,y) a function of two variables. x = 2u y = u-3v Using a change of variables, transform the equation (d²f/dx²) + (df/dy) = 0 into the coordinates system {u,v}.Homework Equations We have kind of a replacement teacher for the session and it is his first time giving the...

[SOLVED] Integrating with Change of variables -method Homework Statement Solve by "Change of variables"-method ∫(dx)/(eˆx + eˆ(-x)) Homework Equations ∫f(x)dx=∫f(x(t))*x'(t)dt The Attempt at a Solution ∫(dx)/(eˆx + eˆ(-x))=∫(eˆx)/(1 + eˆ2x) t=eˆx x=ln...

can someone explain 'change of variables' to me? when do you use it, and why? (not to mention HOW you use it!) I'm taking an intro to DE course, and the textbook only mentions the idea in an exercise, not in the text itself. my prof never covered it in class, either. for example, i was trying...

I need guidance regarding PDE. If u have a nonlinear PDE as Ut+Us+a*U*Us*b*Usss=0 where U is function of (s,t) and a,b are constants. by introducing new variable x=s-t we will get Ut+a*U*Ux+b*Uxxx=0 Ut means partial derivative w.r.t time Us means partial derivative w.r.t s. How can we...

I am wondering if someone could help me with the following? I am supposed to show that ∫∫f(x+y)dA evaluated from the triangular region with the vertices (0,0), (1,0) and (0,1) is equal to ∫∫uf(u)du. This triangular region has the equations, x = 0, x = 1, and y = -x + 1. If I set x+y = u...

Hi. I know the title is not very informative. Here's what I'm trying to do: I have f(x,y). I want to perform a change of variables to obtain a pre-defined g(u,v). How can I work out the actual expressions u(x,y) and v(x,y) so that it works out (including the Jacobian as well)? I have a...

Our math professor gave us this take-home project: Consider a solid in the shape of the region D inside the surface x^2 / (z^3 - 1)^2 + y^2 / (z^3 + 1)^2 = 1 If the density of the solid at the point (x,y,z) is x^2 + y^2 + z^2 then determine the mass of this solid. A GOOD CHANGE OF VARIABLES...

Does change of variables generalize to situations other than integration?

hi, i am having difficulty trying to find a change of variables to solve this partial differential equation \frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2} not sure how to pluck out a change of variables by looking at the equation as its definitely not obvious to the...

Am I insane or is this a typo: Consider the nonlinear DE dy/dx = (y-x)^2 + 1 Show that the change of variables, u=x, v = y-x transforms this DE into the seperable DE: dv/du = v^2 dv = dy du=dx dv/du = dy/dx = (y-x)^2 + 1 = v^2 + 1 not v^2 am i wrong ?

There is an example in my textbook which I´m having trouble with. The example is like this. " Find the area of the finite plane region bounded by the four parabolas, y=x^2 , y=2x^2 , x=y^2 , and x=3y^2 The region is called D. Let u=y/x^2 and v=x/y^2 The the region D corresponds to...

Hi, I would like some help with the following question. Q. Let f be continuous on [0,1] and let R be the triangular region with vertices (0,0), (1,0) and (0,1). Show that: \int\limits_{}^{} {\int\limits_R^{} {f\left( {x + y} \right)} dA = \int\limits_0^1 {uf\left( u \right)} } du...

Hi, I have the following integral. \int\limits_{}^{} {\int\limits_R^{} {\left( {\sinh ^2 x + \cos ^2 y} \right)} \sinh 2x\sin 2ydxdy} Where R is the part of the region 0 <= x, 0 <= y <= pi/2 bounded by the curves x = 0, y = 0, sinhxcosy = 1 and coshxsiny = 1. In the hints section...

define L[u] = a \frac{\partial^2u}{\partial t^2} + B \frac{\partial^2 u}{\partial x \partial t} + C \frac{\partial^2u}{\partial x^2} = 0 show that if L[u] is hyperbolic then and A is not zero the transofmartion to moving coordinates x' = x - \frac{B}{2A} t t' = t tkaes L into a...

Hmm, I can't seem to get this double integral transformation: int(limits of integration are 0 to 3) int (limits of int are 0 to x) of (dy dx)/(x^2 + y^2)^(1/2) and i need to switch it to polar coordinates and then evaluate the polar double integral. i sketched the region over which i am...

Hi, I'm having trouble understanding the solution to a question from my book. I think it's got something to do with the chain rule. The problem is to prove the change of variables formula for a double integral for the case f(x,y) = 1 using Green's theorem. \int\limits_{}^{} {\int\limits_R^{}...

Hi, I'm having trouble evaluating the following integral. \int\limits_{}^{} {\int\limits_R^{} {\cos \left( {\frac{{y - x}}{{y + x}}} \right)} } dA Where R is the trapezoidal region with vertices (1,0), (2,0), (0,2) and (0,1). I a drew a diagram and found that R is the region bounded...

let f be continuous on [0,1] and R be a triangular region with vertices (0,0), (1,0) and (0,1). Show: the double integral over the region R of f(x+y)dxdy = the integral from 0 to 1 over u f(u)du I recognize it is a change of variables problem but I'll be damned if I can create a set of...

I asked this in another thread, but I think this forum might be a better place for it (not trying to spam the same question). When deriving the formula for relativistic kinetic energy, we start with KE = \int_{0}^{s} \frac{d(mv)}{dt} ds = \int_{0}^{mv} v d(mv) So I figure that since v =...

Which change of variable should I use to find: \int\int_D (x^4-y^4) dxdy Where D is in the first quadrant with" 1 \leq x^2-y^2 \leq 3, 2\leq xy \leq 3

let be the integral: \int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx i make the change of variable xy=u y=v whose Jacobian is 1/v but then what would be the new limits?...

Hi, I'm not sure how to do this question. Any help would be great. Let B be the region in the first quadrant of R^2 bounded by xy=1, xy=3, x^2-y^2=1, x^2-y^2=4. Find \int_B(x^2+y^2) using the substitution u=x^2-y^2, v=xy. . Use the Inverse Function theorem rather than solving for x...

Hi I'm having trouble with using change of variables. \int\limits_{}^{} {\int\limits_R^{} {f(x,y)dA = \int\limits_{}^{} {\int\limits_{}^{} {f(x(u,v),y(u,v))\left| {\frac{{\partial \left( {x,y} \right)}}{{\partial \left( {u,v} \right)}}} \right|dudv} } } } I've got two examples to...

Hi. I have a problem with a question. Basically, I have an integral that goes from x=0 to x=1, and I'm supposed to make a change of variables like this: Let x = 1 - y^2. The problem I'm having is trying to find the limits of integration after the change of variables. Since y = +/-...