Change of variables Definition and 219 Threads

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...

Homework Statement I will use d as a partial derivative symbol du/dt-du/dx=0 use a change of variables to solve The Attempt at a Solution v=ax+bt w=cx+dt so then I use the chain rule (du/dv)(dv/dx)+ (du/dv)(dv/dt)+(du/dw)(dw/dx)+(du/dw)(dw/dt) ok then you are supposed to plug...

suppose i want to find the following integral: 7 \intx 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(u2+4u+5)(2u+4)du u0 u1 \int2u3+12u2+26u+20 du u0 the indefinite integral is...

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!

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 Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations: x=(1/3)(4u-v) y=(1/3)(u-v) Homework Equations The Attempt at a Solution 3y=u-v 3y+v=u place this into the x equation 3x=4(3y+v)-v 3x-12y=3v...

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...

I am having a problem with change of variables problems in GENERAL. What I don't understand at this point is how one determines what the lower and upper bounds are in terms of the transformation variables. That is, if you have some weird shaped region that can't be evaluated as either a type I...

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 Solution: Define x=G(u)=(u1,u2/2,u3/3,u4/4) ∫ ∫ ∫ ∫ f(x1,2x2,3x3,4x4) d4x C by change of variables theorem...

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 <---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...