Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [change-of-variable]

This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals

1
vote
0answers
14 views

Flux of $\vec F=\langle x^2,-y^2,z^2\rangle$ over region between two parallel planes in the first octant

Use the divergence theorem to compute the net outward flux of $\vec F(x,y,z)=x^2\,\vec\imath-y^2\,\vec\jmath+z^2\,\vec k$ over the region $D$, where $D$ is the space between the planes $z=3-x-y$ and $...
1
vote
1answer
34 views

Change of variables in QCLP

Is there any change of variables that makes the following optimization problem easier to solve? \begin{align} \max_{x\in\mathbb{R}^n,t\in\mathbb{R}}\quad & c^\top x,\\ \mbox{s.t.}\quad\quad & ...
1
vote
2answers
44 views

Change of variables for double integral

Problem: Using the change of variables $$x=\sqrt2u-\sqrt\frac{2}{3}v,y=\sqrt2u+\sqrt\frac{2}{3}v$$ Calculate the double integral $$\iint_Rx^4-2x^3y+3x^2y^2-2xy^3+y^4dA$$ where $R$ is the region bound ...
0
votes
0answers
18 views

A question on a change of coordinate related to the Gibbons-Hawking ansatz

This question comes from reading the article "Hyperkähler Quotient Construction of BPS Monopole Moduli Spaces", by Gibbons, Rychenkova and Goto. In $\mathbb{H} \simeq \mathbb{R}^4$, with its standard ...
1
vote
1answer
57 views

The $n^{th}$ derivative in change of variable

I want to change variable $x$ in my differential equation to $t=g(x)$. I would like to create pattern for $n^{th}$ new derivative $\frac{d^ny}{dt^n}$ for any transformation depends on old derivatives $...
0
votes
0answers
12 views

Change of variables to double integration over infinite domains

While writing julia scripts, I came across following improper integral which I cannot figure out how to apply change of variable method on it to turn it into finite domain such as [0, 0] to [1, 1]: $$ ...
3
votes
0answers
65 views

Post-Composition By Diffeomorphism And Integrability

Let $d,D$ be positive integers, $p \in [1,\infty)$, $U$ an (non-empty) open subset of $\mathbb{R}^D$, and suppose that $f$ is: In the Bochner-Lebesgue space $L^p_{\mu}(\mathcal{B}(\mathbb{R}^d);\...
0
votes
0answers
23 views

Is there a better way to evaluate the flux of a sphere not centered at the origin?

I am supposed to evaluate the flux with the vector field: $F=x^{2}i+y^{2}j+z^{2}k$, outward across the boundary of the given solid region the ball: $(x-2)^2 + (y)^2 +(z-3)^2 \le 3^2$ The reason i'm ...
0
votes
1answer
36 views

'$\wedge$' in the limit of a double integral

I am confused with how to change the variables of the double integral below after invoking Fubini's Theorem; I am not very familiar with the wedge 'meet' notation which seems to arise. How is it that ...
0
votes
1answer
37 views

Geometry of mean value theorem? [duplicate]

Usually the mean value theorems (including the general one) is extended from the rolle theorem by introducing a function. Does this function has anything to do with geometric transformation? Can ...
0
votes
1answer
29 views

Change of variables in triple integral

Let $D$ be the region in $xyz-$space defined by inequalities $1 \le x \le 2, 0 \le xy \le 2 $ and $0\le z \le 1$. I want to evaluate $\displaystyle \int\int\int_D (x^2y + 3xyz) \text{dxdydz}$ by ...
3
votes
2answers
96 views

solving a $p$-adic integration involving maximum function

I am always struggling when it comes to dealing with maximum functions. I am trying to find the solution to this integral $$\int_{\mathbb{Z}_p^3}||xy,xz,yz||_p^sd\mu(x,y,z),$$ where $||xy,xz,yz||_p^...
1
vote
0answers
30 views

What is the formal definition of change of variable?

Change of variable is a very common and elementary technique in introductory mathematics. But it do know the formal definition of it. First of all What is the formal definition of variable in ...
1
vote
3answers
97 views

problem in $p$-adic integration

I am working on the $p$-adic integration and I am trying to find how to integrate $$\int_{\mathbb{Z}_p^2}||x,y||_p^sd\mu (x,y),$$ where $d\mu$ is the haar measure and $||x,y||_p^s=\sup\{|x|_p^s,|y|_p^...
0
votes
1answer
29 views

Affine Change-of-Variables Formula in $p$-adic Haar measure integration.

Let $p$ be a prime number, let $V$ be an arbitrary non-empty subset of $\mathbb{Z}_{p}$, and let $d\mu$ be the Haar measure on $\mathbb{Z}_{p}$ subject to the normalization $\int_{\mathbb{Z}_{p}}d\mu=...
1
vote
1answer
39 views

Change of variable theorem for the curvilinear integral

I have to proof the change of variable theorem for the curvilinear integral, but I have no idea about how to do that, and I haven't found information in Google. The theorem states: Let be $\Omega$ $\...
1
vote
2answers
61 views

Rudin Principles of Mathematical Analysis Chapter 10, Exercise 8

I'm working on exercises of chapter 10 in Baby Rudin. I refer to R. Cooke's solutions manual to Baby Rudin while I'm solving those exercises.(https://minds.wisconsin.edu/handle/1793/67009) But I ...
0
votes
0answers
16 views

The Change-of-Variable Theorem for Lebesgue Integral

My question concerns an application of the standard integration by substitution tecnique. I am aware of the following result. Let $V \subseteq \mathbb{R}^d$ be an open set and $\varphi \colon V \...
1
vote
1answer
27 views

Differential Equation using change of variables $(xy+2xy(\ln y)^2+y\ln y)\mathrm{d}x+(2x^2\ln y+x)\mathrm{d}y=0$

Solve this differential equation using given change of variables $(xy+2xy(\ln y)^2+y\ln y)\mathrm{d}x+(2x^2\ln y+x)\mathrm{d}y=0$ , change of variable $t=x\ln y$ I get that: $t' = \mathrm{d}t/\mathrm{...
0
votes
0answers
30 views

Transforming a stochastic Fokker Planck equation in a deterministic one

I have the following stochastic Fokker Planck equation, coming from a double well potential $d p = \left[ (x^3 - a x) \frac{\partial p}{\partial x} + (3 x^2 -a ) p + \frac{1}{2} \sigma^2 \frac{\...
1
vote
1answer
42 views

Changing the variable in a double sum

I want to chase up some aspects of the question Change of variable in a double sum, which so far has not received an accepted answer. To recall what is discussed there, we have the double sum: $$\...
0
votes
1answer
23 views

Evaluate a double integral using change of variables

Evaluate $$\int_{x=0}^{1/2}\int_{y=x}^{y=1-x}\frac{y-x}{(x+y)^2\sqrt{1-(x+y)^2}}dydx$$ using change of variables $r=x+y$, $s=y-x$. I found the Jacobian of transformation to be $J^{(x, y)}_{(r, s)...
1
vote
0answers
51 views

Change of variables in stochastic PDE

I have the following stochastic partial differential equation (SPDE): $d v = -\mu \frac{\partial v}{\partial x} dt + \frac{1}{2} \frac{\partial^2 v}{\partial x^2} dt - \sqrt{\rho} \frac{\partial v}{\...
0
votes
1answer
30 views

Finding the change of variables that transforms given domain into another one

I was practicing some integration problem until I came upon this one. To be honest I am quite confused as to how to proceed with these question: Let find the change of variables that transforms the ...
2
votes
0answers
84 views

How to solve the given Partial Differential Equation?

I need help on solving the following PDE by Maple or Mathematica: \begin{equation} D \Big( D - \frac{1}{\alpha} \frac{\partial}{\partial t} \Big) f + \frac{\sin\theta}{\alpha} \bigg( \frac{\...
0
votes
0answers
37 views

Find the integral $\int_{R}(x+y)dA$.

I am trying to attempt the following question: But I am confused how to determine the bounds. I was thinking $u=xy$, but I am not sure.
0
votes
0answers
28 views

Change of Variables For Multiple Sums?

Say you have an infinite multi-sum like $$\sum_{n_1=-\infty}^{\infty}\sum_{n_2=-\infty}^{\infty}\sum_{n_3=-\infty}^{\infty}f(n_1,n_2,n_3)$$ Presume for the moment this has all of the nice properties ...
0
votes
1answer
41 views

Double integral (Cauchy principal value integral)

I am currently reading a book of Hilbert transforms and I have found with the following equality: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{isx}\int_{-\infty}^{\infty^*}\frac{\phi(y)}{x-y}\text{ d}y\...
0
votes
2answers
42 views

How to transform $\int_0^\infty x(1+x^2)^{-2}dx$ to $0,1$ as limits of the integral?

Consider $\displaystyle\int_0^\infty x(1+x^2)^{-2}dx$ If I consider $y=\frac{1}{1+x^2}$, then $dy=\frac{-dx}{(1+x^2)^2}$ and $x=\pm\sqrt{\frac{1}{y}-1}$ and then $\displaystyle\int_1^0 \mp\sqrt{\...
1
vote
2answers
66 views

Why is $\int_0^1e^{e^x}dx$ equal to $\int_1^e{e^uu^{-1}}du$?

Why is this equality $\displaystyle\int_0^1e^{e^x}dx=\int_1^e{e^uu^{-1}}du$? I don't see the change of variable that was used to pass from one integral to the other? Could someone please explain? ...
1
vote
0answers
31 views

How to find the volume of the part of a sphere that protrudes from a square prism?

I have a square prism with width $W$ and height $H$. So it's $W \times W \times H$, where in this case $W$ is less than $H$. The center of mass of the prism is at the origin and the center of a sphere ...
0
votes
0answers
28 views

Finding correct projective change of coordinates $M$.

Find a $3 \times 3$ matrix $M$ such that, under the change of varibles $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = M^{-1}\begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix},$...
0
votes
0answers
18 views

How to change variables in a PDE using Maple?

I have the following equation $$ \frac{\partial m(t, x)}{\partial t}+\frac{\partial}{\partial x}\left(\left[u(t, x)^2-\left(\frac{\partial u(t, x)}{ \partial x}\right)^2\right]m(t, x)\right) = 0 $$ ...
0
votes
3answers
47 views

Laplace Transform: Indeterminate Form in Definite Integral Change of Variables Calculation

I was trying to find the Laplace transform of $e^{3t}$: $$\int^\infty_0 e^{3t}e^{-st} \ dt = \int_0^\infty e^{3t - st} \ dt = \lim_{x \to \infty}\int_0^x e^{3t - st} \ dt$$ So if we then attempt to ...
0
votes
1answer
16 views

change of variables with limits

If I have the following function: $$ 1 = \int_0^{R_M}g_s(R)R^{-1}dR$$ where $x_M = R_M/\sigma_0$, and $x = R/\sigma_0$ how would I perform sub. of variables on the limits? I know I would have ...
1
vote
0answers
32 views

change of variable problem

Let $\Delta\geq 1$ be a constant. I need to prove that $$ 2\int_1^T x\int_{T/2x^2}^{2T/x^2}\left(\frac{\sin(\frac{\Delta}{4}\log\frac{2\pi}{t})}{\frac{\Delta}{4}\log\frac{2\pi}{t}}\right)^2 dt\,dx = \...
0
votes
0answers
36 views

Prove that there is a point $c\in(a,b)$ such that $f'(c)=0$.

Let $f:[a,b]\rightarrow\mathbb{R}$, $0<a<b$, a function that is differentiable and bijective such that $\int_{f(a)}^{f(b)}f^{-1}(x)dx=0$. Prove that there is a point $c\in(a,b)$ such that $f'(c)=...
11
votes
2answers
192 views

Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation

Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below The statement of the theorem: Theorem Among all second-order homogeneous PDEs in two dimensions ...
0
votes
1answer
16 views

How to do change of variables of a j.p.d.f with N pdf(s)?

Given that I have a joint probability distribution(jpdf) of: $$P(x_1,...,x_N) = C_N \prod_{j=1}^{N}(1-x_j)^a(1+x_j)^b \prod_{1\leq j <k \leq N} |x_k - x_j|^2$$ where $$\prod_{1\leq j <k \leq N} |...
0
votes
1answer
26 views

Change Variables in a Multiple Integral

Let $D$ be the region that's bound by $y=x^2, y=2x^2, x=y^2, x=3y^2$. $D$ corresponds to the region $E$ where $u=\frac{x^2}{y}$ and $v=\frac{y^2}{x}$. Sketch $D$ and $E$ Find the solution ...
0
votes
1answer
16 views

change of variable for integral to calculate posterior distribution

I'm working through an example which can be found here (p. 36), if someone is interested. I have an integral of the form: $$P(x|\mu)=\int d\sigma P(x|\mu, \sigma)P(\sigma)=\int d\sigma \frac{1}{\sqrt{...
0
votes
1answer
20 views

Change of Variable Bounds of Integration

One of my practice problems asks us to compute the volume of the region enclosed by the unit sphere $\{(x,y,z): x^2+y^2+z^2=1\}$ and the set $\{(x,y,z): z= |x|\}.$ My first intuition is to use ...
2
votes
0answers
25 views

Integration by substitution in $f: \mathbb{R^2} \to \mathbb{R}$ with Jacobian method proof

I've searched everywhere on Google for a somewhat formal proof for the integration by substitution in double integral with the Jacobian determinant method but I can't find any. Anyone knows any ...
1
vote
1answer
35 views

Multiple integration with constraints on variables

I have a function $(x_1, x_2)\mapsto g(x_1, x_2)$ where $x_1$ and $x_2$ are both 3D vectors. I would like to integrate function $g$ over the whole space but with some constraints on $x_1$ and $x_2$ ...
1
vote
0answers
23 views

>Calculate the volume of the $n$-dimensional ball using change of variables.

Calculate the volume of the $n$-dimensional ball using change of variables. My attempt. Lemma 1. For each $k \in \mathbb{N}$ we have $$\int_{0}^{\pi}\sin^{k}xdx = \frac{k-1}{k}\int_{0}^{\pi}\sin^...
1
vote
0answers
92 views

About Theorem 6.19 on pp.132-133 in “Principles of Mathematical Analysis” by Walter Rudin

Thank you very much, Saaqib Mahmood, for your text. I copied and pasted it: Theorem 6.19 on pp.132-133: Suppose $\varphi$ is a strictly increasing continuous function that maps an interval $[ ...
1
vote
0answers
18 views

Application of change of variables (volume of balls)

Let $A \subset \mathbb{R}^{n}$ be an open set and $f: A \to \mathbb{R}^{n}$ a $C^1$ function with $Df_{a}$ an isomorphism. Show that for each $a \in A$, $$\lim_{r \to 0}\frac{\mathrm{vol} f(B_{r}(a))...
1
vote
0answers
32 views

Analytic posterior likelihood of bayesian logistic regression with uniform prior

I am trying to show that the posterior of the following model is proper. Assuming, $p(\alpha, \beta) \propto 1$, and $n,x, y$ given. $$ \begin{align} p(\alpha, \beta) & \propto p(\alpha, \beta | ...
2
votes
3answers
46 views

Undo this transformation

I have two variables $x$, $y$ and calculate the following: $a = \frac{x}{\sqrt{x^2+y^2}}$, $b = \frac{y}{\sqrt{x^2+y^2}}$ Using $a$ and $b$ is there a way I can derive my original $x$ and $y$?
1
vote
2answers
49 views

Using Hamilton's principle to derive Newton's equations of motion in parabolic coordinates

I have recieved a very hard (optional) assignment on variational calculus, and I have not got a clue where to start other then stating the Euler-Lagrange equations. Here is the problem: According to ...