Skip to main content

Questions tagged [change-of-variable]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
13 views

Compactness argument in Browder's proof of the change of variables theorem for multiple integrals

I'm reading Browder's proof of the change of variables theorem for multiple integrals, which begins with the following lemma: I'm having trouble understanding the highlighted part. That is, I don't ...
sebpar's user avatar
  • 345
0 votes
1 answer
17 views

Change of variables by scaling.

I'm going through the book Applied Partial Differential Equations: with Fourier Series and Boundary value problems by Richard Haberman. In chapter 7 this is said about this equation (Next step is ...
reklem2's user avatar
3 votes
3 answers
276 views

Changing equation into elliptic curve

We can change $s^2 = \frac{t^2 - 1}{2t}$ to $$2ts^2 = t^2 - 1$$ $$16t^2s^2 = 8t^3 - 8t$$ and making the change of variables $(x, y) = (2t, 4ts)$, we get the elliptic curve $y^2 = x^3 - 4x$ Can we do ...
Ravikanth Athipatla's user avatar
0 votes
0 answers
28 views

Using the Chain Rule to Switch Between Coordinate Axes in a PDE

I have a system of PDEs that are written in terms of two coordinates: $(x,z)$ which are the usual Cartesian coordinates and $(l,n)$ which are tangential and normal components to some deformable ...
Mjoseph's user avatar
  • 1,019
0 votes
0 answers
40 views

Change of variable in the Besov norm

Assume that it is true for $t =1, $ I wonder how the change of variable leads to the following estimate : where: $B^{sq}_r$ is the Besov space defined using the dyadic decomposition. Let $\varphi \...
A. PI's user avatar
  • 639
1 vote
1 answer
27 views

Show that $\langle(f\circ\varphi_{\lambda})k_{\lambda}, (g\circ\varphi_{\lambda})k_{\lambda}\rangle=k_{\lambda}(\lambda)\langle f,g\rangle.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
Anacardium's user avatar
  • 2,612
0 votes
0 answers
34 views

Why is this proof about integration correct? [duplicate]

I have already asked about this particular integral, but I am not sure if this reasoning makes sense. From the equality $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-x^2-y^2}\ dxdy=\int_{-\infty}...
MSU's user avatar
  • 185
0 votes
1 answer
121 views

A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz

The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being $$ \frac{1}{a_1\, a_2 \, \cdots\, a_n} = \int_0^1 \int_0^{1-x_1} \cdots \int_0^{1-x_1 -x_2 ...
Noix07's user avatar
  • 3,679
1 vote
0 answers
51 views

Change of Variable Why Does It Have To Be Monotonic?

I understand, that for the cases where there are multiple solutions to $y = g^{-1}(x)$, then we need to account for that transformed density from the x distribution to the y distribution. That is, I ...
BurgerMan's user avatar
0 votes
1 answer
66 views

How to explain why $ \lim _{m+1 \rightarrow \infty} x^m=\lim _{m \rightarrow \infty} x^m=\lim _{n \rightarrow \infty} x^n$ here?

I am reading Tao's Analysis I for my own pleasure, and in this example he gave in his section Why do Analysis I was confused what the (non-rigorous) reasoning employed there could be, where $n = m+1$: ...
Princess Mia's user avatar
  • 3,029
2 votes
1 answer
32 views

Expressing integral based on the change of variables in double integral

Let B be the region in the first quadrant bounded by the curves $xy=1$,$xy=3$,$x^2−y^2=1$, and $x^2−y^2=4.$ Evaluate $\iint_{B} (x^2 + y^2 ) dx dy$ using the change of variables $u=x^2−y^2$,$v=xy.$ ...
Homer Jay Simpson's user avatar
1 vote
1 answer
61 views

Evaluate the product of (j^n + 1) for j in a finite field

I stumbled across an exercice and this product came up, with the claim: $$\displaystyle\prod_{i \in F_{p}}(i^n+1) = \left\{ \begin{array}{ll} 0 & \mbox{if }\; \dfrac{p-1}{\gcd(p-1, n)} ...
Bij2u's user avatar
  • 104
1 vote
0 answers
31 views

Change of variable in functional Fokker-Planck equation

Starting from the following (steady-state) functional Fokker-Planck equation for the probability of a path $\{\phi(t)\}_{t=0}^T$ \begin{equation} 0 = \frac{\delta}{\delta{\phi}} \left\{ \frac{\...
Ludens's user avatar
  • 71
0 votes
0 answers
62 views

Change of variable in gamma function

This is a continuation of my question here This one is about calculus and change of variables. Instead of continuing the discussion in comments I decided to ask a new question. Before the change of ...
zeynel's user avatar
  • 447
7 votes
5 answers
178 views

To integrate $\int_0^{2\pi}\sqrt{\theta^2+1}\ d\theta$, why choose the change of variables $u=\theta+\sqrt{\theta^2+1}$?

In order to find the length of the curve $r=\theta,\ \theta\in[0, 2\pi]$, the integral that must be solved is $$\int_0^{2\pi}\sqrt{\theta^2+1}\ d\theta$$ For which my proffesor opted to use the ...
MSU's user avatar
  • 185
-1 votes
2 answers
69 views

Changing variables in double integral.

To evaluate the integral $$\int_{0}^{1}\int_{0}^{1-x}(y-x)\sqrt{x+y}dydx$$ I have to change the variables to $u=y-x$ and $v=x+y$. I know that $$x=-\frac{u}{2}+\frac{v}{2},\qquad y=\frac{u}{2}+\frac{v}{...
mvfs314's user avatar
  • 2,082
2 votes
0 answers
56 views

Variable change in Copula for the joint pdf of correlated random variables

Let $f_{X,Y}(x,y)$ be the joint probability density of correlated random variables $X$ and $Y$ based on a Copula $C$ (Gaussian in my case) where $f_X(x)$ and $f_Y(y)$ are the marginal probability ...
Yakari Dubois's user avatar
0 votes
1 answer
60 views

Find the transformation of $\iint y^2$ inside $y=x, y=2x, xy=2, xy=4$

I am stumped on this math problem. Find an appropriate change of variables u=u(x,y), v=(x,y) of $$ \iint_r y^2 dA $$ when A is given by $$ y =x, y=2x, xy=2, xy=4 $$ My current attempts start well but ...
nolanpestano's user avatar
0 votes
0 answers
16 views

Asymptotic analysis of a linear advection-diffusion equation

Consider the linear advection-diffusion equation for $t,x>0$ \begin{equation} \frac{\partial c}{\partial t} + f(x)\frac{\partial c}{\partial x} = \varepsilon \frac{ \partial^2 c}{\partial x^2} \tag{...
Giraffes4thewin's user avatar
0 votes
0 answers
36 views

Unsure about this changing variables for a limit question. [duplicate]

Show that $$\lim_{x\to 0}\frac{\sin(\frac{x}{2})}{\frac{x}{2}}=1$$ Now I understand that we can change this limit, by computing $z=\frac{x}{2}$, so that the function becomes $$\frac{\sin(z)}{z}$$ but ...
Luke's user avatar
  • 99
0 votes
1 answer
52 views

Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$

Say I have a smooth vector-valued function $h:\mathbb{R}^n\to\mathbb{R}^n$ and a smooth diffeomorphism $\varphi:\mathbb{R}^n\to\mathbb{R}^n$. Consider the gradient of the composition $h\circ\varphi^{-...
Stuck's user avatar
  • 1,734
0 votes
2 answers
58 views

Is there a relation between Fubini's theorem and change of variable theorem?

In an exercise, it asks to use the change of variable theorem to calculate a double integral, but then it asks to redo the work using Fubini's theorem. Is there a way to benefit from previous work? In ...
Alia's user avatar
  • 79
0 votes
1 answer
46 views

Estimating the parameters of an ellipse (part 3)

This post is a follow up of this and this previous ones. I've found an explanation for the following formulas \begin{equation} \hat{\ell}_1 \triangleq 2\sqrt{\hat{\Lambda}_{11}} \qquad \hat{\ell}_2 \...
matteogost's user avatar
0 votes
1 answer
48 views

Change limits of double integral

Let $0<x<a$ and $0<y<a$ if I want to calculate the area of that square the integral is $$\int_0^adx \int_0^ady=a^2$$ now if I want to change variables to sum and difference: $$S= x-y$$ $$s=...
Sagigever's user avatar
  • 1,430
1 vote
0 answers
53 views

Transform to a unit domain each integral in a 5-D integral.

I want to confirm if i can change the integration domains of any 5-D integral to $[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]]$ if the original domains are $[[a,b],[c,d],[e,f],[g,h],[i,j]]$, for $a,b,c,d,e,f,...
Henrique's user avatar
1 vote
1 answer
45 views

function of random variables - change of variables

Consider two independent random variables $X \sim p_X$, $Y \sim p_Y$. Let $Z$ be a deterministic function of $X,Y$, $z = g(x,y)$, with distribution $Z \sim p_Z$. Does the following identity hold for a ...
Rostam22's user avatar
  • 504
0 votes
1 answer
57 views

How to change coordinates of a differential operator?

Say for example, I start with $\frac{\partial}{\partial x}$ and want to change x to the spherical coordinate $$x = \rho\sin(\phi)\cos(\theta)$$ I know this isn't correct, by my brain immediately goes $...
Researcher R's user avatar
2 votes
1 answer
97 views

Weird change of variables (?) I would like to understand formally

In this paper (bottom of p. 4), the authors state the following (I've added all definitions below): Given the deterministic mapping $z=g_{\phi}(\epsilon, x)$ we know that: $q_\phi (z|x)\Pi_i dz_i = p(...
Anon's user avatar
  • 1,791
1 vote
1 answer
42 views

Why is this integral not invariant under change of variables? (differential entropy function / continuous entropy)

Wikipedia states the following (can be found at the bottom of this section): However, differential entropy does not have other desirable properties: It is not invariant under change of variables, and ...
Anon's user avatar
  • 1,791
1 vote
1 answer
98 views

Transformation of a random variable vs. joint transformation of several random variables

Let $X \sim f_X(x)$ and $Y = g(X)$. If $g(X)$ is a differentiable, monotonic function with inverse such that $X = g^{-1}(Y)$ then the PDF of $Y$ can be described: $$ f_Y(y) = f_X(g^{-1}(y)) \bigg| \...
Joseph's user avatar
  • 373
0 votes
1 answer
29 views

finding a bijection with conditions

$x$ and $y$ are variables in the set $\mathbb R$. With bijective functions $f$ and $g$, $x$ and $y$ respectively goes to $a$ and $b$ which is in the set $[-1, 1]$. So, $f(x)=a$ and $g(y)=b$.For ...
Zjjorsia's user avatar
  • 143
1 vote
0 answers
58 views

If $f_n(x) \to f(x)$ in the Schwartz space, $f_n(Ax) \to f(Ax)$ for any invertible matrix $A$?

Let $f_n$ be a sequence of Schwartz functions on $\mathbb{R}^N$ converging to another Schwartz function in the Schwartz space $\mathcal{S}(\mathbb{R}^N)$. Now, let $A : \mathbb{R}^N \to \mathbb{R}^N$ ...
Keith's user avatar
  • 7,829
1 vote
1 answer
66 views

Textbook says to integrate a fraction using 'Taylor's formula'?

I don't understand the solution my textbook gives for this problem: $$ \int \! \frac{x^3}{(x+1)^5} \, \mathrm{d}x $$ I thought it had to be done with partial fractions, but I couldn't get it right, ...
user avatar
0 votes
1 answer
78 views

$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$ provided the first of these integrals exists. Munkres Analysis on Manifolds

I am reading "Analysis on Manifolds" by James R. Munkres. (a) Show that $$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$$ provided the first of these integrals exists. ...
佐武五郎's user avatar
  • 1,210
-1 votes
1 answer
36 views

Suppose X,Y are iid normal distribution with $\mu = 0$, what is the distribution of ratio of absolute value of X to Y?

Let X and Y be iid normal distribution with mean $\mu = 0$, let $Z=\frac{|X|}{|Y|}$, what is the distribution of Z? I'm thinking of using the distribution method but I'm stuck by the integration. I'm ...
Chang Henry's user avatar
0 votes
0 answers
61 views

Linear transformation of multivariate normal

A well known fact exists which is that if a multivariate normal distribution undergoes a linear transformation it's also multivariate normal. There are two proofs I have seen, If the transformation is ...
maxical's user avatar
  • 603
1 vote
1 answer
38 views

The last part of Step 5 in the proof of change of variables theorem (Lemma 19.1) in "Analysis on Manifolds" by James R. Munkres.

I am reading "Analysis on Manifolds" by James R. Munkres. I cannot understand the last part of Step 5 in the proof of change of variables theorem (Lemma 19.1) https://archive.org/details/...
佐武五郎's user avatar
  • 1,210
0 votes
0 answers
43 views

This corollary tells us that any theorem we prove about extended integrals has implications for ordinary integrals. (Munkres Analysis on Manifolds)

Corollary 15.5. Let $S$ be a bounded set in $\mathbb{R}^n$; let $f:S\to\mathbb{R}$ be a bounded continuous function. If $f$ is integrable over $S$ in the ordinary sense, then $$(\text{ordinary})\int_S ...
佐武五郎's user avatar
  • 1,210
1 vote
2 answers
61 views

Rotating the axes and changing variables

Without using matrix inversion, what is the easiest way to show that, for variables $u$ and $v$ corresponding to rotating the $x-y$ axes by $\theta$, we have $$ x=u\cos \theta -v \sin \theta\\ y=u\sin\...
sam wolfe's user avatar
  • 3,435
0 votes
0 answers
29 views

Understanding Probability Formulation in paper: Need Help with Original Formulation Before Change-of-Variable Technique

I am trying to understand the probability formulation in this paper. This (and all other papers on this subject) refer to having applied the chage-of-variable technique to arrive at the probability ...
econometrically_challenged's user avatar
0 votes
1 answer
24 views

Is the solution of a quasilinear pde unique if you solve it by transforming it into an ode?

$xu_x + yu_y = 4u$ on the unit disk in $\mathbb{R}^2$. With boundary condition $u=1$ on the boundary. And got $u = (x^2+y^2)^2$ by solving it by change of variables. How do I show it is unique? And is ...
Pontus erikssen's user avatar
0 votes
0 answers
32 views

Basic linear partial differential equations

In the book of Treves “Basic linear partial differential equations”, at page 169, the author writes: “We are going the make the following assumption $$ (19.18) \qquad \hspace{6em} (grad_{\eta} P_{m})(...
Julie14's user avatar
0 votes
1 answer
59 views

How to deduce the new distance function of a manifold after a transformation of coordinates?

I have a 3D manifold $\Omega_3$ that is $\Omega_3=S^1 \times S^2$ for which I know its distance function $d_{\Omega_3}$ (although I don't know its metric tensor): $$d_{\Omega_3} \left( (\theta_\mathrm{...
Balfar's user avatar
  • 163
0 votes
1 answer
69 views

Change of variables and the expected value

We have a non-negative random variable $z$. We apply a simple location/scale transformation with $a$ and $b$ being the transformation parameters, respectively. I am trying to show that this change-of-...
user avatar
1 vote
1 answer
121 views

What does $\int_{B_r} h$ mean? Problem 3-41 in "Calculus on Manifolds" by Michael Spivak. $\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}$.

I am reading "Calculus on Manifolds" by Michael Spivak. I solved (a) and (b). Then, I proved $\int_C h=\int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} rg(r,\theta)d\theta dr$ holds. My question ...
佐武五郎's user avatar
  • 1,210
2 votes
1 answer
64 views

Solve PDE with condition

\begin{equation} \begin{cases} \sqrt{x}u_x - \sqrt{y}u_y = u^2 \\ u_{x=y} = \phi (y) \end{cases} \end{equation} I want to change u(x,y) to v$(\tau, s )$ : \begin{equation} \...
innocent_01_10's user avatar
0 votes
0 answers
67 views

How to solve Problem 3-39 in "Calculus on Manifolds" by Michael Spivak? (Change of variable Theorem, Sard's Theorem)

I am reading "Calculus on Manifolds" by Michael Spivak. I tried to solve Problem 3-39 but I could not solve it. My attempt: By Theorem 3-14, $g(B)$ has measure $0$. Since $g$ in Theorem 3-13 ...
佐武五郎's user avatar
  • 1,210
0 votes
0 answers
27 views

$h(D\times\{x^n\})\subset\mathbb{R}^n$ and the author wrote $\int_{h(D\times\{x^n\})} 1 dx^1\cdots dx^{n-1}$. Is this ok? Calculus on Manifolds Spivak

I am reading "Calculus on Manifolds" by Michael Spivak. In the proof of Theorem 3-13 (Change of variable), the author wrote as follows: Let $W\subset U$ be a rectangle of the form $D\times [...
佐武五郎's user avatar
  • 1,210
0 votes
0 answers
60 views

How to determine a change of variable that makes the Hessian matrix diagonal?

Let say I have a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ that depends on two variables $\mathbf{x}=[x, y]$. The hessian matrix $H$ associated to $f$ $$ H(\mathbf{x}):= \begin{pmatrix} \...
duc4rm3's user avatar
  • 65
1 vote
0 answers
32 views

Why did the author calculate $D_n(g^n\circ h^{-1})(h(a))$ to conclude $k'(h(a))=I$? ("Calculus on Manifolds" by Michael Spivak.)

I am reading "Calculus on Manifolds" by Michael Spivak. In the proof of Theorem 3-13 (Change of variable), the author wrote as follows: Since $$(g^n\circ h^{-1})'(h(a))=(g^n)'(a)\cdot [h'(a)...
佐武五郎's user avatar
  • 1,210

1
2 3 4 5
23