Questions tagged [change-of-variable]

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

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \hat{\ell}_1 \triangleq 2\sqrt{\hat{\Lambda}_{11}} \qquad \hat{\ell}_2 \...
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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 ...
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1 vote
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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 $A : \mathbb{R}^N \to \mathbb{R}^N$ ...
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1 vote
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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, ...
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$\int_{\mathbb{R}^2} e^{-(x^2+y^2)}=[\int_{\mathbb{R}} e^{-x^2}]^2,$ 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. ...
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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 ...
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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 ...
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1 vote
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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/...
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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 ...
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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 ...
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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-...
1 vote
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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 ...
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Solve PDE with condition

$$\begin{cases} \sqrt{x}u_x - \sqrt{y}u_y = u^2 \\ u_{x=y} = \phi (y) \end{cases}$$ I want to change u(x,y) to v$(\tau, s )$ : \...
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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 ...
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