Questions tagged [change-of-variable]
This concerns all problem requesting techniques and tricks about changes of variables in computations of limits as well as integrals.
1,212 questions
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Second Derivative Chain Rule in Matrix Notation
If we have a function $f(x_1,x_2,x_3,x_4)$ and perform a coordinate transformation to $f(y_1,y_2,y_3,y_4)$, then by the chain rule,
$$
\frac{\partial f}{\partial x_1}
=
\begin{bmatrix}\frac{\...
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Change of coordinates in the cotangent space
I am studying differentiable manifolds and I came across the definition of cotangent space. I have a doubt on how we change coordinates in the cotangent space. Let $(A,\varphi)$ and $(B,\psi)$ be ...
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How do I prove the change of variables into polar coordinates using measure theory?
From this answer I have that $ \int_Yf(y)\,\mathrm{d}(g\mu)(y)=\int_Xf(g(x))\,\mathrm{d}\mu(x)$, where $g$ is a map between measurable spaces and $g\mu$ is the image measure.
With $X=[0,r]\times[0,2\...
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How to derive relations between integrals over a triangle without evaluating them explicitly?
Solving a problem I found several integrals that look like
$$I(z) = \int \frac{f(x,y)}{(x y)^{1 + z}} dx dy$$
where the integral is over the triangle $\left\{(x,y) | x>0, y>0, x+y<1\right\}$.
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$\lambda_n$ is the volume of the $n$-dimensional unit ball. Compute $\lambda_n$ in terms of $\lambda_{n-2}$. (Analysis on Manifolds James R. Munkres.)
I am reading Analysis on Manifolds by James R. Munkres.
On pp.168-169:
Exercise 6. Let $B^n(a)$ denote the closed ball of radius $a$ in $\mathbb{R}^n$, centered at $0$.
(a) Show that $$v(B^n(a))=\...
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Surface integration identity
I am confused about the equation
$$
\int_{\partial B(x,t)}u(y)\,d S(y)=\int_{\partial B(0,1)}u(x+t\omega)t^{n-1}\,d S(\omega),
$$
where $B(x,r)$ refers to the ball centered at $x$ with radius $r$. $\...
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Is $g:(-\pi/2,\pi/2)\to (-1,1)$ given by $g(x)=\sin x$ right? ("Analysis on Manifolds" by James R. Munkres.)
I am reading "Analysis on Manifolds" by James R. Munkres.
EXAMPLE 1. If it happens that both integrals in the change of variables theorem exist as ordinary integrals, then the theorem ...
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I think we need to modify the proof of Lemma 19.1 as follows. ("Analysis on Manifolds" by James R. Munkres.)
I am reading "Analysis on Manifolds" by James R. Munkres.
Lemma 19.1. Let $g:A\to B$ be a diffeomorphism of open sets in $\mathbb{R}^n$. Then for every continuous function $f:B\to\mathbb{R}$ ...
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Is it simply because this fact is not needed for the proof of Lemma 19.1, or is it actually the case that $S_i \ne g^{-1}(T_i)$? (Munkres's book.)
I am reading "Analysis on Manifolds" by James R. Munkres.
The author proved $S_i=\operatorname{Support}(\phi_i\circ g)$ is contained in $g^{-1}(T_i)$.
I think in fact, $S_i=g^{-1}(T_i)$ ...
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Is there any easy proof of my proposition? (James R. Munkres "Analysis on Manifolds".)
I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 8.2. Let $A$ be open in $\mathbb{R}^n$; let $f:A\to\mathbb{R}^n$ be of class $C^r$; let $B=f(A)$. If $f$ is one-to-one on $...
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How to solve the definite integral $\int_{-1}^{1} \sqrt\frac{|x+1|}{|x|+1} \, {\rm d} x$?
I have the definite integral
$$ I_1 := \int_{-1}^{1} \sqrt\frac{|x+1|}{|x|+1} \, {\rm d} x $$
and I am having trouble solving it. The following is a plot of the integrand, $x \mapsto \sqrt\frac{|x+1|}{...
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Transformation of partials in accelerating reference frame
I am working on a traveling wave problem and I am struggling to convince myself I've transformed to partial derivatives in a co-moving reference frame correctly. My previous post has a more background ...
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Accelerating reference frame in a traveling wave
I am working with a traveling wave model common in population genetics (see, for example, Neher and Hallatschek 2012). In these we model the evolution of a 1D fitness ‘wave’ as a population ...
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Why this particular change of variables in this PDE?
I am trying to study some methods of resolution of PDEs, for my exam of mathematical methods for physics. Currently I am reading “A guide to mathematical methods for physicists” (volume 2) by Petrini, ...
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Zoom-in Interpretation of derivative in $\mathbb{R}$
My book shows this nice geometric interpretation of the derivative.
Let $f:I\to\mathbb{R}$, say $I=[a,b]$. Suppose that $c\in\text{Int}(I)$, and look at the graph of $f(x),\,x\in I$, zooming in more ...
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