Questions tagged [change-of-variable]

Question 1

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{\...

Question 2

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 ...

Question 3

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\...

Question 4

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 $\left\{(x,y) | x>0, y>0, x+y<1\right\}$. ...

Question 5

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))=\...

Question 6

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 $r$. $\...

Question 7

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 ...

Question 8

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 $f:B\to\mathbb{R}$ ...

Question 9

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, $S_i=g^{-1}(T_i)$ ...

Question 10

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 $...

Question 11

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|}{...

Question 12

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 ...

Question 13

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 ...

Question 14

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, ...

Question 15

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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Questions tagged [change-of-variable]

Second Derivative Chain Rule in Matrix Notation

Change of coordinates in the cotangent space

How do I prove the change of variables into polar coordinates using measure theory?

How to derive relations between integrals over a triangle without evaluating them explicitly?

$\lambda_n$ is the volume of the $n$-dimensional unit ball. Compute $\lambda_n$ in terms of $\lambda_{n-2}$. (Analysis on Manifolds James R. Munkres.)

Surface integration identity

Is $g:(-\pi/2,\pi/2)\to (-1,1)$ given by $g(x)=\sin x$ right? ("Analysis on Manifolds" by James R. Munkres.)

I think we need to modify the proof of Lemma 19.1 as follows. ("Analysis on Manifolds" by James R. Munkres.)

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.)

Is there any easy proof of my proposition? (James R. Munkres "Analysis on Manifolds".)

How to solve the definite integral $\int_{-1}^{1} \sqrt\frac{|x+1|}{|x|+1} \, {\rm d} x$?

Transformation of partials in accelerating reference frame

Accelerating reference frame in a traveling wave

Why this particular change of variables in this PDE?

Zoom-in Interpretation of derivative in $\mathbb{R}$

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