This concerns all problem requesting techniques and tricks about changes of variables in computations of limits as well as integrals.
Questions tagged [change-of-variable]
1204 questions
56
votes
5 answers
Intuitive proof of multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory?
What is an intuitive proof of the multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory?
I think that textbooks overcomplicate the proof.
If possible, use linear algebra and calculus to solve it, since…
Victor
- 8,498
17
votes
3 answers
Curl in cylindrical coordinates
I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating this determinant:
$$\left|\begin{matrix}
e_r &…
Marcus Loken
- 211
15
votes
2 answers
When change of variable makes an empty interval
Please consider the following case:
$$I = \int^1_{-1}x^2dx$$
$$u(x) = x^2 \rightarrow du = 2x\,dx$$
$$u(-1) = 1, u(1) = 1$$
So
$$I = \int^1_1\frac{u}{2\sqrt u} du = 0$$
Obviously the problem here is to only consider the positive root of u. I don't…
Winter
- 946
13
votes
2 answers
Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation
Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below
The statement of the theorem:
Theorem
Among all second-order homogeneous PDEs in two dimensions with constant coefficients, show that the only ones that do…
Hushus46
- 983
11
votes
4 answers
$\arctan{x}+\arctan{y}$ from integration
I was trying to derive the property
$$\arctan{x}+\arctan{y}=\arctan{\frac{x+y}{1-xy}}$$
for $x,y>0$ and $xy<1$ from the integral representation
$$
\arctan{x}=\int_0^x\frac{dt}{1+t^2}\,.
$$
I am aware of "more trigonometric" proofs, for instance…
Brightsun
- 6,963
10
votes
6 answers
How can I evaluate $\int \frac{x^3+2x-7}{\sqrt{x^2+1}}\mathrm dx?$
How can I evaluate the following integral $$\int \frac{x^3+2x-7}{\sqrt{x^2+1}}\mathrm dx$$
My work:
I substituted $x=\tan\theta$, $\mathrm dx=\sec^2\theta \mathrm d\theta $
The integral becomes
$$\int \dfrac{\tan^3\theta+2\tan…
user805532
10
votes
1 answer
How to solve $(y)^{y'}=(y')^{y+c},c \in \mathbb{R}$
In the case when $ c=0 $ this ode will be $(y)^{y'}=(y')^{y}$ , let's assume that $y$ and $y'$ are strictly positive
functions so:
$$(y)^{y'}=(y')^{y} \iff e^{y' \log(y)}= e^{y \log(y')} \iff {y' \log(y)}= {y \log(y')} \iff…
Math Student
- 1,308
9
votes
2 answers
Charcteristic function not in a fractional Sobolev space
I am trying to show that for any Lebesgue measurable set of finite positive measure $E$, the characteristic function $\chi_E$ is not in $H^{\frac{1}{2}}(\mathbb{R}^n)$. I found somewhere that it would be enough to show instead that
$$…
Keen-ameteur
- 8,404
8
votes
1 answer
Change of variables in integration on manifolds: what's wrong?
The problem is this: Let $f:M^m\to\mathbb{R}$ a continuous function with compact support and $h:N\to M$ a diffeomorphism between Riemannian compact manifolds. Then
$$\int_{M}fdV^M=\int_N f\circ h|\det dh|dV^N$$
The proof using differential forms is…
Allan Kenedy
- 120
- 1
- 7
8
votes
1 answer
Change of variable formula for a generic measure applied to classical change of variable formula.
I was reading this interesting post about changing the variables in an integral with a generic measure. I was wondering how this applies to the standard change of variable. In other words, $$\int_{F(\Omega)} f d\lambda = \int_{\Omega} f \circ F…
edamondo
- 1,691
8
votes
3 answers
The volume of the image of a map with vanishing Jacobian is zero
Let $\Omega \subseteq \mathbb{R}^n$ be a nice domain with smooth boundary (say a ball), and let
$f:\Omega \to \mathbb{R}^n$ be smooth. Set $\Omega_0=\{ x \in \Omega \, | \, \det df_x =0 \} $
Is there an elementary way to prove that…
Asaf Shachar
- 25,967
8
votes
3 answers
How should I solve this integral with changing parameters?
I can't solve this. How should I proceed?
$$\iint_De^{\large\frac{y-x}{y+x}}\mathrm dx\mathrm dy$$
$D$ is the triangle with these coordinates $(0,0), (0,2), (2,0)$ and I've changed the parameters this way $u=y-x$ and $v= y+x$ and the Jacobian is…
khoshrang
- 139
7
votes
5 answers
To integrate $\int_0^{2\pi}\sqrt{\theta^2+1}\ d\theta$, why choose the change of variables $u=\theta+\sqrt{\theta^2+1}$?
In order to find the length of the curve $r=\theta,\ \theta\in[0, 2\pi]$, the integral that must be solved is $$\int_0^{2\pi}\sqrt{\theta^2+1}\ d\theta$$
For which my proffesor opted to use the following change of variable:…
MSU
- 185
7
votes
1 answer
When does there exist a change of variables such that it maps a compact subset of $R^d$ to a d-cube?
I recently learned about Jacobians and how the change of variables is used to ease out the calculations of an integral in context of double integrals.
This led me to wonder when there's a change of variables possible for every integral such that it…
DatBoi
- 4,097
7
votes
1 answer
Need help understanding solution for using change of variables to evaluate integral
Use the change of variables $x=u^2-v^2$, $y=2uv$ to evaluate $$\iint_{R}y dA$$ where $R$ is the region bounded by the x-axis, the parabolas $y^2=4-4x$ and $y^2=4+4x, y\geq0$
I'm following along with this solution:
I don't understand why the…
user8290579
- 914