Questions tagged [change-of-variable]

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

919 questions
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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 &…
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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
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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…
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$\arctan{x}+\arctan{y}$ 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
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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…
7
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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 $$…
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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…
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Given that $X,Y$ are independent $N(0,1)$ , show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}}$ are independent $N(0,\frac{1}{4})$

It is given that $X,Y \overset{\text{i.i.d.}}{\sim} N(0,1)$ Show that $\frac{XY}{\sqrt{X^2+Y^2}},\frac{X^2-Y^2}{2\sqrt{X^2+Y^2}} \overset{\text{i.i.d.}}{\sim} N(0,\frac{1}{4})$ I was thinking of making polar transformations $X=r \cos \theta, Y=r…
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Use Green's Theorem to Prove Change of Variable for Double Integral

Question: Use Green's Theorem to prove: $$\iint_{R}dxdy=\iint_{S}\begin{vmatrix}\frac{\partial(x,y)}{\partial(u,v)}\end{vmatrix}dudv$$ $\Big(\frac{\partial(x,y)}{\partial(u,v)} $is the Jacobian of the transformation$\Big)$ I found this question from…
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Changing variables in integration over spheres

Suppose we would like to change variables in the integral $$I:=\int_{\mathbb{S}^{n-1}}f(\omega_1,\omega_2,...,\omega_{n})d\sigma_{n-1},$$ where $\mathbb{S}^{n-1}$ is the standard unit sphere in $\mathbb{R}^{n}$, $n\geq 2$, $d\sigma_{n-1}$ is the…
Medo
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Jacobian for polar decomposition

Let $f:\mathrm{Mat}_n(\mathbb{C})\to\mathbb{C}$ be some function and let us suppose we want to make a change of variables in the integral $$ \int_{A\in \mathrm{Mat}_n(\mathbb{C})}f(A)\mathrm{d}{A} $$ from $A$ to $|A| U$, i.e., the polar…
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Weak differentiability and polar coordinates

I have run into a problem/confusion regarding weak differentiability and polar coordinates for which I have found no useful references up to this point. Let $\Omega\subset\mathbb{R}^2$ be the unit disk with angle $\pi < \Theta < 2\pi$, that…
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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…
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How to find a good strategy for tackling change of variables in multivariable calculus

I'm in the process of learning multivariable calculus, and I've been having trouble with the process of changing variables in order to perform an easier integration. I know how to compute the Jacobian and analyze the region once it is given to me,…
Germ
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Change of variables for $\iint_D x\sin(y-x^2) \, dA$

Let $D$ be the region in the first quadrant of the $x,y$-plane, bounded by the curves $y=x^2$, $y=x^2+1$, $x+y=1$, and $x+y=2$. I want to find a change of variables for $\iint_D x\sin(y-x^2) \, dA$. Since $y-x^2=0$, $y-x^2=1$, my inclination is to…
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