28
votes
Accepted
A difficult integral for the Chern number
If Stokes' theorem counts as a standard technique, then here's an answer:
Introduce a "vector potential"
\begin{equation}
A_i = \frac{1-\hat n_z}{\hat n_x^2 + \hat n_y^2}(\hat n_x \partial_i ...
- 628
21
votes
How to generalize the various vector calculus theorems to distributions?
This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial ...
- 12.7k
13
votes
How to generalize the various vector calculus theorems to distributions?
This is an addition to Dirk's answer. It is of course impossible to give an all-embracing answer to your question—there are so many identities. But the general answer is a resounding YES. The ...
- 266
12
votes
Accepted
Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?
Let
$$\begin{aligned} f(x,y) & = x^4 - x^2 y^2 + y^4 \\ & = \tfrac{1}{2} x^4 + \tfrac{1}{2} y^4 + \tfrac{1}{2} (x^2 - y^2)^2 . \end{aligned}$$
Then $f$ is a strictly positive (except at the ...
- 17.2k
9
votes
Accepted
Multivariable higher-order chain rule
You’re looking for the multivariate version of the formula of Faa di Bruno.
Addendum: As the OP notes, the version in Wikipedia is not in sufficient generality, since it takes $f:\mathbb R\to\mathbb R$...
- 47.4k
9
votes
Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?
Let us assume $\alpha\in[0,1)$ (the case of $\alpha\in(-1,0]$ is similar). As $\sin B>0$ for $B\in (0,\pi)$, the inequality $f(A,B)<0$ amounts to
$$
\alpha\sin A<\sin B-\sin(A+B),\quad -[\...
- 3,599
8
votes
Accepted
Proving the simple form of a function from statistical mechanics
We can indeed prove this for reasonable functions, $\log f_0\in C^2$, say.
Let me write $F=\log f_0$. By replacing $F$ by $F(v)-C-d\cdot v$, we can also assume that $F(0),\nabla F(0)=0$.
If $a,v$ are ...
- 24.2k
8
votes
Accepted
Two-variable continuous function which results in an integer if and only if arguments are integer
Such functions as you require do not exist. Your requirements impose that $f(1,1) < f(1,2) < f(2,2)$ (for example, $f(1,1) \leq f(1,2)$ by (2), but $\neg (f(1,2) \leq f(1,1))$ also by (2), so ...
- 32.5k
7
votes
A Curved/Warped Version of Fubini's Theorem
Coarea formula will do this for you. It is a "Fubini formula" relating the integral of a function $u$ on a Riemann manifold $(M_0,g_0)$ to the integrals along the fibers of a smooth map $F:(M_0,...
- 34.7k
7
votes
Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?
$\newcommand{\R}{\mathbb R}$Your conjecture is true.
Indeed,
\begin{equation*}
\begin{aligned}
f(x)&=F(a,u):=a\frac{\sin u}u+4\cos u, \\
g(x)&=G(a,u):=a\frac{\cot(u/2)}{u^2}\,h(u)-4\...
- 128k
6
votes
Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?
Let $x,y$ denote hereafter variables in the interval $I:=[0,\pi]$. Denote $$S:=\{-\sin(y)<\alpha\sin(x)+\sin(x+y)<\sin(y)\}\subset I^2$$ the set to be measured, and $\Delta:=\{x+y<\pi\}\...
- 60.6k
6
votes
Accepted
Definition of Multivariable Antiderivatives
$\newcommand\R{\Bbb R}\newcommand{\1}{\mathbf{1}}\newcommand{\xx}{\mathbf{x}}\newcommand{\yy}{\mathbf{y}}\newcommand{\uu}{\mathbf{u}}\newcommand{\vv}{\mathbf{v}}\newcommand{\0}{\mathbf{0}}$
Your ...
- 128k
5
votes
Accepted
Monotonic dependence on an angle of an integral over the $n$-sphere
That is technically a 2D question. We can assume that $v=e^{-it}, w=e^{it}\in\mathbb R^2$ ($0<t<\pi/4$). Then in the polar coordinates the integral becomes
$$
\int_0^1 \varphi(r)dr\int_0^{2\pi}|\...
- 61.9k
5
votes
Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?
Let $n=1$, $f(t)=t^2 + |t|^{7/2}\sin(1/|t|)$ for $t\ne0$, $f(0):=0$. Then $f'(0)=0$ and $f''(0)=2>0$, so that $0$ is a strict local minimum of $f$. However, $f''(t)\sim-|t|^{-1/2}\sin(1/|t|)$ as $t\...
- 128k
5
votes
Proving the simple form of a function from statistical mechanics
The cited authors assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then if ...
- 188k
5
votes
Accepted
Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?
This vector field is the negative of the gradient vector field of the/a Busemann function $b_u$ associated with $u$. (Below, I am assuming that the Riemannian metric is $C^\infty$.) The function is ...
- 12.3k
4
votes
Accepted
Complex-doubly periodic function in two variables?
The answer is that the only solutions have the form
$$
f = (f_1,f_2) = \bigl(c, h(\,\overline{z}_1, z_2)\bigr)
$$
where $h:\mathbb{C}^2\to\mathbb{C}$ is holomorphic and $c$ is a constant,
which must ...
- 108k
4
votes
Gaussian expectation restricted to a convex polytope
$\newcommand\R{\mathbb R}$Let $S:=\mathbf S$. Let $X$ be any random vector in $\R^n$ with a spherical symmetric distribution (say such that $P(X=0)=0$).
Then $|X|$ is independent of $U:=X/|X|$ and $U$ ...
- 128k
4
votes
Accepted
How to find partial derivatives of the Beta Function?
The input you need to calculate second derivatives of Beta functions is
$$\frac{\partial}{\partial x}B(x,y)=\bigl[\psi ^{(0)}(x)-\psi ^{(0)}(x+y)\bigr] B(x,y),$$
$$\frac{\partial^n}{\partial x^n}\psi ^...
- 188k
3
votes
Accepted
Obtain 3D function from 2D slices
This can be done by a simple homotopy between curves: if we put
$$
\begin{align}
f(x,730\mathrm{A}) &= \frac{501.451}{[1+\exp(-25.544+0.006604\cdot x)]^{\frac{1}{31.698}}}\\
\\
f(x,300\mathrm{A}) &...
- 6,367
3
votes
Accepted
Probability density of a hyperplane for a Gaussian distribution
$\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}$First, one should not denote a random vector in $\R^n$ (which is not actually a vector in $\R^n$ but a function with values in $\R^n$) and a true, ...
- 128k
3
votes
Accepted
Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$
I think it is not possible to do much better than $\frac{1}{\sqrt n}$. More precisely, I believe the best $\alpha$ is $\frac{1}{\sqrt n}$ whenever $n$ is a power of $2$, and therefore (since $\alpha$ ...
- 9,486
3
votes
Integral $ g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $
The answer is
$$g(a)=\frac{\pi}{2} \, \text{sgn}(a) \ln(1+|a|)
$$
for real $a$.
This can be obtained as follows: $g(0)=0$ and
$$g'(a)=\int_0^{\pi/2}\frac{dx}{1+a^2\tan^2 x}=\int_0^\infty\frac{du}{...
- 128k
3
votes
Accepted
Helmholtz decomposition vs Laplacian vector fields on $\mathbb{T}^3$?
The correct generalization to $\mathbb{T}^3 = (\mathbb{R}/\mathbb{Z})^3$ is the Hodge decomposition. Every vector field on $\mathbb{T}^3$ is uniquely expressible as $u = \nabla \times v + \nabla w + h$...
- 21.6k
3
votes
Integrability of modified diagonalizable Jacobian
It has taken me a while to find time to write a more comprehensive answer to the above question. It turns out that for general dimension $N$, the overdetermined PDE system involved is not involutive, ...
- 108k
3
votes
Definition of Multivariable Antiderivatives
This is just to show that differential forms give a valid answer to this question, as in the comments. Let’s work on $\mathbb{R}^n$ and let the volume form be $\omega$. Then the $n$ form $f \omega$ is ...
2
votes
Accepted
Gradient condition implies Hörmander condition
I figured out proving it by using mean value theorem (thanks to @WillieWong). Observe that
\begin{align*}
\int_{|x|>2|y|}|K(x-y)-K(x)|dx &\leq \int_{|x|>2|y|}|\nabla K(tx+(1-t)(x-y))||y| dx\\...
- 239
2
votes
Accepted
Is the minimum of a constraint optimization problem differentiable in the constraint parameter?
The answer to your question is: No, in general $F$ is not differentiable everywhere on $(0,\infty)$.
First, to simplify the notations a bit, consider the change of variables $x=e^u$, $y=e^v$, $s=e^t$...
- 128k
2
votes
A Curved/Warped Version of Fubini's Theorem
You can do this nicely with differential forms: see the chapter on Fubini's theorem in my lecture notes on Stokes's theorem.
- 26.3k
Only top scored, non community-wiki answers of a minimum length are eligible