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Questions tagged [multivariable-calculus]

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A difficult integral for the Chern number

Cross post from Maths stack exchange The integral $$ I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
xzd209's user avatar
  • 333
16 votes
2 answers
1k views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
YuerWu's user avatar
  • 415
11 votes
3 answers
571 views

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?

I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$: $$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$ Numerically, I see that ...
math2021's user avatar
  • 219
10 votes
2 answers
612 views

Proving the simple form of a function from statistical mechanics

I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
LuckyJollyMoments's user avatar
8 votes
2 answers
802 views

Multivariable higher-order chain rule

I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what ...
shuhalo's user avatar
  • 5,327
7 votes
2 answers
567 views

Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?

This is a cross-post. Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$. Let $df^k(x):(\mathbb R^n)^...
Asaf Shachar's user avatar
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6 votes
1 answer
376 views

Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?

I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals) $$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\ g(x)=\frac{\...
user avatar
6 votes
1 answer
379 views

Complex-doubly periodic function in two variables?

I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations $$\partial_{z_2}f_1(z_1,z_2) + \partial_{z_1} f_2(z_1,z_2)=0 \text{ and }$$ $$\partial_{\bar z_1}...
Sascha's user avatar
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5 votes
1 answer
344 views

Monotonic dependence on an angle of an integral over the $n$-sphere

Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral: $$ \int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx. $$ ...
LayZ's user avatar
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5 votes
0 answers
504 views

How to compute the volume of a region transformed by a matrix?

This is a rewrite of the OP's question to emphasize what I think are the research level issues here. Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{...
RyanChan's user avatar
  • 550
4 votes
1 answer
302 views

How to define a function that has these specific properties?

Suppose $x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}$. For $x,y \in \mathbb{Z}^K_{\geq 0}$, we write $x \succ y$ or $y \prec x$ if $x \neq y$ and \begin{align*} x_{i(x,y)} > y_{i(x,y)...
ie86's user avatar
  • 195
3 votes
1 answer
422 views

How to find partial derivatives of the Beta Function?

I was reading the book (Almost) Impossible Integrals, Sums and Series. The author used a method involving taking partial derivatives of the Beta Function to solve some integrals. $$B(x,y)=\int_0^1u^{x-...
Souparna's user avatar
  • 149
3 votes
1 answer
623 views

Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$

Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$. Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$: \...
Toobiz's user avatar
  • 33
3 votes
1 answer
184 views

Generalization(s) of variation diminishing property to multivariate case

Let us first define the variation diminishing property for the Gaussian kernel. Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is sufficiently smooth and define \begin{align} F(x)= \int_{-\...
Boby's user avatar
  • 671
3 votes
2 answers
651 views

Express Dirichlet energy $E_\mu(f) := \int \|\nabla f(x)\|^2 d\mu(x)$ in terms of Fourier information alone

Let $\mathbb R^d$ and let $\mu = p(dx)$ be a probability distribution thereupon, with density $p$ (which maybe assumed bounded, etc.). For a continuously differentiable function $f:\mathbb R^d \to \...
dohmatob's user avatar
  • 6,853
3 votes
1 answer
343 views

Gradient condition implies Hörmander condition

We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition $$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
Lev Bahn's user avatar
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3 votes
0 answers
108 views

Are square configurations the only critical points of the energy on the circle?

$\newcommand{\S}{\mathbb{S}^1}$ $\newcommand{\la}{\lambda}$Let$$M=\{(x_1,x_2,x_3,x_4) \in (\S)^4\,\, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ Define $E:M \to \mathbb{R}$ by $$E(x_1,x_2,...
Asaf Shachar's user avatar
  • 6,741
2 votes
2 answers
383 views

Definition of Multivariable Antiderivatives

In the 1-dimensional case antiderivatives $F(x)$ of a function $f(x)$ have the following properties: $F(x)=\int\limits_0^xf(t)dt$ $\frac{d}{dx}F(x)=f(x)$ $\int\limits_a^bf(t)dt = F(b)-F(a)$ Of ...
Manfred Weis's user avatar
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2 votes
3 answers
806 views

A Curved/Warped Version of Fubini's Theorem

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$. Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
Behnam Esmayli's user avatar
2 votes
1 answer
155 views

Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?

A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
Shin HY's user avatar
  • 23
2 votes
1 answer
213 views

Gaussian expectation restricted to a convex polytope

Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(...
Ye He's user avatar
  • 21
2 votes
1 answer
382 views

Helmholtz decomposition vs Laplacian vector fields on $\mathbb{T}^3$?

I am quite confused between Helmholtz decomposition and Laplacian vector fields in the periodic case. Let $\mathbb{T}^3$ be the $3$-dimensional torus. Then, I thought any divergence-free smooth vector ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
177 views

Obtain 3D function from 2D slices [closed]

I am given a graph of a motor's torque vs RPM values at two different current draws, 730A and 300A (see graph). I need to obtain the 3D function to find current as a function of torque and RPM by ...
AzodineAcid's user avatar
2 votes
1 answer
330 views

Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
etal's user avatar
  • 162
2 votes
1 answer
258 views

Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ...
Johnny T.'s user avatar
  • 3,625
2 votes
1 answer
122 views

Analytical solution for a double integral involving logistic functions and Gaussian distributions

I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows: ​$$...
Charles's user avatar
  • 31
2 votes
1 answer
106 views

Is $f^{-a}$ locally integrable if $f\geq 0$ has a unique stationary point ( a minimum) at which the Hessian is positive definite, $0<a<d/2$

Let $0<a<d/2$, let $B$ be the unit ball in $\mathbb{R}^{d}$ centered at the origin, and let $f:B \to [0,\infty[$ be a a smooth function such that (1) $f(x)\geq f(0).$ (2) $\nabla f(x)\neq 0,\...
Medo's user avatar
  • 852
2 votes
1 answer
326 views

Evaluation of Gaussian multivariable integral

In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears: \begin{equation} \int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
Alex's user avatar
  • 83
2 votes
1 answer
215 views

A 2 dimensional integral in polar coordinate [closed]

Recently I got stuck on a 2 dimensional integral in polar coordinate, the expression is the following: $I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...
NuKuYul's user avatar
  • 71
2 votes
1 answer
154 views

Is the optimum of this problem convex in the constraint parameter?

Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that $|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
Asaf Shachar's user avatar
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2 votes
0 answers
154 views

Applying 1D integral to matrix integral

In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
Alex's user avatar
  • 83
2 votes
0 answers
48 views

Hessian generating functions

I am looking for a characterization of functions $\Phi: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ such that $\Phi(\mathbf{x}) = \nabla^2 f(\mathbf{x})$ for a function $f$ which is twice continuously ...
neticin's user avatar
  • 121
2 votes
0 answers
110 views

The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse

Conjecture. Let $$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$ Then for any ...
Vlad the magnificent's user avatar
1 vote
1 answer
126 views

Integrability of modified diagonalizable Jacobian

I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as $$ J_f(x)=S(x)\Lambda(x) {S(x)}^{-1}, $$ where the ...
Shock Captor's user avatar
1 vote
1 answer
187 views

Practical calculation of Canterbury approximants

I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...
gmvh's user avatar
  • 3,065
1 vote
1 answer
119 views

Is it possible to simplify the coefficient matrix for large values of $x$?

If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
charmin's user avatar
  • 111
1 vote
1 answer
161 views

Multiple integral and integral with respect to a function of variables

This is concerning Eq. (3.7) of C R Rao's 1945 paper (see p.81 of this article). Can someone help me in figuring out the second equality in Eq. (3.7)? His claim is (since $\phi(x,\theta) = \Phi(T,\...
Ashok's user avatar
  • 779
1 vote
1 answer
195 views

Is the minimum of a constraint optimization problem differentiable in the constraint parameter?

Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Let $s&...
Asaf Shachar's user avatar
  • 6,741
1 vote
1 answer
469 views

Does $\sum_{n \geq 0} a_n x^n=\sum_{n \geq 0} b_nx^n$ imply $a_n=b_n$ for vector-tuple power series?

My reference is Infinite series in p-adic fields by Keith Conrad. Corollary 5.6. If $f(x)=\sum_{n≥0} a_nx^n$ has a positive radius of convergence in the $p$-adic field $\mathbb Q_p$ then $f$ is ...
MAS's user avatar
  • 930
1 vote
1 answer
152 views

The monotonicity of the bivariate normal with non-isotropic covariance

Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance. Define $y = (y_1, y_2)$ and let \begin{align} F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
Jon Lebensold's user avatar
1 vote
2 answers
174 views

Does this non-negative function, with no stationary points, have only descent directions close to a constraint set?

Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0} $ is a differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...
ARedder's user avatar
  • 127
1 vote
0 answers
82 views

If $v$ and $w$ are orthogonal smooth vector fields of unit norm and zero divergence, is $\langle w, (\nabla \times v) \times v \rangle=0$?

I am quite confused about manipulating with the curl when the vector field is of unit magnitude and divergence free. For example, let $v, w : \mathbb{T}^3 \to \mathbb{R}^3$ be periodic smooth vector ...
Isaac's user avatar
  • 3,477
1 vote
0 answers
216 views

Bound for the laplacian of a strictly convex function from above by the gradient of it

Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$ I am trying to show that $$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
kumquat's user avatar
  • 185
1 vote
0 answers
260 views

A sufficient condition for multiple differentiability of a function of several variables

While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1: Let $f: \mathbb{R}^n\to\mathbb{R}$, $\...
Alexander Kuleshov's user avatar
1 vote
0 answers
184 views

A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$

Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
Alexander Kuleshov's user avatar
1 vote
0 answers
32 views

What relationship exists between samples of a function and samples of its vector gradient field?

A real function $f(x)$ is defined on $N$-dimensional real space where $N \ge 3$. $f(x)$ is differentiable and its gradient with respect to x is $g(x)$. So $g(x)$ is a vector field. Assume we do not ...
hyu's user avatar
  • 19
1 vote
0 answers
24 views

Creating a Multi-Variable equation to create a "Most Favorable Supplier Index" [closed]

I wanted to create a 'most favorable supplier index', where I want to assign a value to each supplier, ranging from 1-5 or 1-10, whichever, from the most and to the least preferred suppliers. There ...
Jackson Mathews's user avatar
1 vote
0 answers
79 views

Conditions for a function to vanish almost nowhere on its support?

Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support. Under which conditions is it true that $f≠0$ (...
fsp-b's user avatar
  • 463
1 vote
0 answers
123 views

Differential entropy under the change-of-variable with additive Gaussian noise

I have two Gaussian random variables $$X \sim \mathcal N(0, I), \ \ \ W \sim \mathcal N(0, \sigma\cdot I)$$ and I known a parametric change-of-variable $Y(\theta) = T(X; \theta)$. I would like to ...
Ben Usman's user avatar
  • 111
1 vote
0 answers
69 views

Elasticity tensor in terms of principal stretches

Suppose we are given a frame-indifferent isotropic function $W:GL_+(3) \to [0,\infty)$, where $GL_+(3)$ denotes the set of all real $(3\times 3)$-matrices with positive determinant. We can write $W(F)$...
msaBU's user avatar
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