Questions tagged [multivariable-calculus]
The multivariable-calculus tag has no usage guidance.
57
questions
-1
votes
0
answers
21
views
Lagrange Multipliers Not Identifying All Critical Points [migrated]
Suppose that $x,y,z \in \mathbb{R}_{\ge 0}$. Consider $f(x,y,z) = xyz$ (the volume of some cube) and $g(x,y,z) = x+y+z-1$. I wish to maximise $f$ subject to $g=0$ with Lagrange multipliers.
The zero ...
3
votes
1
answer
250
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.
$$B(x,y)=\int_0^1u^{x-...
0
votes
0
answers
33
views
It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
1
vote
1
answer
82
views
Practical calculation of Canterbury approximants
I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...
2
votes
1
answer
116
views
Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?
A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
2
votes
1
answer
168
views
Gaussian expectation restricted to a convex polytope
Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(...
20
votes
1
answer
2k
views
A difficult integral for the Chern number
Cross post from Maths stack exchange
The integral
$$
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 +...
2
votes
1
answer
168
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 ...
1
vote
0
answers
79
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 ...
-2
votes
1
answer
152
views
Two-variable continuous function which results in an integer if and only if arguments are integer
I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...
10
votes
2
answers
589
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 ...
2
votes
1
answer
100
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,\...
1
vote
1
answer
133
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, ...
0
votes
1
answer
111
views
Differential form of the multidimensional "orthogonal dilation" operator
For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
...
0
votes
0
answers
120
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
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
0
votes
0
answers
36
views
Express $Q_0 u + Q_1 \Delta u + Q_2 \Delta^2 u + Q_3 \Delta^3 u=0$ as a conservation law for $u(\vec x, t) : \mathbb R \times \mathbb R \to \mathbb R$
In the study of certain PDEs, it is beneficial to write them as a conservation law so that the energy of the system may be defined. More facts such as causality can be proven by considering surface ...
1
vote
1
answer
113
views
Is it possible to simplify the coefficient matrix for large values of $x$?
If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
0
votes
0
answers
33
views
Specific analytical solution to a multivariate equation
I've encountered the following problem during my research, any help would be highly appreciated.
Let a following multivariate equation be given:
$F(x,P,I) = 0$,
where $x$ is a variable and $P,I$ are ...
2
votes
1
answer
144
views
Obtain 3D function from 2D slices [closed]
I am given a graph of a motor's torque vs RPM values at two different current draws, 730A and 300A (see graph). I need to obtain the 3D function to find current as a function of torque and RPM by ...
2
votes
0
answers
138
views
Applying 1D integral to matrix integral
In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
2
votes
1
answer
295
views
Evaluation of Gaussian multivariable integral
In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears:
\begin{equation}
\int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
3
votes
0
answers
107
views
Are square configurations the only critical points of the energy on the circle?
$\newcommand{\S}{\mathbb{S}^1}$
$\newcommand{\la}{\lambda}$Let$$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,...
6
votes
1
answer
368
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)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...
2
votes
1
answer
219
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$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}$$...
1
vote
1
answer
147
views
Multiple integral and integral with respect to a function of variables
This is concerning Eq. (3.7) of C R Rao's 1945 paper (see p.81 of this article). Can someone help me in figuring out the second equality in Eq. (3.7)?
His claim is (since $\phi(x,\theta) = \Phi(T,\...
6
votes
1
answer
326
views
Monotonic dependence on an angle of an integral over the $n$-sphere
Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...
2
votes
1
answer
198
views
A 2 dimensional integral in polar coordinate [closed]
Recently I got stuck on a 2 dimensional integral in polar coordinate,
the expression is the following:
$I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...
3
votes
1
answer
460
views
Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$
Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$.
Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$:
\...
1
vote
0
answers
201
views
A sufficient condition for multiple differentiability of a function of several variables
While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1:
Let $f: \mathbb{R}^n\to\mathbb{R}$, $\...
0
votes
0
answers
51
views
Solving nonlinear differential multi-variable equation with block-matrices
Here is the problem:
Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...
3
votes
1
answer
174
views
Generalization(s) of variation diminishing property to multivariate case
Let us first define the variation diminishing property for the Gaussian kernel. Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is sufficiently smooth and define
\begin{align}
F(x)= \int_{-\...
11
votes
3
answers
550
views
Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?
I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$:
$$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$
Numerically, I see that ...
1
vote
0
answers
176
views
A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$
Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
3
votes
2
answers
600
views
Express Dirichlet energy $E_\mu(f) := \int \|\nabla f(x)\|^2 d\mu(x)$ in terms of Fourier information alone
Let $\mathbb R^d$ and let $\mu = p(dx)$ be a probability distribution thereupon, with density $p$ (which maybe assumed bounded, etc.). For a continuously differentiable function $f:\mathbb R^d \to \...
16
votes
2
answers
1k
views
How to generalize the various vector calculus theorems to distributions?
Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
0
votes
0
answers
81
views
What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
1
vote
2
answers
169
views
Does this non-negative function, with no stationary points, have only descent directions close to a constraint set?
Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0} $ is a differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...
6
votes
1
answer
351
views
Complex-doubly periodic function in two variables?
I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations
$$\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}...
8
votes
2
answers
536
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 ...
0
votes
0
answers
60
views
Integration question dealing with several variables and Taylor theorem
Dealing with one-variable and smooth function $f$ on a real interval $I$ such that $D^m f\in\mathcal{C}^2$, we have by Taylor theorem centered at $a\in I$
$$ D^mf(y)= D^mf(a) + D^{m+1}f(a)(y-a) + \...
2
votes
1
answer
215
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 ...
4
votes
1
answer
299
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)...
2
votes
0
answers
40
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 ...
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 ...
1
vote
0
answers
22
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 ...
7
votes
2
answers
531
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)^...
1
vote
0
answers
76
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$ (...
1
vote
0
answers
109
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)$$ and I known a parametric change-of-variable $Y(\theta) = T(X; \theta)$.
I would like to ...
3
votes
1
answer
277
views
Gradient condition implies Hörmander condition
We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition
$$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
1
vote
1
answer
172
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&...