Questions tagged [multivariable-calculus]
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65 questions
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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 ...
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0
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22
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Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?
I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
- 708
22
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1
answer
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A difficult integral for the Chern number
Cross post from Maths stack exchange
The integral
$$
I(m)=\frac{1}{4\pi}\int_{-\pi}^{\pi}\mathrm{d}x\int_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\left( \sin^2x+\sin^2y +...
- 333
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1
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Does $\sum_{n \geq 0} a_n x^n=\sum_{n \geq 0} b_nx^n$ imply $a_n=b_n$ for vector-tuple power series?
My reference is Infinite series in p-adic fields by Keith Conrad.
Corollary 5.6. If $f(x)=\sum_{n≥0} a_nx^n$ has a positive radius of convergence in the $p$-adic field $\mathbb Q_p$ then $f$ is ...
- 930
1
vote
1
answer
126
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Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as
$$
J_f(x)=S(x)\Lambda(x) {S(x)}^{-1},
$$
where the ...
- 19
0
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1
answer
75
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Clarification about this optimisation problem
Good morning everybody. First of all, I apologise to ask here the same question I asked on MSE three days ago, but I am in fact re-asking since I obtained no relevant advice. Perhaps I will hear some ...
- 149
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0
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97
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An integral over the sphere in $\mathbb{R}^d$
Let $S^{d-1}$ be the unit sphere in $\mathbb{R}^d$.
Let $|x-y|$ denote the euclidean distance between to points $x$ and $y$ in $\mathbb{R}^d$.
Is there a nice expression for the following (maybe ...
- 4,034
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0
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141
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Multivariate Jackson inequality for Chebyshev approximation
There is an approximation of a multivariate function by a Chebyshev polynomial of degree n. One needs to understand how the approximation error behaves depending on the degree of the polynomial or ...
- 1
10
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2
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612
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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
122
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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:
$$...
- 31
3
votes
1
answer
422
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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-...
- 149
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votes
0
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52
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Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold
Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
- 21
2
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1
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155
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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 ...
- 23
2
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1
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177
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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 ...
- 31
2
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1
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213
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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$ $(...
- 21
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1
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168
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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(...
- 3
1
vote
1
answer
187
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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 ...
- 3,065
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0
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34
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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]...
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2
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1
answer
382
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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 ...
- 3,477
16
votes
2
answers
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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 ...
- 415
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3
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571
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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$$
Numerically, I see that ...
- 219
2
votes
1
answer
106
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Is $f^{-a}$ locally integrable if $f\geq 0$ has a unique stationary point ( a minimum) at which the Hessian is positive definite, $0<a<d/2$
Let $0<a<d/2$, let $B$ be the unit ball in $\mathbb{R}^{d}$ centered at the origin, and let $f:B \to [0,\infty[$ be a a smooth function such that
(1) $f(x)\geq f(0).$
(2) $\nabla f(x)\neq 0,\...
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1
vote
0
answers
216
views
Bound for the laplacian of a strictly convex function from above by the gradient of it
Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$
I am trying to show that
$$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
- 185
1
vote
0
answers
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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 ...
- 3,477
3
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2
answers
651
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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 \...
- 6,853
6
votes
1
answer
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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)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...
user329770
1
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1
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152
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The monotonicity of the bivariate normal with non-isotropic covariance
Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance.
Define $y = (y_1, y_2)$ and let
\begin{align}
F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...
- 113
8
votes
2
answers
802
views
Multivariable higher-order chain rule
I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what ...
- 5,327
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1
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143
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Differential form of the multidimensional "orthogonal dilation" operator
For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
...
- 286
1
vote
1
answer
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Is it possible to simplify the coefficient matrix for large values of $x$?
If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
- 111
5
votes
1
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344
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Monotonic dependence on an angle of an integral over the $n$-sphere
Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral:
$$
\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.
$$
...
- 115
2
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1
answer
326
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Evaluation of Gaussian multivariable integral
In the context of evaluating the propagation of a flattened Gaussian beam, the following integral appears:
\begin{equation}
\int (\mathbf x^T \mathbf F \mathbf x)^n \exp \left [ - \mathbf x^T \mathbf ...
- 83
2
votes
1
answer
330
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$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}$$...
- 162
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2
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567
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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)^...
- 6,741
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2
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174
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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 ...
- 127
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0
answers
154
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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 ...
- 83
1
vote
1
answer
161
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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,\...
- 779
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1
answer
215
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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} \...
- 71
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1
answer
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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$:
\...
- 33
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0
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108
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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,...
- 6,741
3
votes
1
answer
184
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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_{-\...
- 671
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1
answer
379
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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}...
- 536
0
votes
0
answers
39
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 ...
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}$, $\...
1
vote
0
answers
184
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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 ...
2
votes
3
answers
806
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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 ...
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4
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1
answer
302
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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)...
- 195
2
votes
1
answer
258
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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 ...
- 3,625
0
votes
0
answers
55
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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
343
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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 ...
- 239