# Questions tagged [differential-calculus]

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126
questions

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### Prove the integral of multi-variable rational fraction is convergent

I have posted this problem in MSE long ago:
https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this ...

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102 views

### Does this integral have a closed form solution? [closed]

Let $x,y\in \mathbb{R}^2$, $B_r(0)=\{x||x|\leq r\}$. Does the following function (denoted as $f(r)$) have a closed form expression?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$

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32 views

### Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous

Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...

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134 views

### Lower bound of the nth derivative of function

I need to prove that there exists $a> 0$ and $n_0\in\mathbb N$ such that $$\forall n> n_0,\quad \sup\limits_{|x|\leq a} |f^{(n)}(x)| \geq (n!)^{\frac 32}.$$
Where $f$ is defined by $f(x)=\exp(-...

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75 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 $\|\...

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46 views

### Second order partial derivatives of Sobolev functions

This has been asked on Mathematics Stack Exchange but apparently received no attention. The question is very basic in nature:
Is it true that $W^{2,1}_{\text{loc}}$
functions (after possibly modifying ...

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**1**answer

57 views

### Solving a fully nonlinear first order PDE

given a symmetric matrix of Holder continuous functions $A(x)$ such that
$$
\frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2
$$
find a vector field $\Phi$ such that
$$
D \Phi(x)^t D ...

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85 views

### Properness of real analytic maps?

Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...

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204 views

### Gradient Descent for Markov Dynamics [closed]

The closed-loop dynamics of a linear optimal controller are simple but have interesting properties. From a starting state $\mathbf{v}(0)$ the dynamic can be iterated to reach a final state $\mathbf{v}(...

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72 views

### Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]

Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...

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233 views

### Signed distance function and level set

For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \...

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94 views

### The derivation of thin plate spline interpolation energy function？ [closed]

I am trying to derive the "thin plate energy functional". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional
$$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\...

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465 views

### Higher-order derivatives of $(e^x + e^{-x})^{-1}$

I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$
It is fairly straightforward to obtain
$$ \frac{d^n f}{dx^n} = \frac{P_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $...

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124 views

### Inequality involving sigmoid function

Let $\sigma$ denote the sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$, let $x,y \in \mathbb{R}$. Given the following two conditions: $|\sigma(-x) - \sigma(y)| < \epsilon$ and $x - y > c > ...

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**1**answer

157 views

### Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following.
Let $H$ be ...

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**1**answer

542 views

### Example of a function with a curious property

Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$.
$\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that:
$\frac{F(x)}{x}\in ...

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votes

**1**answer

117 views

### Is Sommerfeld radiation condition invariant under translations?

A smooth function $U:\mathbb{R}^3\setminus B_{r_0}(0)\to\mathbb{C}$ (for some $r_0>0$) satisfies the Sommerfeld Radiation Condition with index $k$, denoted $U\in \texttt{SRC}$, whenever
$$
\lim_{r\...

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**1**answer

98 views

### An expression for the $k$-th derivative of $f(x)=x^n\exp(-x)$ [closed]

Is there a finite expression for $k$-th derivative of
\begin{align}
f(x)={x^n}{e^{ - x}}
\end{align}

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42 views

### Solving $\frac{dy}{dx}=1+(a_mx^m+a_{m-1}x^{m-1}+…+a_0)y^2$

I have a problem with the following equation,
$\frac{dy}{dx}=1+P_m(x)y^2$
Where $P_m(x)$ is a polynomial function. I have solution for $P_m(x)=x$ using Mathematica, and Prof @Claude Leibovici solved ...

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**1**answer

102 views

### Bounding the product of lipschitz function [closed]

Assume we know that $f(x,y): \mathbb{R}^{a+b} \to \mathbb{R}^d$ is lispchitz w.r.t. $y$, i.e.
$$\|f(x,y) - f(x,y')\| \leq L \|y-y'\|.$$
Is there a way to bound the product $f(x,y)\cdot f(x,y')^T \in\...

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votes

**1**answer

64 views

### $AC^p$ curves and pointwise metric speed in abstract metric spaces?

For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if
$$
d(\gamma(s)...

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**1**answer

282 views

### A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$.
What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...

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vote

**1**answer

212 views

### A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...

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104 views

### Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...

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271 views

### Does the generalised directional derivative satisfy any version of the chain rule?

Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient.
The generalised directional derivative ...

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**1**answer

63 views

### Does having the derivative in the limit suffice to solve the function at the limit? [closed]

Suppose that I have a function $f(x, \epsilon)$ and I know that
$$
\lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x).
$$
Now let $g(x)$ be the function whose derivative appears above. How can I ...

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53 views

### Density of integers related to the size of its order of appearance in the Fibonacci sequence

Let $z(n)=\min\{k>0 : n\mid F_k\}$. This function is known as the Fibonacci entry point (for example). A result of Sallé gives the sharpest upper bound for $z(n)$, namely, $z(n)\leq 2n$, for all $n$...

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107 views

### Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by
$$
f(x)=x^q-kx^{q-p}-\ell.
$$
This polynomial is related to a family of two-...

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186 views

### Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...

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**1**answer

63 views

### Dominant root of a family of odd degree polynomials

Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences.
In order to try to use some standard ...

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**1**answer

87 views

### What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]

sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there:
How can $\sqrt{2x^2}$ become $4x^2$ ?

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**2**answers

114 views

### How to show that this series of rational functions has a maximum at $x=0$ using the “Descartesschen Regel”?

I am reading an old German paper, and at one step they mention that the function
\begin{equation*}
f(x) := \sum_{k=2}^\infty \frac{(1+x)(k(k-1)^2 + (2+x)(1+x)^2)}{(k+x)^3 (k + 1 + x)^2}
\end{equation*}...

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3k views

### Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix?

In his 1841 article De determinantibus, Jacobi remarked that the notation $\frac{\partial z}{\partial x}$ for partial derivatives is ambiguous. He observed that when $z$ is a function of $x,y$ as well ...

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422 views

### Mathematical Techniques to Reduce the Width of a Gaussian Peak

In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...

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10k views

### Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role
in applications, or in theorems?
Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the ...

**3**

votes

**1**answer

83 views

### Representation of finite differences of order k

We define recursively finite differences $ g_k (x) $ of order $ k $ of function $ f $ as follows:
$g_0(x)=f(x)$, $g_n(x)=g_{n-1}(x+h_n)-g_{n-1}(x) (n\in\mathbb{N})$.
It is known that all arguments of ...

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166 views

### A vector calculus formula

Let me answer my own question, hoping to be forgiven for that.
I asked unsuccessfully that question on Mathematics. Let $A, B$ be vector fields in $\mathbb R^3$.
We have
$$
\text{curl}\bigl((A\cdot \...

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**1**answer

171 views

### About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$

An integral has been pushed me over the edge for several weeks. It reads as:
$$\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$
I tried ...

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**1**answer

79 views

### Existence of a certain type of function

Trying to find functions with the given property:
Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...

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244 views

### Dominant root of a family of polynomials

Let $f(x)=x^5-x^4-x^3-x^2-x-c$, where $c>2$ is a real number. It is easy to prove that there exists a positive real root $\alpha>2$ of $f(x)$ and all the other roots are non real.
Also, by ...

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62 views

### Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros.
I am looking for non-trivial examples of integral transformation
\begin{align}
g(x)= \int f(t) h(t,x) dt
\end{align}
such that $f$ and $g$...

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**3**answers

129 views

### Literature request: Function that depends on a linear optimization problem [closed]

my question is about functions of the following form:
$$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$
where $\mathbf{x},\mathbf{b}, $ ...

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**1**answer

278 views

### A Conjecture about the integral related to Chebyshev polynomial

I am interested in the following integral related to the Chebyshev polynomials
$$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$
where $n,m\in \mathbb{Z}^+.$
It is easy to see ...

**3**

votes

**1**answer

126 views

### A geometric property about certain polynomials in two variables

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$
where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...

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380 views

### An integral in Gradshteyn and Ryzhik

Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...

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**1**answer

611 views

### Properties of the argmin function (continuity, differentiability..) [closed]

Given vectors $x_1,\ldots,x_N \in \mathbb{R}^d$ and a function, say $\psi \colon \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$, one is interested in the properties of the function $$\Phi\colon (x_1,...

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**1**answer

254 views

### If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [duplicate]

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...

**3**

votes

**2**answers

460 views

### Curl as a divergence… Is it possible? [closed]

I want to know if it is possible to express the operation
$$
\nabla \phi \times (\nabla \times \mathbf A)
$$
as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\...

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**0**answers

91 views

### Reference for numerical solutions for differential equations like $f'(x)=f(x+1)+f(x-1)$

One can solve a delay differential equation (like for example $f'(x)=f(x-1)$) if we have a function as a bounded condition (in my example we need to know $f$ on $[0,1)$) and then use a simple forward ...

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131 views

### Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x↦x^{-x}$

The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...