# Questions tagged [differential-calculus]

The differential-calculus tag has no usage guidance.

**0**

votes

**1**answer

72 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 $...

**0**

votes

**2**answers

150 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 ...

**0**

votes

**0**answers

52 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 $g$...

**1**

vote

**3**answers

109 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}, $ ...

**4**

votes

**1**answer

257 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

123 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 ...

**16**

votes

**0**answers

327 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 ...

**0**

votes

**1**answer

141 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,...

**3**

votes

**1**answer

161 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

443 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 $\...

**1**

vote

**0**answers

77 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 ...

**1**

vote

**0**answers

118 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 ...

**1**

vote

**1**answer

212 views

### integrate exponential function of quadratic form over unit norm vectors

Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively.
I need to solve
$$\int_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right)
\, dx$$
where $(\cdot )^\...

**12**

votes

**3**answers

880 views

### Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...

**0**

votes

**2**answers

207 views

### exactness of a 1-form [closed]

This may be a trivial question. If so, apologies in advance.
Let a $1$-form $\omega$ be given. Suppose that for any line segments $C_1=[i,j]$, $C_2=[j,k]$, $C_3=[i,k]$, we have that
$\int_{C_1} \...