# Questions tagged [differential-calculus]

The differential-calculus tag has no usage guidance.

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### Finding the largest possible value of function [on hold]

Suppose for(x) is continuous and differentiable in [-2, 5] and for(5) =4 & f'(x) <=10. Find the largest possible value of f(-2)?

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27 views

### limit derivative convex function exist

Let $U$ be an open convex subset of $\mathbb{R}^{n}$ and $f(y,t)$ are real continuous convex function on $U$. We assume that $x_{n}\rightarrow x$ and $f(x_{n},t_{n})$ is diffrentaible with respect to $...

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48 views

### is it possible to solve $\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$ in closed form? [duplicate]

$$\sum_{i=0}^n (x-\mu_i)e^{-(x-\mu_i)^2} = 0$$
is it possible to solve this equation in closed form ?
if i should use a numerical method, can you show me why i cannot get an exact solution ?

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105 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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146 views

### Is it a weaker condition for the symmetry of mixed derivatives?

Asked in MathStackExchange and I think it may be harder than usual problems.
Conditions:
$f:\mathbb{R^2}\mapsto\mathbb{R}$ has second-order derivatives on some neighborhood of the zero point.
$f_{...

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**1**answer

255 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 ...

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**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 ...

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42 views

### Sensitivity of Lagrangian solution: implicit constraint

just a question about a literature reference. I am writing a paper for engineers.
Usually for the Lagrange multiplier problem ∇f(x)+λ∇g(x)=0
the sensitivity result that the multiplier λ gives the ...

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323 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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104 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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158 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 ...

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441 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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73 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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115 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 ...

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**1**answer

176 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 )^\...

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831 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 ...

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205 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} \...

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69 views

### About Hessians of functions

Say one is given a twice differentiable (at least) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$.
What are some conditions when one can put a lowerbound on the smallest eigenvalue of its ...

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1k views

### Is it always possible to calculate the limit of an elementary function?

I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...

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### Uniqueness of tangent space given local injectivity of orthogonal projection onto it

Definition. Let $X\subset \mathbb R^n$ be a locally Euclidean subset. Say it has a tangent space at $p\in X$ if there exists a linear subspace $V\leq \mathbb R^n$ satisfying the following conditions.
...

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369 views

### Is the space of tangents actually the tangent space?

This is a crosspost of this MSE question.
Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote ...

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### Is there Calculus for (Almost) Continuous functions?

So I asked a similar question to this on Math Stack Exchange a couple of weeks ago, did a bounty, and I didn't receive any answers.
I am struggling a bit with a part of my research (on CS).
Suppose ...

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118 views

### Computing harmonic sum [closed]

I want to show the following equalities for harmonic sum
$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$
Any idea?

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102 views

### Solve 4th order ODE with variable coefficients

I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam:
$u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')...

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518 views

### Question about the validity of Michael Grossman, Robert Katz and Jane Grossman Non-Newtonian / Meta / Multiplicative Calculus [closed]

I am no mathematician by far, but I have studied diff. and integral calc and beyond, in my undergrad years. I recently came upon this book:
"The First Systems of Weighted Differential and Integral ...

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100 views

### Inequality involving joint distribution and marginal distributions [closed]

Let $f(x,y)$ be a joint density of $X$ and $Y$. Let $f_{X}(x)$ and $f_{Y}(y)$ be the marginal density functions. Is the following inequality true?
$$
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}...

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### Equivalent definitions of $C^1-$boundary

I am studying PDE, and I have two definition of $C^1$ open set as follow:
Definition 1. (Evans' PDE book)
An open set $\Omega \subset \mathbb{R}^N$ is $C^1$ if for each point $x_0 \in \partial \...

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371 views

### An inequality inspired by the isoperimetric inequality

Let us consider the simplest isoperimetric inequality. Consider a smooth simple closed curve given by $r=\rho(\theta)$ in polar coordinates, where $\rho(\theta)>0$ can be regarded as a smooth ...

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### Given a continuous function of many variables, how do I know when this function is equals to zero on all the corners of a unit hypercube?

Given a continuous and deriviable function of many variables, how do I know when this function is equals to zero on all the corners (or vertices) of a unit hypercube, i.e. all points, where each ...

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### Maximizing sum of a product of logs

I came across the following note in a paper I'm reading and don't understand how it was derived.
$\max_{\alpha_\ell}\sum_\ell^L\beta_\ell\log\alpha_\ell$ such that $\sum_\ell^L\alpha_\ell=1$ and $\...

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### Existence of local minimizer

For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$
$$\| \nabla F(x) \| < \epsilon$$
and a sufficiently large $\alpha$ where
$$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$
Can ...

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659 views

### “Universal” differential identities

(This is a cross-post from MSE).
Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth ...

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154 views

### Fixed point iteration using derivatives

Let $g(x)= \mathrm{\lim_{n \rightarrow \infty}}\,f^{\circ n} (x) $, where $f^{\circ n} (x)$ is the repeated application of $f$ to $x$, $n$ times. Why doesn't the following procedure work for ...

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169 views

### The limitation of derivation of modified Bessel function of second kind

The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all
$I = \...

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869 views

### How quickly can the derivative of an everywhere differentiable function change sign?

Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...

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159 views

### Time derivative of a quaternion-valued function in exponential form

I was wondering when a quaternion valued function $q(t) = e^{if_1 (t)+jf_2 (t) + kf_3 (t)}$ were written in exponential form, if there was a nice formula to express the time derivative $q'(t)$ in ...

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### Question about optimizing a given function by optimizing an approximation

Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''...

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329 views

### Symmetry of higher order mixed partial derivatives under weaker assumptions

Suppose $U$ is an open subset of $\mathbb{R}^n$, and $f:U\to \mathbb{R}$. When $f$ is $C^2$ we know that the mixed partial derivatives are symmetric, i.e.
$\partial_i\partial_jf= \partial_j\...

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720 views

### Derivative formula

While trying to understand a paper of Cayley, he left something unexplained, I managed to show that it is equivalent to the following formula, which I got stuck at:
$k \cdot (f^k)^{(k-1)} = \sum_{j=0}...

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433 views

### The level sets of a differentiable function is a manifold

I have seen stronger propositions that imply this, but both their statement and their proofs require more advanced tools than I'd like to use in my text, which is aimed at a general scientific ...

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266 views

### nth derivative of error function

Let $f(x)=erfi(a+x)$ and $g(x)=e^{cx}$ with
\begin{align*}
f^{(n)}(x)=\frac{2}{\sqrt{\pi}}e^{(a+x)^2}\sum_{m=0}^{n-1}\sum_{j=0}^{m}\frac{\binom{m}{j}(-1)^j(a+x)^{2m-n+1}}{m!}\\
\prod_{p=1}^{n-1}(2m-2j-...

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### Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $ \mathbb Q_p$).
1) Let $f: K^n \to K$ be a ...

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375 views

### The derivative of an integral function with indicator and max function as integrand

I encounter the following type of problem:
\begin{equation}
F(x) = \int_a^b \mathbf{1}_{\{v+x-h(v)\geq 0\}}\max\{h(v)-y-x,0\}dv
\end{equation}
where $\mathbf{1}_{\{z\geq 0\}}=1$ if event $z\geq 0$ ...

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425 views

### Derivative is Zero on a dense G_delta set

I have the following question:
I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere.
I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G_\delta$-...

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### The Lipschitz continuity of the maximizer of a Liptschitz function

Let $$\phi\left(t,b\right)=-\tilde{c}\left(b\right)\alpha e^{r(T-t)}-\lambda\int_{0}^{+\infty}\exp\left\{\alpha yb e^{r(T-t)}\right\} d G(y),$$
where $G(y)$ is the distribution of a positive random ...

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528 views

### Commuting of exterior derivative and contraction (vector-valued forms)

$\newcommand{\sig}{\sigma}$
$\newcommand{\tr}{\operatorname{tr}_{\eta}}$
$\newcommand{\al}{\alpha}$
$\newcommand{\be}{\beta}$
$\newcommand{\til}{\tilde}$
Let $E$ be a smooth vector bundle over a ...

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63 views

### Combinatorial identity of Derivatives of super-Gaussian function

An asymptotic expansion I stumbled upon has real numbers $c^\alpha_{ij}$ as coefficients, where $i, j \in \mathbb{N}_0$ are non-negative integers and $\alpha \in \mathbb{N}_0^n$ is a multi-index. They ...

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73 views

### Developing a functional equation for log-integral of theta function

I'm studying $u(z,q):=\exp(\left[\tfrac{\pi q}{12}-\ln2-\sum_{k\ge1}\ln(1-e^{-2k\pi q})\right]z+\int_1^z\ln\varphi(x,q)\,dx)$ , with $$\varphi(z,q)=2e^{-\pi qz^2-\pi q/4}\sinh \pi qz\prod_{k\ge1}(1-e^{...

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120 views

### Universal chord theorem for curves

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve.
Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...

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### Approximating a $C^1$ function in $Lip$ norm with piecewise linear

For a continuous function $f:[a,b]\to R$ there is a natural and obvious procedure to approximate it with a sequence of continuous, piecewise linear functions: take $N$ equally spaced points in $[a,b]$ ...