The differential-calculus tag has no usage guidance.

**-6**

votes

**0**answers

36 views

### Mathematics example that didn't quite undestand [on hold]

Find all value k such that the function x^3-3x+k has one real solution ? Thanks

**-2**

votes

**0**answers

16 views

### If derivative of f(x) is f'(x), then what is the integral of (f'(x))^2 in terms of f(x)? [migrated]

Is there some procedure for figuring this out or am I venturing into unsolved territory here?

**1**

vote

**0**answers

45 views

### Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq ...

**0**

votes

**0**answers

30 views

### Reparametrisation of a PDE with arclength

Suppose I have the following PDE:
$\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$
I notice ...

**3**

votes

**0**answers

31 views

### how to study the size of basins of attraction on a graph

I have a certain finite (but huge and without an apparent pattern, so that only numerical studies seem feasible) graph $G = (V,E)$, and a function $f: V \rightarrow \mathbb{R}$. On each edge $e = ...

**0**

votes

**0**answers

50 views

### Monotone functions in ordered Banach spaces

Let $(X,\preceq)$ be a real ordered infinite-dimensional Banach space and $f:X\to \mathbb{R}$ is a Fréchet differentiable function. It is said to be a monotone non decreasing map if
$$ x\preceq y ...

**1**

vote

**0**answers

18 views

### Computing skewness derivative in terms of variance

In the Portilla Simoncelli paper (page 18):
http://www.cns.nyu.edu/pub/lcv/portilla99-reprint.pdf
They go about calculating the derivative of the skewness $\eta(x)$ of a distribution (2D matrix in ...

**0**

votes

**1**answer

157 views

### $P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and
$P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z $ is a complex variable.
$Z$ is eigenvalue of $P(Z )$ if ...

**1**

vote

**0**answers

27 views

### Divergence of a second order tensor [closed]

I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct.
Suppose you would like to compute the components of a ...

**-1**

votes

**1**answer

49 views

### A question on decreasing function [closed]

Let $t\in (0,1)$ and
${a_n}{x^n} + .... + {a_1}{x^1} + f(t) = 0$
$f(t) $ is continuous decreasing function of $t$.
$a_i\ge0$ for all $i$.
$y(t)$ is positive real zero of the first equition.
Can ...

**-3**

votes

**1**answer

208 views

### Does differentiation widen, or narrow, the class of functions?

Let $\cal F^k$ be a set of functions, each of class $C^k$,
i.e., both, for every function in $\cal F^k$:
$k^{\textrm{th}}$ derivatives exist, and
are continuous.
Let $D(\cal F^k)$ be the set of ...

**6**

votes

**1**answer

278 views

### Law of unconsious statistician: application in characteristic function

Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...

**-1**

votes

**2**answers

734 views

### What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$? [closed]

I happened to ponder about the differentiation of the following function:
$$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$
Now, while I do know how to manipulate power towers to a certain ...

**5**

votes

**3**answers

656 views

### A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it?
$$\frac{d^n}{dx^n}f(x) = \lim_{h ...

**2**

votes

**2**answers

296 views

### Minimum of an apparently harmless function of two variables

DISCLAIMER: I already posted this question on Mathematics a month ago, here. However, since it has not been solved yet on that platform, I decided to ask it also here on mathoverflow. At a first ...

**-1**

votes

**1**answer

59 views

### growth of an entire solution of a differential equation

Let $d$ be an integer and $\sum_{k=0}^dP_k(z)y^{(k)}(z)=0$ be a differential equation over $\mathbb C$, where the $P_k$ are polynomials of degree $\le d$. Consider (if it exists) an entire solution ...

**2**

votes

**1**answer

125 views

### Monotonicity of the integral

Let $R(x)$ be the residual function associated to the normal probability density, i.e.
$$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$
Define
...

**0**

votes

**1**answer

89 views

### Why does optimization of a sum of two terms result in “neat” answers? [closed]

This is a somewhat vague and philosophical question.
Consider the following three problems:
Problem 1:
Minimize over all real-valued $x,$ the function $f(x) = bx-ax^2$ where $a,b>0.$
...

**2**

votes

**4**answers

650 views

### Intrinsic definition of arc length [closed]

Is there an intrinsic way of defining the arc length of a curve in $\mathbb{R}^{3}$, that is without resorting to a parametrization of the curve?

**2**

votes

**1**answer

71 views

### Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...

**5**

votes

**2**answers

298 views

### Common roots of polynomial and its derivative

Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots ...

**2**

votes

**2**answers

143 views

### About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...

**2**

votes

**1**answer

188 views

### Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...

**-1**

votes

**1**answer

134 views

### How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable?
In this context PSD means symmetric PSD.

**2**

votes

**0**answers

120 views

### Slice a compact C1 surface in R3 by a moving transverse plane. Does the length of the slice depend C1 on the plane?

To be more precise I am interested in questions similar to the one below
(I asked the question below on math.stackexchange last week but got not answer.)
I have a $C^1$ function $f:[0,1]^2 \to ...

**2**

votes

**1**answer

99 views

### Is first term of my cost function convex?

I have an optimization problem in the form of
[\begin{array}{l}
\mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...

**3**

votes

**1**answer

174 views

### A functional equality

I don't know if this is known, but I was fiddling around with this equality :
$$f:(-1,1)\to (-1,1)\quad \text{satisfies}\quad(f(z)+1)^s=\sum_{j=0}^{\infty}\dbinom{s}{j}f(z^k)
\quad \forall z\in ...

**1**

vote

**1**answer

232 views

### Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function
$h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is:
...

**4**

votes

**0**answers

239 views

### Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...

**3**

votes

**0**answers

167 views

### Looking for author of calculus quote

When I was a lowly calculus student many many years ago, my calculus teacher quoted some famous mathemtician: "Calculus is the last course in arithmetic and the first course in mathematics that one ...

**3**

votes

**1**answer

92 views

### Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$.
I would like to know how many extrema $p$ has on the standard simplex
...

**1**

vote

**1**answer

223 views

### Request for help with two integrals

It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...

**1**

vote

**0**answers

176 views

### Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...

**2**

votes

**1**answer

2k views

### The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In particular, it's the case ...

**8**

votes

**1**answer

1k views

### Differential Calculus and the De Rham Homotopy Operator

Suppose $M$ is a smooth manifold, $\Omega^k(M)$ the vector space (or $C^\infty(M)$-module in case this is better suited) of $k$-differentialforms on $M$ and
$$I: \Omega^k(M\times \mathbb{R}) \to ...

**4**

votes

**0**answers

183 views

### Co-filtered and pro-finite manifolds, filtered algebras, and differential calculus on them

Hello! I've come across a lot of questions (and nice answers) on MO, concerning infinite-dimensional manifolds and differential calculus over them, but nothing suiting the simpler and special case I ...

**22**

votes

**2**answers

879 views

### Is there a convenient differential calculus for cojets?

I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the ...

**17**

votes

**4**answers

7k views

### How did Bernoulli prove L'Hôpital's rule?

To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...

**1**

vote

**0**answers

3k views

### What is the geometric meaning of the third derivative of a function at a point? [closed]

What is the geometric meaning of the third derivative of a function at a point?
This question is now asked on the sister site: ...

**2**

votes

**2**answers

356 views

### Finding the Universal Ideal of a (Covariant) Differential Calculus

Let $(\Omega,d)$ be a differential calculus over an algebra $A$. It is easy to show that $\Omega$ is always equal to a quotient of $\Omega_u(A)$, the universal calculus over $A$, by some ideal $N$ of ...