# Questions tagged [differential-calculus]

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173
questions

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### Does this version of the Wallis Product have a name?

In a particular context I am looking into, it is very natural to express the Wallis product expression for $\pi/2$ in the following form:
$\frac{\pi}{2} = \prod{\frac{\sqrt{-2x}^4}{\left( \sqrt{-2x} +...

-1
votes

0
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91
views

### Derivative of a variable with respect to the function

Assume I have a misfit function defined as
$$
J=\frac{1}{2}\left( \Delta \tau \right)^{2}
$$
where $\Delta \tau$ is defined as the time difference or time shift between two time series $u(t)$ and $d(t)...

0
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0
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28
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### On constructing the canonical boundary operator for a given differential operator

Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...

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0
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37
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### Symmetric expression of boundary term in integration by part

Suppose $\Omega\subset\mathbb{R}^2$ be a smooth domain. $f,g\in C^\infty(\Omega)$. We consider the integration by part here:
$$\begin{aligned}
\int_{\Omega}(\partial_1\partial_2f)g&=-\int_{\Omega}(...

4
votes

1
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461
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### de Rham's Theorem using atlases and colimit preservation

$\DeclareMathOperator\DR{DR}\newcommand\SmoothManifold{\mathrm{SmoothManifold}}\newcommand\ChainComplexes{\mathrm{ChainComplexes}}\DeclareMathOperator\Sing{Sing}\newcommand\DifferentialGradedRAlgebras{...

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0
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169
views

### Reconstructing an object from its shadow

I'm looking into the section "Reconstructing an object from its shadow" in the book Introduction to the Mathematics of Medical Imaging by Charles L. Epstein.
I have two questions
The ...

2
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0
answers

191
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### Is there a geometric or calculus-based reason why the following system of equations should have only one solution?

Let $x_1,x_2,x_3,x_4>0$. Consider the following cyclic system of equations:
$$ 2 + x_2 + x_3 + x_4 + x_2 x_3 x_4 - 2 \left( \frac{x_2}{ \sqrt{x_1 x_2}} + \frac{x_3}{ \sqrt{x_1 x_3}} + \frac{x_4}{ \...

3
votes

1
answer

141
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### Behaviour of the solution of a second order ODE

I am currently studying the following second order ODE
\begin{cases}
\ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\
y(0) = 0\\
\dot y(T) = c
\end{...

1
vote

1
answer

76
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### Joint maximizer of a strongly concave function

I have a question that is arising in my research.
Suppose that $f : \mathbb{R}^ 2 \to \mathbb{R}$ is a strongly concave function, satisfying:
For every $x$, the function $y \to f(x, y)$ is maximized ...

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53
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### Integral of non-Gaussian distributions

In physics, we have an non-Gaussian Distribution which can be simply written as $f(x)=\exp(-ax^2-bx^3)$, and we may need to calculate the integral of this distribution, simply written as $\int_0^\...

2
votes

0
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945
views

### On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...

7
votes

2
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602
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### Interpretation of second order term in Fokker-Planck equation

Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by
$$
\begin{align*}
\nabla^2\cdot G(x)
&=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...

2
votes

0
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81
views

### How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$

Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...

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85
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### Blow-up of solutions to Euler-type ODEs

Let $\ell\in \mathbb{N}$, $a>2$, $C<0$ and $D \in [0,\infty)$. Consider the function $f: [1,\infty) \to \mathbb{R}$ solving
$$[r(r-2/a)f'(r)]' = \frac{f(r) - D}{r(r-2/a)}+ \ell(\ell+1)f(r)$$
$$f(...

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0
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133
views

### Integration on algebraic curves

Consider the plane algebraic curve
$$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$
Its compactification results in a Riemann surface $C_1$ of genus $1$.
Hence, it can be ...

1
vote

0
answers

116
views

### Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...

2
votes

1
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158
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### Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case

I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...

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votes

1
answer

124
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### Can you help me prove this vector identity?

It could be that the preprint where I found this identity has a typo or that it is simply wrong, but I have been trying to see if this is true:
\begin{equation}
\int \left(\nabla\times F_{\bf B}\...

5
votes

3
answers

1k
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### Solving a limit about sum of series

what's the limit of
$\sqrt{1-t}\sum _{n=0}^{\infty}t^{n^2}$ as $t$ goes to the left of $1$? i.e. $t\to 1^{-}$? I tried several times but failed. Here is my thought:
This is a $0\cdot\infty$ problem, ...

3
votes

1
answer

284
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### Are all Helmholtz decompositions related?

Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...

0
votes

2
answers

204
views

### Does surface integral preserve the curl operation?

Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...

3
votes

1
answer

108
views

### Is $\Phi(-a(x+b))+\Phi(-a(x-b))$ log-concave in $x$ over the interval $x \in [0, \infty)$?

Let $f(x)=\Phi(-a(x+b))+\Phi(-a(x-b))$, where $\Phi(\cdot)$ is the c.d.f of the standard normal, and $a>0$.
I would like to know if $\partial^2 \ln(f(x))/\partial x^2<0$ over the interval $x \...

1
vote

1
answer

320
views

### Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential ...

0
votes

1
answer

150
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### Conditions for surface area of surface of revolution to be product of arclengths

Given a circle $C$ in the xz-plane which does not intersect the $z$-axis, we can build a smooth 2-torus with surface area $(2\pi a)(2\pi b)$ where $a$ is the radius of the circle $C$ and $b$ is the ...

3
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1
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169
views

### Example of homeomorphism that lifts to real blow up but not C^1?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(...

1
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0
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28
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### Finding variance-minimizing weights [closed]

I'm trying to solve the following matrix calculus problem:
$\text{argmin}_{v \in R_+^K}(v'\Sigma v) \hspace{0.5pc} \text{subject to} \hspace{0.5pc} 1'v=1$
where $\Sigma$ is a well-behaved (symmetric, ...

2
votes

0
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### What are the limits of what the theory of time-scale calculus can capture?

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article 1 and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...

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vote

0
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146
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### Time-scale calculus (an similar approaches - measure chains) on more general "time" sets

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...

6
votes

0
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134
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### Injectiveness of a monotonic surjective mapping $\mathbb R^n \to \mathbb R^n$ with $\det J \neq 0$

Consider a surjective mapping $F \colon \mathbb R^n \to \mathbb R^n$, $F\in C^1$, $\dfrac{\partial F_i}{\partial x_j} > 0$, and $\det \left(\!\left( \dfrac{\partial F_i}{\partial x_j} \right)\!\...

0
votes

1
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89
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### Rotation of the coordinate system for multi-index differentiations

Let $\mathbf{f} = (f_1,\dotsc, f_n)$ be a $C^l$-map from an open subset $U$ of $\mathbb R^d$ to $\mathbb R^n$, and let $x_0 \in U$ be such that $\mathbb R^n$ is spanned by partial derivatives of $f$ ...

3
votes

0
answers

73
views

### How well do Gauss-Legendre quadrature methods fare on "fractal" functions?

The context
I'm making your tipical Mandelbrot set viewer, and I have a function $f: ℂ → ℕ$ that counts how many iterations of
$$
z_0 = 0 \\
z_{i+1} = z_i^2 + c
$$
it takes for a particular point $c$ ...

4
votes

0
answers

135
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### Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root

Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...

0
votes

0
answers

65
views

### Integration of matrix form of Vasicek variance (Python/Matlab)

$X_t$ is a vector and follows the following Vasicek process.
$$
dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\
$$
What is the variance of $X_t$?
In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...

1
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0
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### How to relate this integration with the integral expansion of special functions?

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, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...

4
votes

1
answer

241
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### Does the homeomorphism have a non-negative or non-positive determinant?

Let $ \Omega_1 $ and $ \Omega_2 $ be domains (open and connected) in $ \mathbb{R}^2 $. $ \psi:\Omega_1\to\mathbb{R} $ and $ \phi:\Omega_1\to\mathbb{R} $ are $ C^1 $ functions with two variables. ...

0
votes

2
answers

118
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### The relation between the convergence of the infinite integral of xf' and f

Question:
Let $ f $ be a real-valued function that differentiable on $ [a,+\infty) $. Suppose that $ f $ is monotonically decreasing, $ \lim_{x\to+\infty} f(x) = 0 $ and the integral $ \int_{a}^{+\...

0
votes

1
answer

248
views

### $f'(x)>f(f(x))$ implies $f(f(f(x)))\leq0$ for nonnegative $x$

If $f\in C^1(\mathbb R)$ satisfies $f'(x)>f(f(x))$ for all $x\in\mathbb R$, then $f(f(f(x)))\leq0$ for all $x\geq0$.
I have some trouble to prove this. I wonder if there's some relations between ...

4
votes

1
answer

235
views

### Ratio of the first squared and the second moment

Let $G(t)$ be a probability generating function of some integer and non-negative r. v. $X$. Suppose that
$$\lim_{t\to1}G'(t)=+\infty.$$
That is
$$
\mathbb{E}X=+\infty.
$$
Can you show that
$$
\lim_{t\...

3
votes

1
answer

91
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### Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$

The motivation for the following is to convert the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=-kx+\beta\int_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds,
\end{equation}
into a ...

5
votes

3
answers

571
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### Osculating circle

(This question may be too elementary for this site — I'm fine if it needs to be moved to math.stackexchange.)
If I approximate a nice planar curve by a straight line, the tangent, then the second ...

0
votes

1
answer

164
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### Solution of this differential equation [closed]

I wonder if it is possible to solve analytically the following equation
$$
\dot{\alpha}_t = -\frac{2}{m} \alpha^2_t + \frac{1}{2m} (\alpha_t - \alpha_t^*)^2
$$
Where $\alpha_t$ is a complex function, $...

4
votes

1
answer

531
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### Hegel's disproof of Newton [closed]

I know it's not a very comprehensive question but I've nowhere else to ask. A friend relayed to me a portion of a book from Hegel where he seemingly disproves Newton's way of finding a differential. I ...

1
vote

0
answers

84
views

### In matrix product, differentiate one element with respect to another element

Background
Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have
$$ AX_{t+1} = CX_t + M $$
where matrix $M$ is a ...

3
votes

1
answer

84
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### Existence and uniqueness of an Euler-type ODE with varying parameters part 2

I am working on some non-local differential equations that appear in geometric analysis.
One of which I posted here and was answered by @WillieWong and @losifPinelis.
Consider this non-local ...

6
votes

2
answers

402
views

### Existence and uniqueness of an Euler-type ODE with varying parameters

Consider this ODE on $[1, \infty)$
$(r^2 - 2ar)f''(r) + 2(r-a) f'(r) - ({4a} + m(m+1))f(r) = -4af(1) $
with initial conditions
$\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$
...

4
votes

1
answer

1k
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### Laplace-Beltrami of the mean curvature

For a surface $S$ defined in 3D space, denote its mean curvature as $H$, and the Laplace-Beltrami operator as $\Delta_S$. I know that there is a result for the Laplace-Beltrami of coordinate functions:...

3
votes

0
answers

156
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### Extension of normal vector field to a domain

Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface ...

7
votes

1
answer

379
views

### A property of $C^2$ functions

Let $f\in C^2(\Bbb R^m), f\geq 0$, Hessian matrix of $f$ is upper bounded by some constant $C$. Do we have $|\nabla f|\leq \alpha \sqrt{f}$ for some $\alpha$, even if the Hessian matrix is degenerate?

1
vote

1
answer

125
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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 ...

1
vote

0
answers

120
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### 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?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$