# Questions tagged [differential-calculus]

The differential-calculus tag has no usage guidance.

163
questions

-1
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### Concern about the degree of rational functions that are quasi-homogeneous in differential Equations [closed]

I came across an interesting set of problems as follows:
$$
\begin{align}
\dfrac{dy}{dx} &= \dfrac{1-xy^2}{2x^2y} \label{1}\tag{1}\\
\dfrac{dy}{dx} &= \dfrac{2+3xy^2}{4x^2y} \label{2}\tag{2}\\
...

0
votes

0
answers

111
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

75
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 ...

0
votes

0
answers

33
views

### Does the gradient theorem holds for a continuous function with weak derivatives on a convex set?

Let $\Omega$ be a convex open set in $n$-dimensional Euclidean space whose closure is compact.
Let $f$ be a real-valued continuous function on $\overline{\Omega}$ which also belongs to the Sobolev ...

2
votes

1
answer

124
views

### 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 ...

0
votes

1
answer

113
views

### 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
views

### 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

275
views

### 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

188
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

106
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

256
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

119
views

### 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
votes

1
answer

158
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
vote

0
answers

28
views

### 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, ...

0
votes

0
answers

93
views

### The problem: what should be the value of $(2n^2+1)^2$

I know that this question is not at the research level, but I am getting no response to this question on mathstackexchange (even after a zillion days) , so I thought that asking it here might help. I ...

0
votes

0
answers

108
views

### For a vector field $f$ and a measure $\mu$, does conservativity $\mu$-almost everywhere have a sense?

$\DeclareMathOperator\Jac{Jac}$
Let $\Omega$ an open star shaped subset of $\mathbb{R}^d$ and $f : \Omega \rightarrow \mathbb{R}^d$ a differentiable vector field. For $x \in \mathbb{R}^d$, we denote ...

2
votes

0
answers

30
views

### 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\...

1
vote

0
answers

67
views

### 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
answers

129
views

### 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
answer

75
views

### 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

59
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

103
views

### 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

55
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
vote

0
answers

28
views

### 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

228
views

### 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

91
views

### 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

218
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

206
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

83
views

### 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

451
views

### 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

156
views

### 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

515
views

### 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

83
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

69
views

### 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

338
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

631
views

### 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

130
views

### 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

264
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

91
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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

117
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?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$

2
votes

0
answers

39
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.
$...

3
votes

1
answer

186
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(-...

0
votes

0
answers

80
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 $\|\...

2
votes

0
answers

65
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 ...

-1
votes

1
answer

86
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 ...

4
votes

0
answers

109
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, ...

-2
votes

1
answer

219
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}(...

-1
votes

1
answer

98
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 ...

5
votes

1
answer

527
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 \...

0
votes

1
answer

189
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{\...