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Questions tagged [divergence]

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point. The divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

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Minors of a matrix $\nabla^{2}\phi$ are divergences of vectors?

Let's consider minors of a matrix $\nabla^{2}\phi$. Let's $M_{ij}$ is the minors ($1\leq i\leq n$, $1\leq j\leq n$). Is each $M_{ij}$ a divergence of a vector? I want a hint for this. Thanks in ...
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Space of divergence-free vector fields on a Riemannian manifold

Let $(M,h)$ be a smooth Riemannian manifold of dimension $d\geq 1$ with smooth metric. Set $X:=\{A= \mbox{smooth vector field s.t. } div_h A=0 \}$. Then $X$ is an infinite dimensional vector space. ...
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DGL: variational problem in $H^1$

I've given $f \in L^2(\Omega), g \in H^1(\Omega)$. I want to find $u \in H^1(\Omega)$ such that $$ -div (A \nabla u ) + <b, \nabla u> + cu = f$$ in $\Omega$ $$u = g$$ on $\Gamma$. In order to ...
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How to evaluate $\nabla \frac 1r$, $\nabla^2 \frac 1r$, $\int_S \nabla \frac 1r . ndS$

Let S be a smooth closed surface in a three-dimensional xyz-space, n, be the unit outward normal vector on S, and r be the distance between the origin and a point (x, y, z). Solve the following ...
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How to use the Divergence Theorem in this question?

Question: Let $r=\sqrt {x^2+y^2+z^2}$ and $\mathbf E = -\mathbf \nabla \big(\frac kr \big)$ where $k$ is a constant. Show that $$ \iint_S \mathbf E \cdot d \mathbf S = 4\pi k$$ where $S$ is any ...
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Jensen–Shannon divergence relation between marginal and joint

I have the following distributions- Joints: $p_1$(x,y) = $\rho_1$(x) $q_1$(y|x), $p_2$(x,y) = $\rho_2$(x) $q_2$(y|x) And the marginals: $\rho_1$(x), $\rho_2$(x). Is it possible to provide any ...
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what is the different between using $w(x)=\text{div} \int_D{g(y)\Phi(x,y)dy}$ and $w(x)= \int_D{\text{div} g(y)\Phi(x,y)dy}$?

Theorem: Let $D$ be a bounded domain of class $C^2$, $v$ denote the unit normal vector to the boundary $\partial D$ directed into the exterior of $D$ and let $w \in C^2(D)\cap C^1(D)$, then we ...
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Speed of vector field must be uniform if divergence is 0?

This is what I initially thought. 1.If $div F = 0$ 2.no compression is allowed in the field 3.velocity must be uniform at all points of the vector field at a specific time I knew that this ...
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Divergence of $1/r$ in cylindrical coordinates

In classical textbooks, like "Introduction to Electrodynamics" by J.D. Griffiths, it is given that $$\nabla\cdot\left(\frac{\widehat{r}}{r^2}\right)=4\pi\delta^3(R).$$ To prove this equality, ...
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is there a way to calculate KL diverrgence if you have mean and std of two distributions?

I have only two information. For an example: N(0,1) and N(0.1,0.2). Is it possible to calculate KL divergence between these two distributions? I would appreciate your help.
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Chain rule of multi-dimensional function

$A : \mathbb{R}^d \rightarrow \mathbb{R}$ and $y \in \mathbb{R}^d$ and $P$ is an invertible matrix. I want to calculate : $\text{div}_y {A(P^{-1} y)}$. How does the chain rule work here?
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Derivation of ELBO upon the Existence of Conditional Latent Variable Model

I am reading the recently published paper from DeepMind, "Neural Scene Representation and Rendering" and especially its "Supplementary Materials". Following is the page 1 and it's pretty hard for ...
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Calculation with divergence.

Note that this question is related to Divergence free is inherited under congruence? Let's $\sum_{j=1}^{d}a_{ij}(x)=A_{i}$ where $x\in\mathbb{R}^{d}$ and this satisfies $\text{div}A_{i}=0$. Of course $...
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Eigenvalues of the Divergence Operator

I am researching the spectrum of eigenvalues for the divergence operator on Riemannian manifolds and how they deform tensor fields. This is mainly motivated by trying to understand dynamical systems ...
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Divergence free is inherited under congruence?

Let's define "Divergence" $$ \text{Div}A(x)=\left(\sum_{j=1}^{d}\frac{\partial a_{ij}}{\partial x_{j}}\right)_{1\leq i\leq d}, $$ where $A$ is a $d$ by $d$ matrix and $x\in\mathbb{R}^{d}$. I use ...
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Is a sequence diverging almost surely to infinity almost surely positive?

I have proved that a sequence of random variables $(M_n)_{n\in\mathbb N}$ diverges to $+\infty$ almost surely. I.e I have proved that $$\bigcap_{c\in\mathbb Q^+}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^\...
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$\nabla^{2}\phi$ is $[0,1]^{d}$-periodic implies $\phi(x)=\frac{1}{2}x^{T}Sx+\text{linear}+\psi(x)$ where $\psi$ is $[0,1]^{d}$-periodic?

Note that $x\in\mathbb{R}^{d}$. $\nabla^{2}\phi$ is $[0,1]^{d}$-periodic implies $\phi(x)=\frac{1}{2}x^{T}Sx+\text{linear}+\psi(x)$ where $\psi$ is $[0,1]^{d}$-periodic? An article says the above ...
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$\nabla\int_{S_1(0)} f(|x|w)d\sigma(w) = \int_{S_1(0)} \nabla f(|x|w)d\sigma(w)$

Let $U=\{x:x\in \mathbb{R}^n, x\neq 0\}$ and $f\in C^2(U)$. Define, for $x\in U$, $$f_{\#}(x) = \int_{S_1(0)} f(|x|w)d\sigma(w)$$ Show that $\nabla(f_{\#}) = (\nabla f)_{\#}$ Hint: use the ...
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Why divergence of radius vector equals 3?

Ok, I know $div \vec r=3$. $\vec r = r_x\vec i+r_y\vec j + r_z\vec k$. $(\frac{\partial}{\partial x}i+\frac{\partial}{\partial y}j+\frac{\partial}{\partial z}k, r_xi+r_yj+r_zk)=\frac{\partial r_x}{\...
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Calculation of flux through sphere when the vector field is not defined at the origin

I am trying to calculate the flux through the unit sphere centered at the origin given a vector field $F:\mathbb{R}^3 \setminus \{(0,0,0)\} \rightarrow \mathbb{R}^3$ with $\operatorname{div} F=1/(x^2+...
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Total Generalized Variation [closed]

While reading the paper Total Generalized Variation, I came across the following formula (Equation 1.5), which describes the regularization term: Total Generalized Variation formula While there are ...
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What are the gradient, divergence and curl of the three-dimensional delta function?

The three-dimensional delta function is defined as follows: $$\delta(\mathbf{r}-\mathbf{r'})= 0 \;\; \mathrm{for} \;\;\mathbf{r}\neq\mathbf{r'} $$ $$\delta(\mathbf{r}-\mathbf{r'})= \infty \;\; \...
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Gradient of the loglikelihood for the RSM (contrastive divergence)

I'm actually implementing an RSM in TensorFlow and I've realized that when I used the energy function and let TensorFlow compute the gradient, the ouput of my RSM (Replicated Softmax Model) is pure ...
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What difference does it make between a single integral and a double integral in surfaces?

I am a little confused what difference is there when it comes to first and double integrals when dealing with surfaces. For example if have to find $$\int_S (2xy\textbf{i} + yz^2\textbf{j} +xz\...
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Stochastic Neighbor Embedding (SNE) - How to Understand the Cost Function of the Kullback Leibler (KL) Divergence

In the paper 'Stochastic Neighbor Embedding', the cost function in term of K-L divergence is $$ C = \sum_i \sum_j p_{ij} \log \frac{p_{ij}}{q_{ij}} = \sum_i \operatorname{KL}(P_i \| Q_i). $$ ...
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What is the value of $\nabla\times (\vec a\times \vec r)$?

Given that $\vec r=xi+yj+zk$ & $\vec a$ is a constant vector then what is the value of $\nabla\times (\vec a\times \vec r)$ ? a) $\ -2\vec a\quad $ b) $\ 2\vec a\quad $ c) $\ 3 a\vec r\quad $ ...
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2D Divergence Theorem (example)

Let $\Omega \subseteq \mathbb{R^2}$ be an open set, $\partial \Omega=C$ a piecewise smooth, simple closed curve and $f,g \in C^2(\Omega).$ To show that $$\oint_C\bigg[f\frac{\partial g}{\partial n}-g\...
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Stokes equations and change of variable

Let $\Omega$ be domain of $\mathbb{R}^n$ and $\Phi : \Omega \to \Phi(\Omega)$ a deformation. Consider the Stokes equations written in the deformed configuration \begin{align} - 2\mu \operatorname{div}...
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Divergence of a radial function [closed]

Is there any proof of the relation $$\nabla\cdot f(r)= \frac{1}{r^2}\frac{\partial}{\partial r}({r^2}f(r))$$ Is it true for any radially directed function or for some specific function?
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Pole in limit expression

I would like to compute $$ f(E,q):=\lim_{\epsilon\to 0}\text{Im}\left[ \frac{(E^2-q^2)^2-i\epsilon}{(E^2-q^2)^4+\epsilon^2}\left( E^2 + \alpha(m+in) \right) \right] $$ with an infinitesimal ...
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Is the divergence of the curl of a $2D$ vector field also supposed to be zero?

In three dimensions, it seems pretty straightforward to prove the identity that for any vector field $\mathbf{A}$, $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ Does this identity still hold true ...
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Understanding this very generic divergence theorem where the open set have border $C^k$

I'm at a PDE class and my teacher gave a very generic definition of the divergence theorem. I can't find it anywhere. It's something like this: Definition: let $k\in \{1,2,\cdots,\infty\}$, $N\ge 2$ ...
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Fourier transform of product of Bessel functions

I need help finding the Fourier transform of the function $$ \rho(\vec{r}) = \alpha \delta_{\vec{r},0} \left(\lambda\lambda' J_1 (\beta |\vec{r}|)Y_1(\beta |\vec{r}|) - \pi^2 J_0 (\beta |\vec{r}|)Y_0(...
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Are all Distance Metrics, Divergence Functions?

Quite simply, as the question states. Are all distance metrics, divergence functions? The wikipedia defintion for a divergence function is given as Suppose S is a space of all probability ...
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How to show if a sequence diverges to infinity then it has no limit point? [closed]

Consider a set of numbers in a divergent sequence $x_n$. My deduction: If a set has no limit point, then there does not exist any sequence in the set that converges. Then if {$x_n$} has no limit ...
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$\varphi$ verify that $\nabla \cdot\varphi=0$ but doesn't exist $G:\Bbb R^3\to \Bbb R^3$, $\mathcal C^1$ such that $\nabla \times G=\varphi$

I have this problem: Probe that $\varphi:\Bbb R^3-[0] \to \Bbb R^3, \varphi=\frac{(x,y,z)}{\vert\vert(x,y,z)\vert\vert^3}$ verify that $\nabla \cdot\varphi=0$ but does not exist $G:\Bbb R^3\to \Bbb R^...
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Evaluate the integral $ \ \iint_S F \cdot dS \ $ where $ \ S: \ x^2+y^2+z^2=16 \ $

Evaluate the integral $ \ \iint_S F \cdot dS \ $ where $ \ S: \ x^2+y^2+z^2=16 \ $ and $ \ F(x,y,z)=\left\langle z,y,x \right\rangle \ $ Then verify that, $$ \iint_S F \cdot dS=\iiint_V div(F) dV \ ...
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Laurent series of an integral with parameter

To find the Laurent series of function $f(a)$ at point $a=0$ $$ f(a)=\int^1_0 \frac{d x}{x^2+a^2} $$ one can first do the integral $$ f(a)=\frac{1}{a}\arctan(1/a) $$ then expand $\arctan(1/a)$ and ...
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Differentiate an energy containing integral in a region to derive curve evolution

I mainly aim to understand the following paper: Tomographic reconstruction of piecewise smooth images https://ieeexplore.ieee.org/document/1315083/ Here we want to minimize the energy where $p$ is ...
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About the definition of divergence and curl

I was wondering about a simple detail. As you probably already know, there are two definitions curl and divergence can be defined in the following way (in $\mathbb{R^{2}}$ but the question is also ...
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Physical interpretation of divergence for a velocity vector field

Suppose we have a velocity vector field given by $\mathbf v=k_1x^2\ \mathbf{i}+k_2y\ \mathbf{j}+k_3z\ \mathbf{k}$. $\displaystyle\text{div }{v}=\frac{\partial(k_1x^2)}{\partial x}+\frac{\partial(...
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Calculating divergence of function containing mollyfier

Let $(\phi_\alpha)_{\alpha>0}$ be a familiy of mollyfiers, $\phi_\alpha:\mathbb{R}^n \rightarrow \mathbb{R}$ defined as: \begin{align} \phi_1(x)=\left\{\begin{array}{rcl} c \cdot exp(\frac{-1}{1-\...
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How do I prove that if the divergence of a field is 0, then it's a curl of another field? [closed]

The title says it all, Thank you very much in advance, having an exam tommorow :)
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Computing $\int_{S} \vec{F}.\hat{n}\,dS$ where $S:\{(x,y,z)\in R^3:x^2+y^2+2z=2,z\geq 0\}$

Let S be the surface $\{(x,y,z)\in R^3:x^2+y^2+2z=2,z\geq 0\}$ and let $\hat{n}$ be the outwards unit normal to S. If $\vec{F}=\langle y,xz,x^2+y^2\rangle$ then find the value of $\int_{S} \vec{F}.\...
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How to calculate or expand the divergence: div [mu* grad(U)^T ]

We know that the viscous stress term in the Navier-Stokes Equations for in-compressible Newtonian flow is: $\nabla \cdot \tau = \nabla \cdot \big[\mu \nabla \textbf{U} + \mu (\nabla \textbf{U})^T \...
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Equivalence of notation in Divergence Theorem

Normally I read the Divergence Theorem written as (\oiint doesn't exist here): \begin{align} \oint_{\partial \Omega} \vec{F} \cdot \hat{n} \; dS = \iiint_{\Omega} \...
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Finding the flux of a cylinder using the Divergence Theorem

I have to calculate the flux of the $$F(x,y,z)=(x+\ln(yz^{2}),y+e^{-(x^{2}+z^{2})},2z)$$ through the cylinder (without its bases): $$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ for $z\in[0,4]$. I know that ...
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Implying continuity from knowing divergence/curl/laplacian of vector function equals zero

my question comes from a physics question but is math related. In an electrodynamics related question I've seen the following declaration: Link to a screenshot . In the problem the author is using ...
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25 views

Is it possible to evaluate this integral for general volume $V$?

If, for a volume $V$ in three dimensions, we have $$\nabla ^2 u=0 \qquad \text{ in }V \\ u(x,y,z) = ax+by+cz \qquad \text{ on } \partial V$$ then is it possible to find the value of the integral $$\...