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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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Ramanujan's 1/pi Series: Proving that $a_{n+1} < La_n$ (for $n \geq 1$) implies that $a_n< L^{n-1}a_1$ for $n \geq 2$.

I don't know how to go about this question regarding Ramanujan's formula: $$\frac{1}{\pi} = \sum_{n=0}^{\infty} \frac{\sqrt{8}(4n)!(1103+26390n)}{9801(n!)^4396^{4n}}$$ Let $a_n$ denote the nth term ...
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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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58 views

Does $\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$ converge or diverge?

Does this series converge or diverge? If it converges, determine its limit. $$\sum_{n=1}^{\infty}\frac{1}{2^n} + \frac{3}{n}$$ So far I said that $\frac{1}{2^n}$ is a geomotric series that converges,...
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1answer
45 views
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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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41 views

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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23 views

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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2answers
51 views

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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2answers
43 views

$\varphi$ 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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51 views

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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14 views

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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46 views

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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1answer
24 views

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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1answer
22 views

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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35 views

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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2answers
55 views

Does an alternating sequence converge or diverge or none?

How come this sequence does not approach any limit? $\{\max((-1)^n,0)\}^\infty _{n=1} : {0,1,0,1,0,1,0,1,...}$ I read that since this alternates between 0 and 1 this does not approach any limit. ...
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1answer
40 views

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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1answer
37 views

Assessing Whether the Series $\sum_\limits{n = 0}^\infty (-1)^n x^{2n} = 1 - x^2 + x^4 - x^6 + \dots$ Converges (conditional/absolute) or Diverges?

It's been a while since I've used the various tests for convergence (conditional/absolute) and divergence, and I can't remember which test needs to be used and how to assess whether the series $\sum_\...
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1answer
32 views

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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1answer
30 views

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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1answer
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 $$\...
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1answer
52 views

Divergence and Curl of a vector field $\vec{F}$

When given a two dimensional vector field $\vec{F}(x,y)$ drawn in the plane, how can one tell the sign of the divergence of $\vec{F},$ namely whether it's positive, negative, or zero? Another ...
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1answer
83 views

Understanding some linear algebra for KL derivation

Having some trouble understanding this proof in certain steps, even after trying to consult the matrix cookbook. For two multivariate Gaussians $P_1, P_2 \in R^n$: $KLD(P_1 || P_2) = E_{P_1}[\log ...
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What are the common ways to measure the distance/divergence between two stochastic matrices?

Let say two matrices $A$ and $B$ are stochastic matrix with the same size. How to measure the distance/divergence between them? For example: $\|A - B\|^2_F$ may work. Any other smart choices?
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2answers
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Every $L^2$ function is the divergence of a $L^2$ vector field

As in the title, I am struggling with the following statement: For any $f\in L^2(\mathbb R^n)$ there exists a vector field $F\in L^2(\mathbb R^n, \mathbb R^n) $ such that $f=div F$. Apparently ...
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1answer
62 views

Verify the divergence theorem for this surface and vector field.

Define the vector field $F(x,y,z) = (2xy+z)\hat i + y^2\hat j - (x+3y) \hat k$ and let the surface S be the closed surface consisting of the boundry of the solid enclosed by the four planes: $2x+2y+z=...
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1answer
69 views

Relative entropy (divergence) of sum of two Gaussian random variables

Let $X$ and $Y$ be two Gaussian randoms with $X \sim N(m_1, s_1)$ and $Y \sim N(m_2, s_2)$ and density functions $f_1(x)$ and $f_2(x)$. Let $f(x)$ be the density function of $X = X_1 + X_2$. I want ...
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15 views

Computing surface integral over a given domain.

Let $\Omega$ be a smooth plane domain of unit area. Let $u(x, y) = 3x^2 + y^2$. If $\frac{\partial u}{\partial n}$ denotes its outer normal derivative on $\partial \Omega$, the boundary of $\Omega$, ...
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1answer
32 views

Finding vector differential operators with arbitrary coordinates using differential forms

Given a vector field $F=(f_1,f_2,f_3)$, we can associate it to a differential 2-form $\omega = f_1 dy\wedge dz+f_2 dx\wedge dz+f_3dx\wedge dy$. Taking the exterior derivative is sort of an equivalent ...
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Prove the following set is compact in $\mathbb{C}$

Let $f$ be a polynomial of degree greater than 1. Prove that the following set is compact: $K(f)= ( z \in \mathbb{C}: f(f(f(...f(z)...)))$ does not diverge ) Alternatively, the set of all complex ...
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Divergence theorem version of Green's Theorem

$$\iint_D \left( \frac{\partial P}{\partial x}+ \frac{\partial Q}{\partial y} \right) \, dx \, dy = \iint_D \operatorname{div} F \, dA$$ is the divergence theorem version of Green's Theorem I believe,...
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102 views

Why does divergence represent expansion or contraction?

Why does $\mathrm{div}\ V$ represent how much $V$ is expanding or contracting? By its definition I know that diverging means deviating from its original path, but what about $\mathrm{div}\ V$ makes it ...
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1answer
43 views

Is the sum of the reciprocals of Ramanujan primes divergent?

I have read many a wonderful proof that the sum of the reciprocals of the primes is divergent and I know that the sum of the reciprocals of twin primes does not diverge, but do we know any results ...
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Example of complex power series boundedly divergent on some set.

I am looking for an example of complex power series that is boundedly divergent on some set, e. g. one point (and convergent in other points of unit circle). Does anyone know one? Such example is ...
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1answer
20 views

Show that divergence is a function of r only

In Boas mathematical methods it says that $$\nabla \cdot [ e_{r}F(r)] $$ is a function of only r for any F(r). I've tried to investigate why this is but I can't seem to get it right. When I do $$\...
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Does Helmholtz Decomposition make sense for $\hat{r}/r^2$?

Consider an inverse square vector field $\vec{F}$ and it's curl and divergence shown below $$\vec{F} = \hat{r}/r^2\\ \nabla \times \vec{F} = \vec{0}\\ \nabla \cdot \vec{F} = 0 $$ $\vec{F}$ is ...
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3answers
43 views

If $a_n\ge 0$ and $\sum a_n$ diverges, then $\sum \frac{a_n}{1+a_n}$ also diverges

Prove that if $a_n\ge 0$ and $\sum_{n=1}^{\infty} a_n$ diverges. Then $\sum_{n=1}^{\infty} \dfrac{a_n}{1+a_n}$ also diverges. I know that if $\sum_{n=1}^{\infty} a_n$ diverges then we do not know if $...
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1answer
37 views

Find scalar function so that vector field has no divergence

Given is the vector field $\vec{w} = f \left( |\vec{x}| \right)\vec{x} $. How do I find the scalar function $f$ so that $\vec{\nabla} \cdot \vec{w} = 0$ ??
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2answers
40 views

Divergence of the cross product of two gradients

Given are the following functions: $$\varphi_j(\vec{x}), j=1,2$$ and the vector field $\vec{w}$ is the defined as following: $$\vec{w} = \vec{\nabla}\varphi_1 \times \vec{\nabla}\varphi_2$$ what can ...
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1answer
31 views

Limit of a sequence with real and natural number variables

Currently learning about limits and I'd like to check out if I calculated that all correctly: number 1: SOLVED number 2: UNSOLVED $$a_n=\frac{c_2n^2+c_1n+c_0}{b_2n^2+b_1n+b_0} \text{with $c_1,c_2,...
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4answers
49 views

Vector Field, so that Divergence is a specified constant

To find is vector field $\vec{w}(\vec{x})$ so that the divergence results in a specified constant $C$. I set up the folling the equation: $$ \frac{\partial x_1}{\partial x_1} + \frac{\partial x_2}{...
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18 views

Why does a u-sub go wrong when finding values for q such that $\int_{1}^\infty x^q e^{x^{q+1}} dx$ converges or diverges?

I'm trying to determine for which values of $q\in\mathbb{R}$ this improper integral converges and diverges. I have discrepancy in my solutions, so there appears to be something wrong, or some subtle ...
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0answers
34 views

Short Proof of the Divergence Theorem

I'm attempting to prove that for a closed, path-connected, volume $V$ of $\mathbb{R}^3$, $$\iiint_{V} (\operatorname{div} F) \, \text{d}V = \iint_{\partial V} F \cdot \mathbf{\hat n} \, \text{d}S$$ ...
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0answers
49 views

My Generalized Mean Value Theorem

I'm trying to prove the following without the divergence theorem(for soon to be obvious reasons): Let $F: \mathbb{R^3} \to \mathbb{R^3}$ be a differentiable function. In $\mathbb{R}^3$, let $V$ be a ...
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1answer
74 views

Inverse-Square vector fields have both a divergence and curl of $0$?

Consider an inverse- square vector field $$ \vec{F} = \frac{x}{r^3}\hat{x} + \frac{y}{r^3}\hat{y} + \frac{z}{r^3}\hat{z} = \frac{\hat{r}}{r^2}$$ where $r = \sqrt{x^2 + y^2 + z^2}$. The curl $\nabla \...
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1answer
39 views

Convergent integral of divergent exponential function

$$$$ The following integral is a divergent integral: $$\int_{0}^{\infty} \frac{\mathrm{e}^{- a\, x}\, \sin\!\left(x\right)}{x^5} \,d x $$ However, the following solution is provided (though ...
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1answer
57 views

Integration of $\int_0^4 \frac{1}{\sqrt{|x-3|}} dx$ [closed]

I believe it might diverge, but how do you evaluate/show that it diverges (if it does)? I am especially not sure about how to deal with the absolute value sign when integrating.
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2answers
162 views

How to find the divergence rate of a recursive sequence defined by $s_n = s_{n-1}(1 + c_n s_{n-1})$

So my question is regarding a sequence of numbers defined recursively by $s_n = s_{n-1}(1 + c_n s_{n-1})$, where $(c_n)$ is some sequence of positive numbers when $s_0 = \epsilon > 0$ is small (how ...