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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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Divergence theorem over non-smooth manifolds

The divergence theorem on smooth surfaces is well understood and changes the integrand from div(X) to dot(X,n) where n is the normal of the boundary of the surface. Suppose my surface is peicewise ...
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Euclidean Norm of Curl / Divergence

Are there any general statements of the curl or the divergence of a 3-dimensional vectorial function, e.g. for the magnetic field: $$|\nabla\times\boldsymbol B\left(t,\vec{x}\right)| = ?$$ $$|\nabla\...
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Does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$ converge or diverge?

I'm having trouble figuring out if the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$$ Here's my thinking: $$\frac{2}{n} \leq \frac{3+(-1)^n}{n}$$ Since $\sum_{n=...
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Find the “region of interest” of an unknown function

Given an unknown function $f:\mathbb{R} \rightarrow \mathbb{R}$, is it possible to find it's region of interest? By that I mean either the range in which $f$ does not converge or diverge, e.g. $f(x)=(...
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Divergence theorem for vector field over top half of sphere [on hold]

$$F= 2xz^2i + (2(y^3)/3 + \tan z)j + (2(x^2)z + 5y^2)k.$$ Evaluate $∫FdS$ over the surface $S$, where $S$ is the upper half of the sphere centered at the origin, with radius $1$, oriented upwards.
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Does $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}-\frac{2}3}$ converge or diverge?

Does this series converge or diverge? $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}-\frac{2}3}$$ I tried using the limit comparison test with $\frac{1}{\sqrt{n}}$, which diverges. $$\lim_{n\to\infty}{\...
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Divergence Theorem - Cone

Here's the question: Evaluate the surface integral $\iint _S F\cdot n \space dA$ by the divergence theorem. $ \mathit F = [xy, yz, zx]$, S the surface of the cone $x^2 + y^2 \le 4z^2, \space \space 0 ...
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Taking divergence of the gradient of a scalar field that depends only on the position vector in $\mathbb{R}^3$.

The scalar field $f$ depends only on $r=|\underline{\mathbf{r}}|$ which is the position vector in $\mathbf{R}^3$ and I need to calculate the quantity, $$\nabla \cdot \nabla f$$ i went about ...
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Intrinsic definition of Jacobian matrix on manifolds

For a vector field $X:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, the Jacobian matrix at $p\in\mathbb{R}^{3}$ is defined as $$\mathcal{J}_{p}X:=\begin{bmatrix}\left.\frac{\partial X^{i}}{\partial x^{j}}\...
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Series convergence: $sin (n \frac{\pi}{2})$

Determine whether the following series : $$\sum_{n=1}^\infty \sin \left(\frac{n\pi}{2}\right) \frac{n^2+2}{n^3 +n}$$ converges absolutely, conditionally or diverges. I know that for even natural ...
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Surface integral of curl

Let $\vec F(x,y,z)=(x^2+y^2+\sin(xy)$, $e^x$+$2xy, -yz$). Calculate the flow of $\nabla\times\vec F$ through the following surface: $$ S=\{(x,y,z) x^2 + 2y^2 + 3z^2=10, y\ge 0\} $$ I figured that if ...
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Divergence and Curl

I was given a function $F(x,y,z)=(z^c,x^c,y^c)$ and asked to find divergence and curl. My initial answer was $0$, but i don’t think that’s right. I noticed the brackets weren’t the typical $\langle,\...
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Operation on a vector with a special property

I have a vector $\vec u$ with the following property: $$\vec u (tx, ty, tz) = t^n \vec u (tx, ty, tz)$$ Now I have to prove that: $$(\vec r.\vec \nabla)\vec u = n.\vec u$$ How can I do this? I tried ...
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Difficult Region of Integration Involving Gauss's Theorem

I'm told to use Gauss's Theorem to compute the flux of a field $\vec F = <x,y^2,y+z>$ along the boundary of the cylindrical solid $x^2+y^2 \le 4$ below $z=8$ and above $z=x$. I know by Gauss's ...
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Magnetic field at points on the circuit

I know magnetic field lines due to a circuit always form closed loops. Therefore $\nabla \cdot \vec{B}=0$ everywhere (even at points on the circuit). However due to singularity, magnetic fields are ...
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How to show $\vec{\nabla} \cdot \vec{P}=0$ over the whole domain of ${\mathbb R}^3$

If in ${\mathbb R}^3$, $r=\sqrt{x^2+y^2+z^2}\text{ and }\vec{P}= \dfrac{\partial}{\partial z} \left( \dfrac{1}{r} \right) (\hat{j}) -\dfrac{\partial}{\partial y} \left( \dfrac{1}{r} \right) (\hat{k}...
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If $\vec{\nabla} \cdot \vec{V} \neq 0$ at only one point, will this prevent us from saying that $\vec{V}=\vec{\nabla} \times \vec{U}$?

This question has an answer in the language of high level mathematics. Can somebody explain this in the language of vector calculus. Part I: Let us consider Cartesian coordinate system with origin $O$...
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Surface Integral of a Cone

I've been staring at Q1b) for ages, I've tried using cylindrical co-ordinates. But I keep on ending up with the area of the top circle being (pi * h3) /2 which is obviously wrong so I can't possibly ...
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Find the volume of the solid region $T$ enclosed by $S$

Let $S$ be the surface obtained by rotating the curve $$ x=\cos (t), \ y=0, \ z=\sin (2t), \ -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} $$ about the $z$- axis. Find the volume of the solid region $T$ ...
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Developing a cross product of tensors within integrals

I read in a book the following unproven statement: $\int_{s} u\times A n \, ds = \int_v ( u\times \nabla\cdot A + \mathcal{E}: A^T ) \, dv$ with a: 1st order tensor, n: normal vector of s, A: 2nd ...
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Why is divergence a measure of “outwardness”?

Why is divergence a measure of "outwardness"? Since it's a dot product of gradient and a function. However, the gradient is measure of change. What makes the change "outward"?
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Verify divergence theorem of paraboloid beneath a plane

Verify the divergence theorem for the vector field $\mathbf F =3x^2y^2\mathbf i +y\mathbf j -6xy^2z\mathbf k$ for the volume bounded by the paraboloid $z=x^2+y^2$ and $z=2y$ . I tried to compute ...
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Does every Riemannian manifold has a local orthonormal divergence free frame of vector fields?

Let $(M,g)$ be a smooth Riemannian manifold, and let $p \in M$. Is there an open neighbourhood $U$ of $p$ that admits an orthonormal frame of divergence-free vector fields? Edit: At least for ...
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Cauchy-Riemann conditions and vector fields

Can Cauchy-Riemann conditions of Complex Functions valid for vector fields, I observe when vector fields are irrotational and incompressible they possess a result similar to Cauchy-Riemann conditions. ...
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Will $\vec{f}(x,y,z)=\nabla \times \vec{g}(x,y,z)$ here?

Suppose we have a vector field $\vec{f}(x,y,z)$ defined everywhere in space except a certain region say $(y=3)$. We also know divergence of $\vec{f}(x,y,z)$ is zero at all points where $\vec{f}$ is ...
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How do subsequent steps follow? (Divergence and partial derivatives)

Let $(x,y)$ evolve to $(X,Y)$ under the velocity field $\vec v=(u,v)$ in time $t$. Thus, $$X=X(x,y,t), Y=Y(x,y,t), t=t$$ Then, $$J(x,y,t)= \begin{bmatrix} \frac{\partial X}{\partial x} & \...
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Verification of Gauss' Divergence Theorem visualisation

I'm having trouble visualising what the information provides specifically this part: Φ : [0, 1] × [0, 2π] → R^3 Does this mean the hemisphere has height of 1 in the z-axis and a radius of 2pi in the ...
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Flux density using limits-integration and divergence— Why both are not same?

Consider a vector field v$(x,y)=xy$i +$(x^2+y^2)$j Now I tried to do the following , but I cannot see why this limit does not come equal to $3y_1$ ?? Please help.. I would be very thankful for your ...
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How to apply integration by parts or the divergence theorem to a quantity involving the derivative of a vector field?

Is there a way to apply integration by parts or the divergence theorem to the integral: $$ \int_{\Omega} \langle F'(x)n(x),n(x)\rangle dx $$ where F: $\mathbb{R}^m\rightarrow\mathbb{R}^m$ n: $\...
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How to find a vector field by Gauss's divergence theorem.

enter image description hereI know how to find the value by knowing the vector filed but I want to find out if you can do it the other way round an example would be double integral over closed loop F ...
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Find Location of a Charged Particle

An electric current is $\vec{J}=y^2 \hat{j} + z \hat{k}$. At time = 0, a charged particle is at (0, 0.25, 1). Where will it be at $t=11$ and $t=26$? The divergence of the current is $\nabla\cdot \vec{...
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Divergence theorem to calculate flux through an open cylinder

Calculate the flux of $\vec F=\dfrac{1}{x^2+y^2}(x,y,z)$ through the cylinder $\{(x,y,z)\in\mathbb{R}^3∣x^2+y^2=2,−2\leq z\leq2\}$ by using Gauss Law (divergence theorem). By calculation I obtain ...
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What is the intuition behind the gravitational field having zero divergence?

So I know this has been asked before, but I really do not get why the divergence of the gravitational field $\mathbf{g}$ that exists due to a point mass $M$ at the origin is zero. I mean, I get it ...
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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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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 ...