Questions tagged [divergence-operator]

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. For questions about divergent sequences use [tag:convergence-divergence]

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A laplacian working on an equation containing a laplacian and a gradient

I have an equation as follows: $a \Delta \vec{u} + \vec{\nabla}(\vec{\nabla} \cdot \vec{u}) = 0$. If I take a Laplacian of the above equation, the answer is supposed to be: $\Delta \Delta \vec{u} = 0$....
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How to derive divergence operator in complex spaces [closed]

I would like to find a convinient way to understand divergence and gradient in complex space. Lets for example take a simple function f=u+vi. It might be the case that the are not harmonic conjugates. ...
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Divergence of transposed gradient of vector quantity

I would appreciate some help with this. Is: $$ \nabla \cdot \left( \nabla \vec{v}\right)^T= \nabla \left( \nabla \cdot \vec{v}\right) $$ How can I show this? Is the gradient of a vector mathematically ...
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integral of vector field

I'm wondering how to prove this : if $F$ is a vector field and $g$ is a scalar function (continuously differentiable function) and $$ \nabla g$$ is not zero. assuming that $D= {x: b≥g(x)≥a}$ how can ...
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Writing equations in the form of a conservation law

I'm presented with the following internal energy equation: $$ \frac{\partial (\rho e)}{\partial t} + \nabla \cdot (\rho e \vec{v}) = -p\nabla \cdot \vec{v} + \nabla \cdot (K\nabla T) + \varepsilon_V + ...
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Exact solutions to a kind of 3D wave equation

Consider the mapping ${\bf{u}}: [0,T] \times \mathbb{R}^3 \mapsto \mathbb{R}^3$, defined by ${\bf{u}} = {\bf{u}}(x,y,z,t)$. I am looking for general solutions ${\bf{u}}$ of the following PDE $$ \frac{\...
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Surface integrals in Force Fields

So I am given the following question: On $\mathbb{R}^3\backslash \{\underline{0}\}$ consider the vector field \begin{align*} \underline{F}(x,y,z) = (0,\frac{-2yz}{r^4}, \frac{-1}{r^2}+ \frac{2y^2}{...
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Why does vector field <x, -y> have 0 divergence?

enter image description here Numerically, the divergence is 0. However, when looking at the vector field, it appears like the flow is away from the origin. Hence, why does it have 0 divergence just by ...
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Define vector field with known curl and div [duplicate]

I have a problem with vector analysis and can't figure out. Can vector field be determined if we know its divergence and curl? If so, what is the function of vector field? Let's say div v = f(x, y, z) ...
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Is there any condition under which $\nabla\cdot F=0$ implies $F=0$?

On a physics course it was stated that $$ \nabla\cdot\vec{D}=\rho_f=\nabla\cdot(\varepsilon_0\vec{E}) $$ and then it follows that $$ \vec{D}=\varepsilon_0\vec{E} $$ I know this is not generally true, ...
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Prove that $n \cdot \frac{\partial u}{\partial n} = 0$ for divergence-free functions that vanishes on boundary

Prove that $n \cdot \frac{\partial u}{\partial n} = 0$ for divergence-free functions that vanishes on boundary. I just read that if $u$ is a divergence-free (vectorial) function in $\Omega$ with $u = ...
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Question about divergence of $\vec{F} = \frac{1}{r^2} \hat{r}$

I am looking at the divergence of this famous expression: $\vec{F} = \frac{1}{r^2} \hat{r}$ i saw this calculation which looks reasonable ... $\hat{r} = (x,y,z)/\sqrt{x^2 + y^2 + z^2}$ $$ F(x,y,z) = \...
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Eigenvalue stability and Divergence of Vector field

Consider a 2 dimensional Autonomous system \begin{equation} \dot {\boldsymbol{x}}=\begin{bmatrix} a_{11}& a_{12}\\ a_{21}& a_{22} \end{bmatrix} \boldsymbol{x} \end{equation} with the origin as ...
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intuition behind differential operators

I discover these concepts in my math class, and I want to understand the intuition behind the Laplacian, the curl and the divergence operators, how do you explain this to a student who is discovering ...
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Vector calculus singularity

Let $$ \mathbf{A} = A_{\theta} \hat{\theta} + A_{\phi} \hat{\phi} $$ be a vector field in $\mathbb{R}^3$ expressed in spherical components, where $\theta$ is the polar angle and $\phi$ is the ...
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How to show $\delta(h(\omega, \cdot)) = \langle \delta h, \omega \rangle -\langle h, \nabla \omega \rangle $?

Picture below is from Topping's 37th page of Lectures on the Ricci flow. I try to show $$ \delta(h(\omega, \cdot)) = \langle \delta h, \omega \rangle -\langle h, \nabla \omega \rangle $$ $\delta$ ...
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Simplifying divergence expression using product rule

I have the following expression $$ \textrm{div} \, \frac{\vec{r} - \vec{a}}{\lvert\vec{r} - \vec{a}\rvert^n},\quad n \in \mathbb{N} $$ where $\vec{r} \neq \vec{a}$ and $\vec{a}$ is a constant vector ($...
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Construct vector field in $H^1$ with given divergence (and not only)

I'm trying of understand if the following is true. Let $\Omega$ and $\Omega_0$ be some regular open bounded sets of $\mathbb R^d$ such that $\overline\Omega\subset\Omega_0$. I do not have constraints ...
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Compute $\operatorname{div}(C\nabla g)$ for $g(x):=e^{{\rm i}\langle x,\:y\rangle}$

Let $d\in\mathbb N$, $y\in\mathbb R^d$, $$g(x):=e^{{\rm i}\langle x,\:y\rangle}\;\;\;\text{for }x\in\mathbb R^d$$ and $C\in\mathbb R^{d\times d}$ be symmetric. How can we compute $\operatorname{div}(...
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Writing an expression as divergence (vector algebra)

Let $F(\mathbf{r})\equiv F$ is a a field, and $F^\dagger$ is its conjugate field. And there is an expression: $$ \mathbf{\nabla} \cdot \mathbf{J_{r}} = (\nabla^2 F^\dagger)(\nabla F\cdot\mathbf{A}) + (...
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On divergence freeness of unit vectors and Navier-Stokes

Let $\mathbf F: \mathbb R^3 \rightarrow \mathbb R^3$ be a smooth, nonzero, divergence-free vector field. A simple computation shows that a sufficient condition for $$ \operatorname{div} \left( \frac{\...
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Can we show the Voss-Weyl formula "the other way"?

I've recently come across the Voss-Weyl formula: $$\operatorname{div}\boldsymbol u=\frac{1}{\sqrt{\left|\det\mathbf{g}\right|}}\partial_i\left(\sqrt{\left|\det\mathbf{g}\right|} ~~u^i\right)$$ For a ...
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Proof with div and curl

Let $$F, G:\mathbb{R}^3 \to \mathbb{R}^3$$ Prove that: $$\nabla\cdot(F\times G)=G\cdot (\nabla \times F)-F \cdot(\nabla\times G)$$ I found that: $$∇\cdot(F×G)=(∂(F_2 G_3-F_3 G_2))/∂x+∂(F_3 G_1-F_1 G_3 ...
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Does the classical expression for divergence hold for (not continuously) differentiable functions?

The question is simple. I state it for $\mathbb{R}^3$, although I am pretty sure that dimension 3 should not be relevant here. It is well known that the divergence of a vector field at a point is ...
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Divergence of a radial vector field

I am reading Modern Electrodynamics by Zangwill and cannot verify equation (1.61) [page 7]: \begin{equation} \nabla \cdot \textbf{g}(r)=\textbf{g}^{\prime}\cdot \mathbf{\hat{r}}, \end{equation} where ...
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Calculating the Divergence of a Tensor

I am working through a fluid dynamics paper and came across this equation: $$ \frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}=\nabla\cdot T - \frac{1}{\rho}\nabla \phi\tag1$$ where T ...
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Divergence of field in spherical coordinates does not match Cartesian

I have a vector field $$\vec{F}(x,y,z)=\frac{x \hat{x}}{(x^2+y^2+z^2)^{3/2}} + \frac{y \hat{y}}{(x^2+y^2+z^2)^{3/2}} + \frac{z \hat{z}}{(x^2+y^2+z^2)^{3/2}}$$ Solving this in Cartesian coordinates, I ...
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Wrong computation of divergence in cylindrical coordinates. Where is my error?

The divergence of a vector field $X$ in $\mathbb{R}^3$ in Cartesian coordinates is given by $$ \operatorname{div}X:=\frac{\partial X_x}{\partial x}+\frac{\partial X_y}{\partial y}+\frac{\partial X_z}{\...
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Calculate the Kullback–Leibler divergence by comparing samples from both distributions

How can one calculate the KL-divergence of an unknown distribution Q from which one can sample and a standard normal distribution P? Let's say that I can sample from Q with $z_i = T(\mu + \sigma * \...
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Invariance of divergence under arbitrary coordinate transformations

I've found the following proof which seems to have the conclusion, that the divergence is invariant under a general coordinate transformation when defined with the derivatives of the respective (...
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Why can the divergence be defined equivalently as ${\rm div}(X)(x)=\sum_j \langle\nabla_j X,e_j\rangle_x$ and $d(i_X dV)=({\rm div}X)dV$?

Let $M$ be an oriented Riemannian manifold with volume form $dV$, and let $X$ be a smooth vector field on $M$. Recall that the divergence of $X$ is characterized by the formula $d(i_X dV)=(\text{div}X)...
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Divergence is coordinate independent

I am working on a project about Spectral Geometry. One of the main goals of the project is to be able to define the Laplacian on a Riemannian Manifold. As such, Let $(M,g)$ be a Riemannian Manifold, ...
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Kernel of the operator $\int_{\Omega} \operatorname{div} vq$

Let $B:H^{\operatorname{div}}_0(\Omega) \rightarrow L^2_0(\Omega)$ the operator defined by $$\langle Bv,q\rangle = \int_{\Omega} \operatorname{div}(v) q$$ where $H^{\operatorname{div}v}(\Omega) = \{v \...
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Property of vector field whose divergence is a compactly supported function with vanishing integral.

Given a function $f\in{C^1_c}(\mathbb{R}^3)$ such that the Lebesgue integral over $\mathbb{R}^3$ vanishes: \begin{equation}\int_{\mathbb{R}^3}f\mathrm{d}\lambda=0.\end{equation} I am trying to prove ...
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Solenoidal field that is not a curl

A vector field in $\mathbf{A}$ of class $\mathcal{C}^2$ in $\Omega=\mathbb{R}^3\backslash\{a\} \subset \mathbb{R}^3$ where $a\in \mathbb{R}^3$. $S$ is a closed surface whose bounded region contains ...
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If the flux of a vector field $F$ is zero for ANY surface $S$ then is $F$ zero?

If $\vec F$ is a vector field defined in a open (contractible) set $U$ of $\Bbb R^3$ such that $$ \int_S\vec F\cdot\hat ndS=0 $$ for ANY surface $S$ then in particular the last equality holds for the ...
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Divergence of multiplication: what order?

Really quick question: Suppose you're asked to calculate $\text{div}(f\cdot F)$ where $f$ is a scaler function and $F$ is a vector field (both depending on same variables). Do I just multiply $f$ and $...
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Apply divergence theorem for $\int_R r \times \operatorname{div}(v \otimes v)dV$

I'm trying to show the following integral equality, but I really can't come up with a proof. The context here is the one of an introductive book to continuum mechanics, so everything is smooth and ...
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Inconsistent calculation of divergence in spherical coordinates [duplicate]

I want to calculate $\nabla\cdot(\frac{\hat{r}}{r^2})$ Naively applying the expression for divergence in spherical coordinates gives $\nabla\cdot(\frac{\hat{r}}{r^2})=\frac{1}{r^2}\frac{\partial}{\...
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Divergence of first Piola-Kirchoff stress tensor: how is it computed?

studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor $$S=\left(\frac{\pi + \mu_0}{\lambda_1} + \mu_0 \lambda_1\!\right)...
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Sobolev spaces with respect to divergence and their properties

Let $n \in \mathbb{N}$, $\Omega$ a non-empty bounded open set of $\mathbb{R}^n$ with Lipschitz boundary and $p \in [1,\infty]$. Define $$V_p:=\bigg\{\overrightarrow{q}\in L^p(\Omega;\mathbb{R}^n) \mid ...
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Convexity of Bregman divergence

I'm studying Bregman divergence and there is one of the properties that i dont quite understand. The definitions of the Bregman divergence is the following: $$ $$ Let $d_f : C x C $ $\xrightarrow{}$ $...
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Divergence of a first Piola-Kirchoff stress appearing in elasticity

I'm studying elasticity theory and for the problem "Finite Bending of an incompressible elastic block" I have the following first Piola-Kirchoff stress tensor: $$S=\left(\frac{\pi + \mu_0}{\...
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Convective derivative vs Divergence of velocity

What is the physical significance or difference between Convective derivative : $\vec{v} \cdot \nabla $ and the Divergence of velocity $\nabla \cdot \vec{v}$? I have understood the convective ...
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Div, Grad Curl and all that,Chapter-2 Problem-17: Deriving divergence in cartesian coordinates using a prism as volume infinitesimal

In the text we obtained the result of of $\nabla \cdot F$ in cartesian coordinates, by integrating over the surface of a small rectangular parallelepiped. As an example of the fact that this result ...
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Proving $(\nabla \times \mathbf{v}) \cdot \mathbf{c} = \nabla \cdot (\mathbf{v} \times \mathbf{c})$ using cylindrical coordinates

Assuming the form of divergence in polar coordinates is known, I am attempting to use the following definition of the curl of a vector field to determine the form of the curl in cylindrical ...
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Is it possible to recover a vector field from its divergence equation?

I have the following vector field $p(x,y):\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ $$\nabla\cdot p=1$$ Is it somehow possible to deduce the vector field function $p$ just from the divergence equation?...
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Change of Variables in Partial Derivatives - Multivariable Calculus

So I am trying to solve the divergence of F $$ {F(x, y, z) = z \mathbf i + y \mathbf j + x \mathbf k} \\\\ {\nabla F(x, y, z) = {\partial z \over \partial x} + {\partial y \over \partial y} + {\...
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Question about Divergence formula derivation posted a while ago

As a new user, I am not allowed to comment on someone else's answer to a question. My only choice was to ask a new question about an old answer to a question. User @Kcronix mentioned, in this question,...
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Is it even possible to integrate a two-dimensional vector field over the interior of a square?

As you know, the concept of divergence of a vector field $\mathbf{f}\colon\mathbb{R}^n\to\mathbb{R}^n$ is a measure of how closely certain parts of $\mathbb{R}^n$ resemble sinks or sources, so that ...

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