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Questions tagged [grad-curl-div]

For questions on the vector operators: gradient, curl and divergence.

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Computing flux integral in two ways what is my mistake?

So doing it this way is easy $ \int_D \nabla F dV $ which gives me $8/3$ But doing it via $ \int _{\delta D}F\cdot nds=\int _{\delta D}F\cdot n\left|r\left(t\right)\right|dt$ gives me troubles. After ...
user832075's user avatar
10 votes
1 answer
553 views

Numerically compute and clear divergence of discrete vector field

I have a fluid simulation that represents velocity as a vector field in a grid of cells. The cells all have the same width and the same height, but the height is not necessarily equal to the width. I ...
Gyoo's user avatar
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Abundance of divergence-free vector fields in noncompact manifolds

Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n \geq 2$. How big is the space $D_b(M)$ of (pointwise) bounded divergence-free vector fields on $M$ of noncompact support? I ...
Eduardo Longa's user avatar
2 votes
1 answer
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Volume-preserving vector fields in noncompact manifolds

Let $(M,g)$ be a complete, connected, oriented and noncompact Riemannian manifold. If $M$ were compact, then a vector field $X$ is called volume-preserving when its associated flow $\phi : M \times \...
Eduardo Longa's user avatar
1 vote
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72 views

Curl in non-orthogonal coordinates

How can I transform the curl operator into general non-orthogonal coordinates? I have tried to transform its orthogonal expression using the determinant but to no avail. I can't get the same results ...
Petr Bulušek's user avatar
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If the pullback of $\phi_1(x)=\exp_x(X_x)$, preserves the volume of the Riemannian manifod $M$, then $\operatorname{div}(X)=0$?

Let $M$ be an orientable compact Rimeannian manifold whithout boundary with volume form $\operatorname{vol}_M$. Take $X\in \mathfrak{X}(M)$ such that if $\phi_t(x)$ is the flow of $X$ and $\phi_1:M\to ...
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Do line integrals of closed curves depend on the orientation of the curve or of the vector field?

I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how ...
Elena's user avatar
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What is $\nabla$ when finding the curl/divergence of a vector field?

You are supposed to do $\nabla\times\vec{F}$ for the curl, and $\nabla\cdot\vec{F}$ for the divergence where $\nabla$ is defined as $[\frac{\partial}{\partial x}\frac{\partial}{\partial y}\frac{\...
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Who are the divergence free vector fields of a compact Lie group?

Let $G$ be a compact Lie group and $X\in\mathfrak{X}(G)$ a divergence free vector field. Is there a characterization of such fields? For example, if $G=S^1$, from the fact that it is parallelizable ...
Gomes93's user avatar
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Calculating curl in cylindrical and cartesian coordinates

So I have this vector function $\mathbf{R}=\mathbf{i}r\cos\omega t + \mathbf{j} r\sin\omega t$, where $x=r\cos\omega t$ and $y=r\sin\omega t$. I want to find the curl of it's time derivative, $\frac{\...
Vebjorn's user avatar
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How do I make my formula for $\nabla \times \mathbf{F}(x,y)$ correct?

Apparently, the curl of a vector field is a function that outputs the "rotationality" of the vector field at some point, as a function of that point's coordinates. I want to go from this ...
user110391's user avatar
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What is the Laplacian of the Gradient? Is $\boldsymbol{u}(\nabla\cdot\nabla p) = \nabla (\boldsymbol{u}\cdot\nabla p)$?

I am supposed to find out whether for a scalar function $p$ and a divergence-free vector function $\boldsymbol{u}$ we have that $$\nabla\cdot\Big [\boldsymbol{u}(\nabla\cdot\nabla p) - \nabla (\...
user1313292's user avatar
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Confusion about divergence theorem for flux computation

I want to compute the flux of the vector field $$ F = \frac{\langle x,y,z\rangle}{(x^2+y^2+z^2)^{3/2}} $$ over the unit sphere $x^2+y^2+z^2=1$. I know this is $$ \iint F\cdot n \, dS $$ where $n$ is ...
user13121312's user avatar
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In layman's terms, what are curl and divergence? [duplicate]

As the title says, I'm wondering what $curl(\Bbb F)$ and $div(\Bbb F)$ mean, assuming $\Bbb F$ is a vector force field. Today in class I learned that if $\Bbb F$ is conservative, $curl(\Bbb F) = \vec ...
Bob Flinn's user avatar
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Is this the correct equation for divergence of a horizontal vector field in spherical coordinates?

There is a Horizontal vector field $\textbf{V} = <u\hat{\lambda}+v\hat{\theta_{lat}} + 0\hat{\textbf{r}}>$ which is gridded along Earth's surface. Physically speaking, the vector field's ...
Researcher R's user avatar
3 votes
1 answer
142 views

Pushforward of vector field and its divergence

Let $X:U_1 \rightarrow \mathbb{R}^2$ be a smooth vector field defined in an open subset of $\Bbb{R}^2$ and $\phi:U_1\rightarrow U_2$ a diffeomorphism between open subsets of $\Bbb{R}^2$. Let $Y = \...
Guilherme Costa's user avatar
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The relation between KL-divergence and Hellinger distance or Chi-square distance

From "Gibbs, On Choosing and Bounding Probability Metrics", we know that, in general, $$TV^2(p,q)\leq H^2(p,q)\leq KLD(p,q)\leq d_{\chi^2}(p,q)$$ where $TV(,),H^2(,),KLD(,),\ \text{and}\ d_{\...
jerry's user avatar
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Confusion over the representation of a curl of a vector field

Learning vector calculus and I'm still confused over what the curl represents for a vector field. It is stated that the curl represents the magnitude of rotation of surrounding vectors to a given ...
Anson Pang's user avatar
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How do i prove that the these two definitions of divergence are equivalent?

In class we were given this definition: $Div (\vec{F}):= lim_{r \rightarrow0} \oint_{C} \vec{F} \cdot \vec{n} \:ds$ (where r is the radius of the circle C and $\vec{n}$ is the outward pointing normal ...
Minimo's user avatar
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Am I correct in terms of understanding gradient and covector?

Well actually I know what gradient is and what that means. My professor once said that gradient is actually a covector in passing. As I know, covector is a linear map that maps vector to a scalar but ...
posfn0319's user avatar
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Integral of $\nabla\cdot\left(\hat{\mathbf{r}}/r^{2}\right)$

$\nabla\cdot\left(\hat{\mathbf{r}}/r^{2}\right) =0$ where $r$ is in spherical coordinates and represents the distance from the origin. In the Griffith' Electrodynamics pg 46, first line it is stated ...
Samar Sidhu's user avatar
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Vector rules used on del operators

I know you can use vector rules [BAC CAB and $ \vec a \cdot (\vec b \times \vec c) = \vec c \cdot (\vec a \times \vec b)$] on del operator as long as you keep track of what terms each $\nabla$ is ...
Artin8476's user avatar
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1 answer
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Kurzweil-Hensotck integral, an unclear point in a book for undergraduates by Fonda

I would like to understand how from $$d\omega_2(x)=(\frac{\partial g_{2,3}}{\partial x_1}-\frac{\partial g_{1,3}}{\partial x_2}+\frac{\partial g_{1,2}}{\partial x_3})dx_{1,2,3}$$ follows $$\...
user122424's user avatar
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2 answers
120 views

Prove that $\operatorname{curl}({\vec a\over r})=\frac{[\vec a \times\vec r]}{r^3}$

Prove that $\operatorname{curl}({\vec a\over r})=\frac{[\vec a \times\vec r]}{r^3}$ where $r$ is the radius vector and $a$ is a constant vector. I break this problem down into $\nabla \times ({1\over ...
Koldis27's user avatar
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Regularity conditions needed for divergence to equal "flux density"? (I.e. for "coordinate-free definition" to be valid?)

Background: Let $\vec{V} : \mathbb{R}^n \to \mathbb{R}^n$ be treated as a vector field, $V_i$ denote the corresponding scalar coordinate functions $\mathbb{R}^n \to \mathbb{R}$, and then if these ...
hasManyStupidQuestions's user avatar
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3 answers
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Proof that $\frac{n^6+n^2}{n^7+3}$ converges

I'm looking at a problem that follows: Test if $$\sum_{n=1}^\infty \frac{n^6+n^2}{n^7+3}$$ converges or diverges. I think I have a proof but it seems a bit awkward. You take out the 3 in the ...
fiftytwo's user avatar
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31 views

How can I find $\mathbf{J}$ from the equation for $\nabla\cdot\mathbf{J}$?

How can I find $\mathbf{J}$ from the following equation $$ \nabla \cdot \mathbf{J}= ∇\left(∇^2 (∇ψ^†⋅\mathbf{A})+∇⋅(\mathbf{A} ∇^2 ψ^† )\right)⋅∇ψ+∇ψ^†⋅∇\left(∇^2 (∇ψ⋅\mathbf{A} )+∇⋅(\mathbf{A} ∇^2 ψ)\...
Luqman Saleem's user avatar
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Piecewise mooth closed curve in simply connected space is a boundary of surface

I'm interested in a simple proof of the following fact: Let $V \subset \mathbb{R}^3$ be a bounded, open, connected, simply connected set. Let $\gamma$ be a piecewise-smooth simple closed curve in $V$....
sansae's user avatar
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1 answer
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Meaning of a scalar surface integral of a vector field?

I apologize if the title is confusing but after going through some assignments I was able to correctly derive that $\iiint_{V}(\overrightarrow{\nabla}\times A)dV=-\oint_{\partial V} (A \times \hat{n})...
Marcos Mejia's user avatar
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0 answers
76 views

Divergence with respect to pull-back metric

Let $(M,G)$ be a Riemannian manifold of dimension $n$ and $S$ be a submanifold of $M$ of dimension $m$. Consider a vector field $X$ on $S$. $X$ is also a vector fieldd on $M$, so we can compute its ...
DavideL's user avatar
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1 answer
80 views

A paradox on curl equations in cylindrical and spherical coordinates

Let $\mathbf{A}=\sin(\theta)\hat{\phi}$ be an azimuthal vector field in either cylindrical (cylindrical radial, azimuthal, vertical)=$(\rho,\phi,z)$ or spherical (spherical radial, colatitude, ...
Aria's user avatar
  • 422
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0 answers
41 views

Calculating the flux over a non simply connected surface

Here is the question: The surface $S$ shown here has boundary the circle of radius $2$ in the $xz$-plane. With respect to the normal vector field indicated, compute the flux of $G = \langle 0, 3, 0 \...
Skop's user avatar
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Does $P_{z} \operatorname{curl} F = P_z \operatorname{curl} P_{xy}F$?

Let $F: \mathbb R^3 \to \mathbb R^3$ and let $P_{xy}F$ be the projection of $F$ onto the $xy$ plane and $P_zF$ the projection of $F$ onto the $z$ axis. Is it true that $$P_{z} \operatorname{curl} F = ...
SRobertJames's user avatar
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1 answer
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Divergence of two orthogonal vector fields [closed]

I have the following question. Let $b_1, b_2$ two suitably regular vector fields such that $\operatorname{div}b_1=0$ and $\langle b_1, b_2\rangle=0$. What can we say about $\operatorname{div}b_2$? Is ...
Jeji's user avatar
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1 answer
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What is the Divergence of a Spherically Symmetric Vector Fields?

A vector field is spherically symmetric about the origin if, on every sphere centered at the origin, it has constant magnitude and points either away from or toward the origin. A vector field that is ...
SRobertJames's user avatar
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1 vote
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How to solve this kind of surface integral with Hamilton Operator?

In $\mathbb{R}^3$, $f=\left(\frac{x}{2}\right)^2+\left(\frac{y}{2}\right)^2+\left(\frac{z}{4}\right)^2$, Surface $S$ is defined by $S=\{(x,y,z)|f(x,y,z)=1, z>0\}$, and the vector field $A$ is ...
Furina's user avatar
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3 votes
1 answer
88 views

How to prove that $\vec{\nabla}\times\vec{\nabla}\times(e^{i\vec{k}\cdot\vec{x}}\vec{\omega})=k^2(e^{i\vec{k}\cdot\vec{x}}\vec{\omega})$

In a physics paper I have encountered the following claim: $\vec{\nabla}\times\vec{\nabla}\times(e^{i\vec{k}\cdot\vec{x}}\vec{\omega})=k^2(e^{i\vec{k}\cdot\vec{x}}\vec{\omega})$ where $\omega=\vec{\...
Wild Feather's user avatar
0 votes
1 answer
91 views

What's wrong with this integration over the volume of a sphere (from Gauss' Theorem)?

I'm struggling to find my mistake in the following problem: Let $S$ be a sphere of radius $R$ with center at the origin. Let $f(x,y,z) = 3z^2$. Find the integral of $f$ over the volume of the ...
SRobertJames's user avatar
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2 votes
0 answers
52 views

Inverse of gradient of divergence, in two-dimensions?

Given a two-dimensional vector field $U(x,y) \in C^2(\mathbb{R}^2, \mathbb{R}^2)$, consider the expression \begin{align} \operatorname{grad} \operatorname{div} U(x,y) = V(x,y) \end{align} for some ...
Jonathan Lindbloom's user avatar
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0 answers
25 views

Help understanding the use of tetrahedrally arranged vectors to compute the gradient of a function

As a long-time user of the free, open-source raytracer POV-Ray, I'm trying to understand some of the source code used to compute and perturb surface normals. The method uses the evaluation of 4 ...
Bald Eagle's user avatar
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0 answers
80 views

Dot Product with Partial Differential Question

Given: $$\vec{A}(r,\theta)$$ $$\vec{B}(r,\theta)$$ Is it always true that: $$ \left(\vec{A}\frac{\partial}{\partial{\theta}}\right)\bullet\vec{B}\overset ? =\bigg(\vec{A}\bullet\frac{\partial\vec{B}}{\...
John's user avatar
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0 answers
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Intuition about the divergence of a vector field in non-orthogonal basis

My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way: $$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
Krum Kutsarov's user avatar
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0 answers
17 views

When is valid to first evaluate a function in a point and then do the DIvergence than to first the divergence and then evaluate?

I have been having this doubt when applying the Divergence of a Electrical Potential evaluated in a point, and then doing the Divergence to get the Electric Field in that point. But is this valid? or ...
Oswaldo RO's user avatar
0 votes
1 answer
76 views

Prove a particular case of product rule for divergence

I am trying to verify the vector calculus identity in appendix B.2 in "Finite Elements Methods: A Practical Guide" (Whiteley 2017) $$ \begin{aligned} \nabla \cdot (vp\nabla u) &= v\...
Jared Frazier's user avatar
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0 answers
32 views

How do I find orthogonal vectors using the del operator in an arbitrary coordinate system?

enter image description here The image shows the full question, part (b) is what I'm interested in. I'm stuck on how to use the del operator here to find out if $\vec{B}_0$, $\vec{E}_0$ and $\vec{q}$ ...
Jack Tiler's user avatar
8 votes
3 answers
1k views

Why does it seem the inner curls within a surface always cancels in order for greens theorem to be true

Im trying to learn aerodynamics in general for my course. Every video i see to derive the concept of greens and stokes theorem shows how the inner curls within a surface area cancel to 0 and its only ...
George kirby's user avatar
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0 answers
37 views

Curl of normal unit vector of a smooth and closed surface?

Let's say we have a curvilinear coordinate system $(\rho,\theta,\zeta)$. Also, let's say we have a smooth and closed surface $\Gamma$ parameterized as $\Gamma: \mathbf{S}(\rho(\theta,\zeta),\theta,\...
Francisco Sáenz's user avatar
1 vote
0 answers
62 views

Local stability estimate for divergence free vector field

For a divergence free field $w\in [L^2(\mathbb{R}^d)]^d$, especially for $d = 3$, it is well known that there exists a skew-symmetric matrix $\psi \in [W^{1,2}_{loc}(\mathbb{R}^d)]^{d\times d}$ with $\...
Ryan Li's user avatar
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0 answers
82 views

Formula for the divergence of a tensor

I need to express the divergence of a tensor $\nabla \cdot (a \times b)$ to a different form. I found here that $$\nabla \cdot (a \times b) = (\nabla \cdot a)b + a \cdot \nabla b$$ which is exactly ...
epselonzero's user avatar
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0 answers
68 views

Green's theorem example application for engineers and physicists.

I am looking for example applications of Green's theorem (in $2D$) that appeal to physicists or engineers. It's to come up with example for the divergence theorem in fluid dynamics, but finding a very ...
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