Questions tagged [curl]

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.

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Physical significance of $\vec{w}$ $\times$ $($curl $\vec{v})$ [migrated]

I think if curl of a vector field $\vec{v}$ corresponds to an applied rotation, it's cross product with a velocity vector field $\vec{w}$ (say) should give something analogous to the resulting torque. ...
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Calculating the Curl of a Velocity Field

The curl of a (velocity) field can be defined as: $\nabla\times\vec{u}=(\frac{\partial\,\omega}{\partial\,y}-\frac{\partial\,v}{\partial\,z})\vec{i}+(\frac{\partial\,u}{\partial\,z}-\frac{\partial\,\...
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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 ...
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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 $V$....
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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})...
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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, ...
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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 \...
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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 = ...
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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 ...
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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{\...
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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 $\vec{q}$ ...
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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 ...
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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,\...
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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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What is the adjoint of the curl operator?

Let $\Omega \subseteq \mathbb{R}^3$ be a bounded, connected domain, and set $$\mathcal{V} = \{ \vec\phi = (\phi_1, \phi_2, \phi_3) \in C_c^\infty(\Omega): \nabla\cdot \vec\phi=0\}.$$ Denote $V$ to be ...
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Curl without integrals

I have a hard time following the integral derivation of a curl in a given coordinate system and it seems somewhat meandering from the entities involved. I came up with this method, I'm wondering if ...
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Can you characterize a vector field $v$ with uniform curl as $v(x) = a + b \times x$?

Let $v:\mathbb R^3 \to \mathbb R^3$ a vector field such that $\forall x\in\mathbb R^3, (\nabla \times v) (x) = \rho~$ where $\rho$ is a constant. Example here. Can you characterize $v$ simply? What I ...
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Divergence theorem with normal component of a curl to a surface

Let $\mathbf{A}$ be a vector function in $\mathbf{R}^3$ and we want to find the normal and tangent components of $\nabla \times \mathbf{A}$ on a smooth and closed surface $\Gamma$. $\mathbf{n}$ is the ...
Francisco Sáenz's user avatar
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Conservative vector field given in polar coordinates

Given a vector field $F$ in polar coordinates, for the example the field $$\vec F(r,\theta)= -r \hat r + (r^2\sin \theta ) \hat \theta $$ I am asked to check if the field is conservative. is it right ...
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In studying waveguide mode vectors I have difficulty in interpreting the expression

The components of $E$ waves are, \begin{equation} e_x=k_x k_z C_E^{\mathrm{rect}} \mathrm{cos}\left(k_x x\right)\mathrm{sin}\left(k_y y\right) \end{equation} \begin{equation} e_y=k_y k_z C_E^{\mathrm{...
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Visualizing the curl of curl

In some differential equations from phyiscs, e.g. in elastodynamics, terms with the curl of the curl of a vector field appear. For example $(\lambda + 2 \mu) \nabla(\nabla\cdot \mathbf{x}) - \mu \...
DiggingDeep's user avatar
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Holomorphic in $\mathbb{C}$ vs divergence-and-curl-free in $\mathbb{R}^2$

I was reading this post in order to grasp what holomorphic mean. Did I understood correctly that the analog of a holomorphic function (if we consider $\mathbb{R}^2$ as a $\mathbb{R}$-vector space) is ...
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$\vec \nabla \times \vec F=0 $ $\implies$ $\exists$ a scalar potential $\phi$ such that $\nabla \phi=\vec F$. Prove it.

If $\vec F$ be defined in a simply connected region, then $\vec \nabla \times \vec F=0 $ $\implies$ $\exists$ a scalar potential $\phi$ such that $\nabla \phi=\vec F$. Prove it by using Helmholtz ...
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Show that $\nabla\cdot [(\vec u\times \vec v)\times \vec w]=\vec w\cdot [(\vec v\cdot \nabla)\vec u-(\vec u\cdot \nabla)\vec v]$

If $\vec u, \vec v~ \text{and} ~\vec w$ are vectors, show that $$\nabla\cdot [(\vec u\times \vec v)\times \vec w]=\vec w\cdot [(\vec v\cdot \nabla)\vec u-(\vec u\cdot \nabla)\vec v]$$ if $\...
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Is $(f_1(x)+f_2(y)+f_3(z))\vec{r}$ irrotational?

If we define $F(x,y,z)\equiv f_1(x)+f_2(y)+f_3(z)$, the curl of $\vec{f}(\vec{r})=F(x,y,z)\vec{r}=(F\cdot x,F\cdot y,F\cdot z)$ would be \begin{align*} \nabla\times\vec{f}&= \begin{vmatrix} \hat{i}...
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If $\vec F=\frac{\vec r}{r^3}$ then $\exists$ is no $\vec G ∶ \mathbb{R}^3 ⧵ {0} \to \mathbb{R}^3$ such that $\vec F = curl \,\vec G$.

If $\vec F=\frac{\vec r}{r^3}$ then show that $\textrm{div} \vec F = 0$ but $\exists$ is no $\vec G ∶ \mathbb{R}^3 ⧵ {(0,0,0)} \to \mathbb{R}^3$ such that $\vec F = \textrm{curl} \,\vec G$. I can show ...
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If $\bar{F}=(xyz)^b (x^ai + y^aj + z^ak)$ be irrotational, then show that $b=0$ or $a=-1$. Find the scalar potential $\phi$.

If $\bar{F}=(xyz)^b (x^ai + y^aj + z^ak)$ be irrotational, then show that $b=0$ or $a=-1$. Find the scalar potential $\phi$. I can solve the 1st part by solving $curl\,\bar{F}=\bar{0}$. By comparison ...
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Prove that $\bar{F} = r^n \bar{r}$ is conservative and find the scalar potential $\phi$ such that $F = grad\,\phi$.

Prove that $\bar{F} = r^n \bar{r}$ is conservative and find the scalar potential $\phi$ such that $F = grad\,\phi$. I can solve the 1st part by showing $curl\,F=0.$. I am facing two problems: Nothing ...
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Find an irrotational function $F(x)$ and a solenoidal function $G(x)$ such that $H = F + G$.

Let $H(x) = x^2y\,\mathbf{i} + y^2z\,\mathbf{j} + z^2x\,\mathbf{k}\,$. Find an irrotational function $F(x)$ and a solenoidal function $G(x)$ such that $H = F + G$. Sol. As $F(x)$ is irrotational ...
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Integration using Stokes' theorem

I have a problem that I can't seem to figure out. Given a surface $S$: \begin{cases} x^2+y^2 \leq 1 \newline z = y^2 \end{cases} Let $C$ be the edge of $S$. $C$ is oriented so that the projection of $...
WatT's user avatar
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Stokes theorem applied to planes

Let $\textbf{F} = (y+z,-xz,y^2)$. Let $S$ be the surface above the $xy$ plane and bounded by $2x + z = 6$, $y = 2$, $y = 0$ and $x = 0$. Calculate $$\iint_S \text{curl } \textbf{F} \cdot d\textbf{S}$$ ...
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Why is curl given by a cross product?

I've googled a lot but in most places, they usually only show how to compute it which I by now can do. But I wanna understand why the cross product between nabla and the vector give us the curl. So ...
Need_MathHelp's user avatar
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Why can the del operator cross product a triple integral be placed inside the triple integral?

Consider the (electric) vector field $$\pmb{E}(\pmb{r})=k_e\iiint_V \frac{\rho(\pmb{r_s})}{\lVert \pmb{r}-\pmb{r_s} \lVert^2}\frac{\pmb{r}-\pmb{r_s}}{\lVert \pmb{r}-\pmb{r_s} \lVert}d\tau\tag{1}$$ ...
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Dot product symbol for divergence, merely a convenience? [duplicate]

As the title says, is it merely a convenience to write the divergence as a dot product? Is there an intuition on the relationship between the geometric interpretation of the divergence and that of the ...
João Pedro's user avatar
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How are the formal definitions of curl interpreted? Where do they appear from and how to derive them?

So here are formal definitions of curl for 2dimensional and 3dimensional. Can someone please explain how to derive these and interpret them? I am familiar with limit definition of curl, however these ...
Aleksejs Tabunovs's user avatar
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Proving the equality between the cross product and integral definitions of curl

Let $F(x,y) = \left(M(x,y),N(x,y)\right)$ be a vector field on real plane. Then the curl of our vector field is something like: $$\dfrac{\partial N}{\partial x}-\frac{\partial M}{\partial y}$$ The ...
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What is the vorticity of a velocity field?

If $u:\mathbb{R}^3\to \mathbb{R}^3$ is a velocity field one defines the vorticity as the curl \begin{align} \omega= \text{curl}(u). \end{align} I just read that vorticity measures the rotation of the ...
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A vector field $V$ is gradient of some function $f$ on an open rectangular solid in $\mathbb{R}^3$

I want to prove that a continuously differentiable vector field $V$ is the gradient of some function $f$ on an open rectangular solid $(a,b)\times(c,d)\times(e,f)\subset\mathbb{R}^3$ if it has curl $0$...
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Curl and rotations

I was looking at this great video, circa minute 13:00, where a relationship between the curl of a 2d-vector field and the rotation-like behavior of the field is mentionned. I tried the usual things, ...
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On a certain "anticurl" operator

I recently found myself curious what explicit formula I would get if I traced through the de Rham cohomology proof that if $\mathbf{F}$ is a vector field defined on all of $\mathbb{R}^3$ which has ...
Daniel Schepler's user avatar
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Potential and Field

So given a field: $$\vec E(r)=\frac{\alpha(\vec p \cdot \vec e_r)\vec e_r + \beta \vec p}{r^3}$$ where $α, β$ are constants, $\vec e_r$ is the unit vector in the direction $\vec r$, and $\vec p$ is a ...
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Given $G:\mathbb{R}^3\to\mathbb{R}^3$, find $F$ such that $G=\operatorname{curl}(F)$

At my calculus II class we are studying multivariable functions and yesterday er talked about the curl operator (we used the definition in "usage" section here. The typical exercise we got ...
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Intuition behind the curl of the curl of a vector field

I believe myself to have an intuition of the basic vector differential operators, $\mathrm{grad} \equiv \nabla$, $\mathrm{div} \equiv \nabla \cdot$, $\mathrm{curl} \equiv \nabla\times$ and the Laplace ...
El Aprendiz's user avatar
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Relation between $|(\nabla\times F)(u)|$ and $|\nabla F(u)|$

Let $F:\mathbb{R}^3\to\mathbb{R}$ be of class $C^\infty$. Let $u\in\mathbb{R}^3$ be nonnull. I am trying to understand if there is a relation between $$|(\nabla\times F)(u)|\quad\text{ and }\quad |\...
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Show that $\vec\nabla\times(\vec\nabla\times\vec A )=- \nabla^ 2\vec A + \vec\nabla (\vec\nabla\cdot\vec A )$ [duplicate]

I just started learning this and I don't understand much so how can I prove this? $$\vec\nabla\times(\vec\nabla\times\vec A )=-\vec\nabla^2\vec A +\vec\nabla (\vec\nabla\cdot\vec A )$$
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How can I simplify $\nabla (X\cdot \nabla u)$?

What is $\nabla (X\cdot \nabla u)$ where $X:U\subseteq\mathbb{R}^2\to\mathbb{R}^2$ is a vector field and $u:U\subseteq\mathbb{R}^2\to\mathbb{R}$ a scalar field?
Joseph's user avatar
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Existence of anti-gradient of a vector field

1. The Original Question (informal) I have a function (vector field) $\mathbf{F}:\mathbb{R}^N\rightarrow\mathbb{R}^N$ with expression $\mathbf{F}(\mathbf{x})=\sigma(\mathbf{Wx+b})$, where $\mathbf{W}\...
BinChen's user avatar
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calculating curl

With $$\tilde{\mathbf{E}}=2j\hat{y}E_me^{-jk_zz}\sin(k_xx)\quad\text{for}\quad 0<x<a$$ the phasor form of Faraday's law $\nabla\times\tilde{\mathbf{E}}=-j\omega\mu_0\tilde{\mathbf{H}}$ leads to $...
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Curl in spherical coordinates on example

On Wiki (https://en.wikipedia.org/wiki/Dipole_antenna) the Vector Potential of a Dipole Antenna is roughly given by: $$A = c\cdot \dfrac{e^{-i\,k\,r}}{r}\,\hat{e}_z$$ Now the curl is computed in ...
Leon's user avatar
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Taking the curl of advective part of navier-stokes equation to get vorticity in index notation

I need to take the curl of $\frac{\partial u_i}{\partial t} = -\frac{1}{\rho}\frac{\partial p}{\partial x_i}-\frac{\partial}{\partial x_j}u_i u_j+\nu\frac{\partial^2 u_i}{\partial x_i^2}$ to get $\...
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