# What is Curl: Definition and 367 Discussions

cURL (pronounced 'curl') is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL", which was first released in 1997.

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1. ### I What's the physical meaning of Curl of Curl of a Vector Field?

So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$ Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise. The laplacian...
2. ### Find the divergence and curl of the given vector field

Been long since i studied this area...time to go back. ##F = x \cos xi -e^y j+xyz k## For divergence i have, ##∇⋅F = (\cos x -x\sin x)i -e^y j +xy k## and for curl, ##∇× F = \left(\dfrac{∂}{∂y}(xyz)-\dfrac{∂}{∂z}(-e^y)\right) i -\left(\dfrac{∂}{∂x}(xyz)-\dfrac{∂}{∂z}(x \cos...
3. ### I How do I format equations correctly? (Curl, etc.)

A question in advance: How do I format equations correctly? Let's say $$\mathbf{k}\cdot\nabla\times(a\cdot\mathbf{w}\frac{\partial\,\mathbf{v}}{\partial\,z})$$ - a is a scalar Can I rewrite the expression such that...
4. ### MATLAB MATLAB: Fluid Flow - Curl of a Vector Field

I am working with some data which represents the fluid position and velocity for each point of measurement as an x, y, u, and v matrix (from particle image velocimetry). I have done things like circulation, and discretizing the line integral involved was no problem. I am stuck when trying to...
5. ### I Ampere's Law For Static Magnetic Field

Hi there! Please refer to the picture below. I would like to understand the equation Curl(H) = J, where H is the magnetic field intensity and J is the current density. So, I inspect a simple problem. There is a wire carrying current I in the z-axis direction. a_r, a_phi, and a_z are the unit...
6. ### I A math confusion in deriving the curl of magnetic field from Biot-Savart

I am recently reading "Introduction to Electrodynamics, Forth Edition, David J. Griffiths " and have a problem with the derive of the curl of a magnetic field from Biot-Savart law. The images of pages (p.232~p233) are in the following: The second term in 5.55(page 233) is 0. I had known...
7. ### Solving Curl A in Spherical Coordinates: Tips & Hints

I've tried writing the curl A (in spherical coord.) and equating the components, but I end up with something that is beyond me: {\displaystyle {\begin{aligned}{B_r = \dfrac{1}{4 \pi} \dfrac{-3}{r^4} ( 3\cos^2{\theta} - 1) =\frac {1}{r\sin \theta }}\left({\frac {\partial...
8. ### I Why do dimensions curl up or expand?

In string theory, the universe can have 9-10 spatial dimensions and the reason why we experience 3 is because those higher dimensions compactify. Under right conditions the extra dimensions can decompactify into the macroscopic dimensions we see. Why do some dimensions curl up or expand? Is...
9. ### A Curl in cylindrical coordinates -- seeking a deeper understanding

I calculate that \mbox{curl}(\vec{e}_{\varphi})=\frac{1}{\rho}\vec{e}_z, where ##\vec{e}_{\rho}##, ##\vec{e}_{\varphi}##, ##\vec{e}_z## are unit vectors of cylindrical coordinate system. Is there any method to spot immediately that ##\mbox{curl}(\vec{e}_{\varphi}) \neq 0 ## without employing...
10. ### A question on the definition of the curl of a vector

The curl is defined using Cartersian coordinates as $$\nabla\times A = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}.$$ However, what are the...
11. ### Finding a vector from the curl of a vector

Consider the following $$\nabla\phi=\nabla\times \vec{A}.$$ Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so? [Moderator's note: moved from a homework forum.]
12. ### A B and A in Curved Space Time: Does \nabla \times A =B?

By definition of the vector potential we may write \nabla \times A =B at least in flat space. Does this relation hold in curved space? I am particularly interested if we can still write this in a spatially flat Friedmann-Robertson-Walker background with metric ds^2=dt^2-a^2(dx^2+dy^2+dz^2) and...
13. ### How to observe if a vector field has curl or not?

These are the vector fields. I really have no idea how to see if there is a curl or not. I have been looking at the rotation of the vector fields. The fields d and e seem to have some rotation or circular paths, but I read online that curl is not about the rotation of the vector field itself...
14. ### A Calculate curl of e_φ: Non-Zero Vector

Why curl(\vec{e}_{\varphi}) is not zero vector? And how to calculate this. Vector ##\vec{e}_{\varphi}## is unit vector in cylindrical coordinates.
15. ### True or false questions about Divergence and Curl

##F = (P,Q,R)## is a field of vector C1 defined on ##V = R3-{0,0,0}## There are a lot of true or false statement here. I am a little skeptical about my answer because it contains a lot of F, but let's go. 1 Rot of F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S...
16. ### Vorticity and Curl of Velocity

This is not homework. I'm studying fluid mechanics/dynamics in the heart/blood vessels and I just want to understand this, so I can have a better appreciation for it's clinical relevance. I'm more of biology/biochem type of person so this has been a bit of challenge. I have basic physics course...
17. ### B What Does "Curl" Mean in Vector Fields?

That is how I understand curl: If I have a vane at some point ##(x,y)## of a vector field, then that vane will experience some angular velocities in points 1 ##(x+dx,y)##, 2 ##(x,y+dy)##, 3 ##(x-dx,y)##, 4 ##(x,y-dy)##. Adding those angular velocities gives me the resulting angular speed of...
18. ### Physics History (Maxwell) Rotary Vectors?

I am reading the text 'Innovations in Maxwell's Electromagnetic Theory'. on page 44 there is a discussion on Ampere's circuital law . The passage is below. I don't understand the final statement. "In general represent a kind of relationship that obtains between certain pairs of phenomena , of...
19. ### I How to obtain the determinant of the Curl in cylindrical coordinates?

I have a vector in cylindrical Coordinates: $$\vec{V} = \left < 0 ,V_{\theta},0 \right>$$ where ##V_\theta = V(r,t)##. The Del operator in ##\{r,\theta,z\}$is:$\vec{\nabla} = \left< \frac{\partial}{\partial r}, \frac{1}{r}\frac{\partial}{\partial \theta}, \frac{\partial}{\partial z}...
20. ### I Deriving Curl of B from Biot-Savart Law & Vector Identity

$$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ using the vector identity: $$\nabla \times (A \times B) = (B \cdot \nabla)A - B(\nabla \cdot A) - (A \cdot \nabla )B + A(\nabla \cdot B)$$ ##A=J## and ##B=\frac{r-r'}{|r-r'|^3}## since...
21. ### I Problem about the usage of Gauss' law involving the curl of a B field

I am trying to derive that $$\nabla \times B=\mu_0 J$$ First the derivation starts with the electric field $$dS=rsin\varphi d\theta r d\varphi$$ $$\iint\limits_S E \cdot dS = \frac{q}{4 \pi \varepsilon_0} \iint\limits_S \frac{r}{|r|^3} \cdot dS$$...
22. ### I Divergence & Curl -- Is multiplication by a partial derivative operator allowed?

Divergence & curl are written as the dot/cross product of a gradient. If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator. is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
23. ### What is happening to the sin(phi) factor in the spherical curl?

This is from my E&M textbook. I'm doing a problem where I need to take the Curl in spherical coordinates but I'm getting the wrong answer. I tried applying the matrix, but it doesn't seem like it make sense with the expansion that they show in the textbook (screenshot below). If I apply the...
24. ### Finding Scalar Curl and Divergence from a Picture of Vector Field

For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...
25. ### Vector potential ##\vec A## in terms of magnetic field ##\vec B##

My solution is making an analogy of the ##\text{Relevant equations}## as shown above, starting from the equation ##\vec \omega = \frac{1}{2} \vec \nabla \times \vec v##. We have ##\vec B = \vec \nabla \times \vec A = \frac{1}{2} \vec \nabla \times 2\vec A \Rightarrow 2\vec A = \vec B \times...
26. ### I'm not getting the curl of vector potential equal to magnetic field

In this image of Introduction to Electrodynamics by Griffiths . we have calculated the vector potential as ##\mathbf A = \frac{\mu_0 ~n~I}{2}s \hat{\phi}##. I tried taking its curl but didn't get ##\mathbf B = \mu_0~n~I \hat{z}##. In this thread, I have calculated it like this ...
27. ### How to find the curl of a vector field which points in the theta direction?

I have a vector field which is originallly written as $$\mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...
28. ### Vector field equality Curl Proof of Moving Magnet & Conductor Problem

The moving magnet and conductor problem is an intriguing early 20th century electromagnetics scenario famously cited by Einstein in his seminal 1905 special relativity paper. In the magnet's frame, there's the vector field (v × B), the velocity of the ring conductor crossed with the B-field of...

34. ### Curl of Polarization in a bar electret

In Griffith's "Introduction to Electrodynamics" says that in a bar electret the curl of the polarization does not equal zero everywhere. Why is that ? Thanks in advance
35. ### Electrodynamics, Curl of P and D

I know that in statics curl of P=curl of D, since the variation of B in time = 0, and I also know that for linear mediums those curls are 0, but I don't know why, and I don't know if there is any expresion always valid. I would like to know where this curl comes from like I know where the curl...
36. ### Expressing the magnetic vector potential in terms of its curl

We have the retarded magnetic vector potential ##\mathbf{A}(\mathbf{r},t) = \dfrac{\mu_0}{4\pi} \int \dfrac{\mathbf{J}(\mathbf{r}',t_r)}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 \mathbf{r}'## And its curl ##\mathbf{B}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \left[\frac{\mathbf{J}(\mathbf{r}'...
37. ### I How is rotation related to the curl of a vector field?

If the curl of a vector is 0 e,g ##\vec \nabla×\vec A=0## the vector A is said to be irrotational,can anyone please tell how rotation is involved with ##curl## of a vector??
38. ### Magnetic field of vector potential

So I was able to do out the curl in the i and j direction and got 3xz/r5 and 3yz/r5 as expected. However, when I do out the last curl, I do not get 3z2-3r2. I get the following \frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}...
39. ### I Curl in cylindrical coordinates

So, let me derive the curl in the cylindrical coordinate system so I can showcase what I get. Let ##x=p\cos\phi##, ##y=p\sin\phi## and ##z=z##. This gives us a line element of ##ds^2 = {dp}^2+p^2{d\phi}^2+{dz}^2## Given that this is an orthogonal coordinate system, our gradient is then ##\nabla...
40. ### Small question about maxwell's equation curl of H

The first page of this short pdf from MIT sums the starting point to formulate my question: https://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/lecture-notes/lecture29.pdf We can see that ∇xH = Jfree and ∇xB =μo (Jfree +∇xM) ∇xB =μo (Jfree +JB) And now my...
41. ### A Angular Moment Operator Vector Identity Question

In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere...
42. ### MHB Equality with curl and gradient

Hey! :o I want to show that $\nabla\times (f\nabla g)=\nabla f\times \nabla g$. We have that $f\nabla g=f\sum\frac{\partial g}{\partial x_i}\hat{x}_i$, therefore we get \begin{align*}&\nabla\times (f\nabla g)=\nabla\times \left (f\sum\frac{\partial g}{\partial x_i}\hat{x}_i \right )\\ &...
43. ### I Why Is the Curl of a Conservative Force Field Zero?

Why the curl of a conservative force field is zero everywhere?
44. ### Calculate the Curl of a Velocity vector field

Homework Statement The velocity of a solid object rotating about an axis is a field \bar{v} (x,y,z) Show that \bar{\bigtriangledown }\times \bar{v} = 2\,\bar{\omega }, where \bar{\omega } is the angular velocity. Homework Equations 3. The Attempt at a Solution [/B] I tried to use the...
45. ### I Is the curl of a vector perpendicular to the vector?

I am asking this because ∇ is also considered as a vector in some cases. Considering it as a vector in this case ,too, ( ∇× E).E = (- ∂ B/∂t).E = - ∂ (B.E)/∂t =0 Since B and E could be arbitrarily dependent on t, B.E = 0 where B and E are magnetic and electric fields respectively. This...
46. ### Magnetic Dipole Field from a Loop of Wire

I am trying to understand the magnetic dipole field via loop of wire. The above pictures show how this problem is typically setup and how the field lines are typically shown. The math is messy but every textbook yields the following: β = ∇xA = (m / (4⋅π⋅R3)) ⋅ (2⋅cos(θ) r + sin(θ) θ) The...
47. ### Curl of a gradient and the anti Curl

Homework Statement Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D? Homework Equations Curl of gradient f = 0 Curl of Vector D = <x,y,z>The Attempt at a Solution Curl of vector D Where vector D=<A,B,C> Cy - Bz = x Az - Cx = y Bx -...
48. ### Curl of the Magnetic Field of an Infinite Wire

I'm familiar with the relationship \nabla\cdot\frac{\hat{r}}{r^2}=4\pi\delta(r) in classical electromagnetism, where \hat{r} is the separation unit vector, that is, the field vector minus the source vector. This is result can be motivated by applying the divergence theorem to a single point...
49. ### Why must this expression for the curl be wrong?

Homework Statement Without explicit calculation, argue why the following expression cannot be correct: $$\nabla \times (\mathbf{c} \times \mathbf{r}) = c_{2}\mathbf{e_{1}}+c_{1}\mathbf{e_{2}}+c_{3}\mathbf{e_{3}}$$ where ##\mathbf{c}## is a constant vector and ##\mathbf{r}## is the position...
50. ### A Impossible Curl of a Vector Field

Let's assume the vector field is NOT a gradient field. Are there any restrictions on what the curl of this vector field can be? If so, how can I determine a given curl of a vector field can NEVER be a particular vector function?