What is Curl: Definition and 367 Discussions

Homework Statement Per Maxwell's equation and my H. H. Skilling EM textbook, curl H = 0 in the absence of current density. But consider a long, thin wire along the z axis carrying time-invariant current I. By Ampere's law, at a point (x,0) outside the wire, H = I/2πx j . But curl H...

Hi All, I'm trying to figure out how the components of the curl transform upon changing the coordinate system. In general coordinates, the contravariant components of the curl (if applied to the velocity field; then the curl is known as vorticity) are defined as \omega^k =...

How does it work, exactly? Assume I have a vector field function and I take the curl of it. If I get a curl of zero, then does that guarantee that there is no potential function? And if I get a curl of non-zero, does that guarantee that there is a potential function? I googled this...

Homework Statement The electric and magnetic fields in a plane wave propagating in free-space in the z-direction can we represented by (in complex-exponential notation) E(x, y, z, t) = E_0 e^i(kz−wt+d ) and B(x, y, z, t) = B_0e^i(kz−wt+d ) Starting with the Ampère-Maxwell law, ∇ x B =μ_0 ε_0...

Homework Statement Homework Equations The Attempt at a Solution For part (c), I showed that the tripple cross product = 2a using einstein notation. Then, I showed that 2∇(a.r) = 2a which is the same as LHS. I don't think this is as elegant as it can get.. How do I prove it...

define curl "rotation per area" When they define curl, they say it is a measure of "infinitesimal rotation", or "rotation per area". What does that mean? Does it mean they measure how much something goes around in an infinitesimal point (which makes no sense), kind of like a whirlwind shrunk...

What does "curl E = const." on Ω say about E on ∂Ω? Assume I have a simply connected domain Ω and a twice differentiable vector field E for which I know that "∇×E = const." (1) and "∇E = 0" (2) on Ω - I am interested in solving a BC Problem on ∏ = (Ʃ ⊃ Ω)\Ω, the remainder of Ʃ less Ω. (1) and...

Homework Statement This question is regarding clarifying some reading in Griffith’s Electrodynamics, page 224. “deriving the curl of B” In particular it’s less on electrodynamics and more on some vectors or vector calculus. The book states: we must check that the second term integrates to...

Homework Statement I put this in the math forum because although it's for my EM waves class, it's a math question. Show that the spin force can be written as: F_{spin}=\frac{-1}{2}Im(\alpha)Im(E\cdot\nabla E^{*})=\nabla\times L_s Find L_s. Where \alpha is complex. I'm using E^{*} to denote...

Homework Statement Not sure if this belongs in homework or general discussion - I found this in reading In studying the divergence and curl of the magnetic field (B), I found a statement that I need some help with. In the derivation of the divergence of B using the Biot-Savart, I have...

hey all i know and understand the component of curl/line integral relation as: curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr where we have vector field F, A(C) is the area of a closed boundary, u is an arbitrary unit vector, dr is an infinitely small piece of curve C my...

This should be simple but I know I'm going wrong somewhere and I can't figure out where. The curl of a electric field is zero, i.e. \vec { \nabla } \times \vec { E } = 0 Because , no set of charge, regardless of their size and position could ever produce a field whose curl is not zero...

Simple question. It came out of lecture, so it's not homework or anything. My professor said that the curl of a vector field is always perpendicular to itself. The example he gave is that the magnetic vector potential A is always perpendicular to the direction of the magnetic field B. (I haven't...

Hi guys! I recently saw on Wiki that given a magnetic potential A(r)=(u/4π)(mXR/r^3) ,( where u is the permeability of free space, m is the magnetic dipole moment of a magnetic field, R is the position vector, and r is the distance from the magnetic field ) upon taking the curl of the magnetic...

Homework Statement I'm trying to understand where the Cartesian components of the rotor and the divergence of a vector field derived. I read that the divergence of a vector field is defined by: \vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V...

For a 2D vector field {F}=P(x,y)\vec{i}+Q(x,y)\vec{j} curl {F} = \frac{\partial Q}{\partial x}+\frac{\partial P}{\partial y}\vec{k} So that's the rate of change of the j component of a field vector with respect to x plus the rate of change of the i component with respect to y...how does...

Homework Statement if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"? Homework Equations Prove this by construction. Let...

Homework Statement compute the curl of: \vec{r} and \frac{\vec{r}}{r^3}Homework Equations \vec{r}=x\hat{x}+y\hat{y}+z\hat{z} r^3=(x^2+y^2+z^2)^\frac{3}{2} The Attempt at a Solution I figured out that the curl of \vec{r} = 0 as my book says it should be... however...I also need to prove...

The vector field \vec{F} = <\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0> has a zero curl, which means its circulation is zero. However \int \vec{F}.d\vec{s} around a unit circle on the xy plane is equal to (+/-)2\pi and not zero Is it because F is undefined at (0,0)? No, because Stoke's...

What is the Physical significance of Divergence and Curl?

Homework Statement Calculate the curl F: F(x,y,z) = cos(x)i + sin(y)j + exyk at point (1,1,1) Homework Equations The Attempt at a Solution After calculating ∇×F, my answer was: curl F = 2.72i + 2.72j + 0k I'd appreciate a confirmation of my answer.

Homework Statement I need to analyze these pictures for my homework and find out the curl of the vector field at the point (red) on the picture. Homework Equations http://i1242.photobucket.com/albums/gg525/sjrrkb/ScreenShot2012-11-26at61615PM.png The Attempt at a Solution basically...

hi whats's the physical meaning of curl? and why it is a vector? it's definition is line integral per volume. i can't understand why this is a vector.

Homework Statement ∫Bdot[∇×A]dV=∫Adot[∇×B]dV Prove this by integration by parts. A(r) and B(r) vanish at infinity. Homework Equations I'm getting stuck while trying to integrate by parts - I end up with partial derivatives and dV, which is dxdydz? The Attempt at a Solution I...

The relation between the vector operator curl and rotation in fluids and vector fields is treated thoroughly in many texts. And the uniform (pure or simple) shear of a solid is adequately described by the strain tensor. I'd like to put the two together. My guess is that an alternative...

I'm looking for a physical proof, something I can understand easily, though a mathematical proof might help too. Apologies if its the wrong section, encountered this while studying mechanics :|

we have a well known and simple equation for curl in cartesian coo. now we want it in let's say cylindrical coordinates. question is...can we transform every thing to cylinderical and then use the formula for cartesian?I mean writing basis vectors of cartesian in terms of r and theta and z and...

In some texts the author tries to interpret operations like Curl. Some say the curl of a vector field shows the amount of rotation of the vector field But some of them say,if you put a wheel in a fluid velocity field which is like the vector field at hand,if it can rotate the wheel,then it has...

http://web.mit.edu/6.013_book/www/chapter2/2.4.html I was going through the curl derivation on the above link. How does equation 3 turn out? Δy is the incremental length. But how do you decide whether it is +Δy/2 or -Δy/2. And why is the line integral taken Δz when the change is in the y...

Let's say I have this relation B=∇XA I know B, now I want A. What ever do I do? Is there some tried and true method out there?

1. Homework Statement I'm having trouble using equation 2.1 or 2.2 in the article to find the curl of the navier-stokes equation. I understand how to find curl, but can't make sense of the explanation/steps in the document provided by the professor. Homework Equations All relavent...

I was looking through a calculus book doing some of the practice problems where I was asked to calculate the curl of a few functions. One of them got me thinking, is there a function whose curl is itself? Much like how e^{x} is it's own derivative, is there a vector field that is it's own curl...

Please Someone explain why: 1.div(F×G)=GcurlF-FcurlG 2.curl(F×G)=F.divG-G.divF+(G.∇).F-(F.∇).G

Suppose we begin with F which is a force.. Does taking the curl of a force such as \nabla \times F state that this expression no longer has dimensions of force? What I mean is, does the nabla operator have dimensions, and so would this change the dimensions of the expression?

A ZERO Divergence Vector Field There is theorem that is widely used in physics--e.g., electricity and magnetism for which I have no proof, yet we use this theorem at the drop of a hat. The theorem is this: Given sufficient continuity and differentiability, every vector function A such that...

I'm confused about what polarization of a dielectric does to its electrical properties. It is clear to me that polarization causes every little atom to get a tiny dipole moment. A measure of the polarization is therefore P = dipole moment per unit volume. However, what is really a dipole moment...

This isn't a homework problem, but it won't let me post on the other page. A well known vector identity is that rot(rot(E)) = grad(div(E)) - div(grad(E)). I've actually used this before without encountering any problems, so I don't know if I'm just having a brain fart or something, but...

I would just like to know what the 4 dimensional curl is? I believe it is a matrix but I am not completely sure. Thanks for any help!

I'm a bit confused. I'm trying to calculate the curl of a 4 dimensional matrix. It's an attempt to use stokes theorem for 4 dimensions. The curl can be written as a antisymmetric matrix from what I understand with entries, Mi,j = d Ai/d j - dAj/di where i and j would be the different...

I look at the curl as how much it causes a stick to rotate. So like suppose we have the force field F = (y,0,0) Then we see the curl is nonzero, because the force in the x direction is increasing as we move perpendicular to the direction of the x-axis. Therefore suppose we place a twig with one...

I would very much like a good intuitive understanding of what the curl of a vector field is. I thought it was a measure of the how much the field tries to rotate something, but that must be wrong because an electric field can have field lines that turn and not just go out radially, but still the...

It doesn't seem like I have seen such a thing. I want to know if there's an inverse Curl operator? Let's say I was given the Curl of X = some function, now how can I solve for X ? How do I inverse the curl on both side of the equation? Thank you very much in advance.

Hello! I'm want to prove a vector identity for (\nabla \times \vec{A}) \times \vec A using the familiar method of levi-civita symbols and the identity \epsilon_{kij}\epsilon{kmn} = \delta_{im}\delta_{jn} - \delta_{in}\delta{jm}, but I don't seem to come up with any usefull answer. I...

Homework Statement Consider the intersection,R, between two circles : x2+y2=2 and (x-2)2+y2=2 a) Find a 2-Dimensional vector field F=(M(x,y),N(x,y)) such that ∂N/∂x - ∂M/∂y=1 Homework Equations none. The Attempt at a Solution There are other parts to the main question but I don't think I will...

Under what circumstances can a conservative field NOT be irrotational?

Homework Statement Give an example of a rigid body that has: Rotation but not curl Curl but not rotation Rotation and curl Neither rotation nor curl The Attempt at a Solution i don't even get what rotation is. Or curl, to be honest...*sigh* Maybe if you help me I can begin to...

I'm studying Curl. And I would like to know if Curl can be correctly understood this way. Imagine a spanner. It's mouth is A and the end is B. Imagine if the spanner was a vector pointing from A to B. If a torque acts on it the spanner will rotate. Can curl be understood as something like a...

Homework Statement Find a vector field \vec{A}(\vec{r}) in ℝ3 such that: \vec{\nabla} \times \vec{A} = y2cos(y)e-y\hat{i} + xsin(x)e-x2\hat{j} The Attempt at a Solution I broke it down into a series of PDE's that would be the result of \vec{\nabla} \times \vec{A}: ∂A3/∂y - ∂A2/∂z...

All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct)...

All necessary information is attached except the answer in Cartesian coordinates, which is -ix-jy+2kz and my work converting back from cylindrical to Cartesian, which I used WolframAlpha for, as the trig is a mess (that is, if the way I am doing this is correct)...