Curl Definition and 359 Threads

Hi, I have a certain NON-orthogonal curvilinear coordinate system in 3D (in the metric only g_{13}=g_{23}=g_{31}=g_{32}=0) and I want to take the curl (\nabla\times\mathbf{v}) of a vector. Any idea on how to do this? The only information I can find is about taking the curl of a vector in...

How to make another interpretation of "curl"? Recently,I've tried hard to find the physical interpretation of "curl". But , most of what I found were the same ,that is,"fluid flow"! I'm now wondering whether there's another annotation so that I can learn more about vecor calculus...

Homework Statement Calculate the (1) divergence and (2) curl of the following vector fields. (a) \widehat{E}(\widehat{x}) = r^{n}\widehat{x} (b) \widehat{E}(\widehat{x}) = r^{n}\widehat{a} (c) \widehat{E}(\widehat{x}) = r^{n}*(\widehat{a} X \widehat{x} where r = |\widehat{x}| and...

Homework Statement Determine the curl on teh surface of the bounded region consisting of the bottom part of the sphere with equation 625=z^2+x^2+y^2 where z<=20, in the force field F(x,y,z)=<x^2 * y,x*y^2 * z,2x> Homework Equations...

Homework Statement Suppose that f is a vector field such that curl f=(1,2,5) at every point in R^3. Find an equation of a plane through the origin with the property that \oint_{C}f dot dX = 0 for any closed curve C lying in the plane. Homework Equations...

Curl Product Rule confusion? Homework Statement In Griffith's Introduction to Electrodynamics, he gives the rule: \nabla\times(\bold{A}\times\bold{B})=(\bold{B}\cdot\nabla)\bold{A}-(\bold{A}\cdot\nabla)\bold{B}+\bold{A}(\nabla\cdot\bold{B})-\bold{B}(\nabla\cdot\bold{A}) Now I know I am...

Can someone give me analogies for each of these? I know the standard ones so try to be creative. I just received an A- in Calc IV and these words are KILLING me (moreso Curl and Div than Flux, as I'm close to understanding those and I've no idea what flux is). It would help if you gave examples...

Hello! I was thinking the other day, of the Earth's rotation around its axis. If one spins a boiled egg, it maintains spin longer than does an unboiled. Eventually both stops because of the friction against the floor, but not at the same time. The Earth has different levels of viscosity...

Homework Statement F(r) = r/!r!^3 (Sorry but the ! is supposed to imply that its scalar) I found the curl using the cartesian coordinate definition of curl. It came out to be zero. Now the question is, is F path independent? Its silly, becuase if the curl is zero then it does imply that...

Hi Everyone, Our class just learned Green's theorem using the Curl of a vector field F. I'm just having a tough time visualizing what in the world this means. I'm trying to view it in a physics perspective but I'm having a tough time. There's a picture in my textbook which has a plane with...

Just for reference, i got this question from reading an online ebook: http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf The bottom equation on page 24 is where i these equations came up. I have been reading some stuff and i keep coming across an annotation which looks exactly like a...

Homework Statement A vector field is defined by A=f(r)r a) show that f(r) = constant/r^3 if \nabla. A = 0 b) show that \nabla. A is always equal to zeroHomework Equations divergence and curl relationsThe Attempt at a Solution I tried using spherical co-ordinates to solve this. But I am not sure...

Hi, does anyone know a link showing how to calculate curl with a Levi-Civita tensor. I can't figure it out but I am sure if I could see an actual example would be able to work out what is going on. Thanks.

(Sorry, the title should read "...why curl of gradient of a scalar "function" is zero) Of course I know how to compute curl, graident, divergence. Algebrically I know curl of gradient of a scalar function is zero. But I want to know the reason behind this...and also the reason why gradient of...

Homework Statement F = \frac{r}{r} Find divF and curl F Homework Equations r = x\widehat{i} + y\widehat{j} + z\widehat{k} r = \sqrt{(x^{2} + y^{2} + z^{2})} The Attempt at a Solution F = \frac{x}{(\sqrt{x^{2} + y^{2} + z^{2}})}\widehat{i} + \frac{y}{(\sqrt{x^{2} + y^{2} +...

1. Evaluate the line integral∫F . dr with F = 3(-y,x,0) from (a,0,0) to (a,0,2πb) along a straight line. 2. Do the same along a circular helix between the two points, parameterised as r = (a cosλ, a sinλ, bλ) 3. Compute the curl of F. How does this relate to the two integral calculations...

Homework Statement Prove: \int\left(\nabla \times \vec{F}\right)\cdot d\vec{V} = \oint \left(\vec{\hat{n}} \times \vec{F} \right) dS Homework Equations In the previous part of the question, we proved that: \nabla \cdot \left( \vec{F} \times \vec{d} \right) = \vec{d} \cdot \nabla...

Homework Statement It can be shown that the line integral of F = xj around a closed curve in the xy - plane, oriented as in Green's Theorem, measures the area of the region enclosed by the curve. (You should verify this.) Use this result to calculate the area within the region of the...

Homework Statement Change to cylindrical coordinates and find the divergence F = <x, y, 0>/(x^2 + y^2) Homework Equations \nabla . F = \frac{1}{\rho}\frac{\partial\rho F}{\partial\rho}+\frac{1}{\rho}\frac{\partial F}{\partial\theta}+\frac{\partial F}{\partial z} The Attempt at...

Homework Statement Prove that \nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - (\nabla \cdot \nabla)\vec{A} using Einstein notation. Homework Equations \nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - (\nabla \cdot \nabla)\vec{A}...

Can anyone explain to me how to expand this expression for curl which I find in the GR book I'm reading (by Hobson, Efstathiou and Lasenby, page 71)? In a section entitled Vector Operators in Component Form they state the curl as a "rank-2 antisymmetric tensor with components": (curl)ab =...

Anyone know what topic, branch of math, book, or subject I should look up in order to find a formulation for Maxwell's equations in higher spatial dimensions? I don't mean having time as a 4rth dimension. I mean a 4rth (and more) spatial dimension. This would require the maxwell exquations...

N is the normal vector of a surface Sb and N dot B=0, div B=0 on Sb. Therefore, let B=curl A where A is a vector field. How can you prove that N cross A=0? Thanks.

Homework Statement Let F be F = ( x^2 z^2 ) i + (sin xyz) j + (e^x z) k.Find \int\int \nabla \times F \cdot n dS where the region E is above the cone z^2 = x^2 + y^2 and inside the sphere centered at (0,0,1) and with radius 1. (so it is x^2 + y^2 + (z-1)^2 = 1).. I know that they intersect at...

Greetings- out of college for 50 yrs and studying H.M. Schey's book. Cannot understand his derivation of the z component of the curl of a vector function F for a part of a sector of a circle in a plane parallel to xy axis. Cylindrical components. Let me describe equation as three parts for ease...

I am trying to think of a non-constant function whose divergence and curl is 0. It seems like this is impossible to me. Any hints?

i am having trouble with understanding the physical significance of these two operators.

In differential geometry, the usual curl operation that we are familiar with from elementary calculus is generalized to \,^*dA (where A is a one-form). In three-dimensions, this gives back a one-form. Now, the components of this one-form are \sqrt{g} \epsilon_{ijk} \partial^j A^k ...

Okey Dokey, so I'm bored and decided to play around with some math. I've got a problem that I can't figure out now; I have the green's function for the laplacian G(\vec{x}, \vec{x'}) = - \frac{1}{4\pi |\vec{x} - \vec{x'}|} There are no boundary conditions. Is there any lazy way to figure out...

When we say condition of a vector field F being conservative is curl F=0,does it mean that F=F(r)?.I know normally it does not look so.Please,then site an example where F is not a function of r,but still curl F=0.

if the grad of the curl of a function is always zero does this mean the magnitude of the curl is constant? or am i way off here?

[SIZE="5"]Problem The velocity of a two-dimensional flow of liquid is given by \textbf{V} = \textbf{i}u(x, y) - \textbf{j}v(x, y). If the liquid is incompressible and the flow is irrotational show that \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial...

What does a divergence calculate? What meaning is it for a vector? And also what is curl? Why we have to definte divergence and curl?

Hi, I am teaching myself electrostatics – something which seems to have been ‘skipped’ over in my electronics degree. I am working from the book: Fudamentals of engineering electromagnetics’ by David K. Cheng. My particular interest is electrostatics – more specifically the study of...

Can someone please point me in the direction of the one form required for my starting point producing the div and curl in 3space? I know grad is simply the d operator and div is *d* curl is *d i want to know what the one form i need to operate on to produce the classical div and curl...

Gradient, divegrance and curl? del operator! in static magnetic and electric fields, the del operator was introduced and then used to describe three different quantities.. i still can't quite figure out the physical meaning and difference between the curl,divergance and the gradient in terms of...

I didn't want to overload the last topic, "The Meaning of Curl in Electrodynamics", but I have a question so I'll do it as a new thread. I'm studying The Feynman Lectures on Physics - Vol 2, Sections 3-5 and 3-6: "The Circulation of a vector field" and "The circulation around a...

Hi I have a hard time understanding what the curl really means in Maxwell's equations, for example in a steady-state you have \nabla\times \textbf{E} = 0 and in a time-varying field you have \nabla\times \textbf{E} = -\frac{\partial \textbf{B}}{\partial t} The meaning of the...

1. If a vector field has zero curl, does it always mean that it is the gradient of some scalar (potential) field? 2. If the vector field is a force field and its curl is zero does that mean that the "potential" scalar field that it is the gradient of is always some form of "potential energy"...

purpose of each of the "operators", divergence, gradient and curl? Hi. Can anybody give me a reasonably simple explanation of what the purpose of each of the "operators", divergence, gradient and curl? (I've been looking but I never found something simple to understand) I know how to evaluate...

Calculate the average value of the curl of the fluid for a rectangular path 15 cm by 10 cm, as shown in the figure (see file attachment). Va=(10i + 5j) Vb = (5i+10j) Vc= (5i + 10j) Vd = (10i + 5j) Could someone help me to get started with this one? Please :smile: Maybe give me an...

Jackson("Classical Electrodynamics", Ch.6) uses the theorem of curl of curl to separate current density into transverse and parallel, \vec J = \vec{J_p}+\vec{J_t} to say, \begin{align*}\vec{J}(\vec{x}) &= \int\vec{J}(\vec{x'})\delta(\vec{x}-\vec{x'})d^{3}x'\\ &=...

A vector field is difined by A = f(r)r. a) show that f(r) = constant/r^3 if divergence A equal to zero. b) show that curl A is alway equal to zero

A fluid rotates with an angular velocity w about the z-axis. The direction of rotation is related to the z-axis by the rigrt hand screw rule. a) Find the velocity v of a point in the fluid, and show that curl v = 2w_k b) If now w is a function of the radius r, show that curl...

I need a physical meaning about "curl" (rotational field) thanks... NOR.

To calculate the divergence of a vectorfield in cartesian coordinates, you can think of it as a dot product, and to calculate the curl, you can think of it as a cross product. But how can you calculate the div and curl when you have spherical or cylindrical coordinates, without explicitely...

Is there any neat way/rule to write: \vec B \times (\vec \nabla \times \vec A) ? I've tried it myself and found (e.g) for the x-component: \left(B_x\frac{\partial A_x}{\partial x}+B_y\frac{\partial A_y}{\partial x}+B_z\frac{\partial A_z}{\partial x}\right)-\left(B_x\frac{\partial...

Hi Guys,, i have just started to study Divergence and curl but this is not at all enetering into my mind...Pls help me out understand this...This also has Divergence and Stokes theorm ..pls help me grasp it...Thx in advance... The Divergence Theorem and Stokes's Theorem provide the...

Hi, I'm having trouble proving the following result: \int_{V} (\nabla\times\vec{A}) dV = -\int_{S} (\vec{A}\times\vec{n}) dS I'm not sure how I should Stokes' and/or the Divergence Theorem in proving this, or if you should use them at all. Thanks in advance.

I'm having a bit of difficulty with this problem: \vec{\nabla} \times \vec{G} = \vec{F} where \vec{\nabla} \cdot \vec{F} = 0 and \vec{F} = <y, z, x> . Find \vec{G} . I'm really at a loss how to solve this. I know the solution must be quick and easy because it was on a quiz. What...