Curl Definition and 371 Threads

Homework Statement Can someone explain the following to me, \nabla x \vec{V} = -k \frac{\partial{\Phi}}{\partial{t}} \hat{a}_n where \vec{V}, \Phi are the wind velocity and pressure respectively.Homework Equations Take the cross product- thus in the matrix we have the unit vectors in the first...

Regarding the equation for curl: Nable E literally means the sum of the differences of certain rates of change with respect to certain coordinates i hat, j hat, k hat. Since Nabla Cross E also is interpreted as the volume of a paralleliped in 3D space... 1. when the volume is zero, does...

Two problems one that I have some idea about solving, the other I have no idea at all about where to start. 1. Find the surface integral of E . dS where E is a vector field given; E = yi - xj + 1/3 z3 and S is the surface x2 + z2 < r2 and 0 < y < b Well Gauss' theorum would be the place...

Homework Statement \vec{F}(x,y,z) = x^2y\vec{i} + y^2z^3\vec{j} + xyz\vec{k} Homework Equations The Attempt at a Solution I got: Curl: (xz - 3y^2z^2)\vec{i} + (-yz)\vec{j} + (-x^2)\vec{k} Div: 2xy + 2yz^3 + xy Are these right?

Homework Statement Three small circles, C1, C2, and C3, each with radius 0.1 and centered at the origin are in the xy-, yz-, and xz-planes, respectively. The circles are oriented counterclockwise when viewed from the positive z-, x-, and y-axes, respectively. A vector field, , has...

Homework Statement http://img4.imageshack.us/img4/4218/divergenceandcurl.jpg The Attempt at a Solution Totally confused on what the question's asking. Wouldn't the divergence of say x_hat be the partial of x_hat over x which is just 0? So every answer would just be 0 or something? Same...

Homework Statement http://img5.imageshack.us/img5/8295/capturewmw.th.jpg Homework Equations The Attempt at a Solution I tried to find the curl first and what i got is y - 3 and then I multiply that by the area of the circle which is 4pi.. am I doing something wrong?

Hello All .. How are you ? I hope you fine Our professor taught as about the meaning of curl , but I was totally confused about it , especially when he used Taylor expansion of two variables and line integrals It’s like this Sorry for the very bad diagram in attachments , where delta...

Given an equation describing the curl of a vector field, is it possible to derive an equation for the originating vector field? The divergence of the field is known to be zero at all points

I'm working with Maxwell's equations, and I have found the curl of a magnetic field at all points. How can I figure out what the magnetic field is at those points?

My notes say that if we know the divergence and curl of a field then that uniquely determines the field. Can somebody give me an example of how, given only the div and curl of a field, we can deduce the field? I considered the electric field where we have, \nabla \cdot...

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...

I'm trying to get a picture of magnetic potential, as in how to relate it to spin precession. (So have I got the right picture so far?) Classically, B is measured as the derivative of curl (which is a circulation integral) "around" a conducting current (that is, perpendicular to the direction...

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?

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 u}{\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...