What is Curl: Definition and 367 Discussions

So I know what they are and I've been given some really vague and weak interpretations, but I want to build up my intuition and know more about the specifics of curl and divergence. To my understanding now I know that curl is similar to a paddle wheel spinning in a direction dependent on the...

Hi, I've been trying to derive the electromagnetic stress tensor on my own, and I've run into a bit of a problem. I have a cross product of a curl (\vec{E}\times(\nabla\times\vec{E})) that I need to expand, and the typical...

In my derivation of Maxwell's 4th equation from the empirical Biot Savart law, for a static electric field, I have that the curl of B is equal to the magnetic permeability times the current density. Now, the source coordinates (i. the circuit) are given by R_1 ( R is a vector). The field...

Homework Statement Without parameterizing the path, determine what the value of the line integral (integral of F dot dr) is, if C is the closed, oriented path that travels around the triangle with vertices (0,0) (5,2), and (-3,6) and F=yi + xj Homework Equations Curl possiblY? The...

Homework Statement Their solution: The Attempt at a Solution My attempt: My query is: where have I gone wrong? Or are the solutions incorrect? Edit: Actually I just figured it out, I am right, just that I never divided by \omega*\epsilon, and I do get the same answer.

Homework Statement F=-ysin(x)i+cos(x)j Homework Equations Can the Curl test be applied to this vector field and state three facts you can deduce after applying the curl test. The Attempt at a Solution

trying to remember rules for curl test applicability. is it just simple closed curve? is F=-ysin(x)i+cos(x)j able to use the curl test?

Homework Statement This question is closely related to physics but it's in a maths assignment paper i have so here it is: By taking curls of the following equations: \nabla \times \bf{E} = -\frac{1}{c}\frac{\partial\bf{B}}{\partial t} \nabla \times \bf{B} =...

Homework Statement Assume the vector function A = ax(3x^{2}2y^{2})-ax(x^{3}y^{2}) a) Find \ointA\cdotdl around the triangular contour shown in Fig. 2-36 [it is a triangle with base and height of one on the x and y axis. the curl travels so that the normal vector is in the -z direction] b)...

given curl E = -1/c*(\partialH/\partialt) div E = 0 div H = 0 curl H = 1/c*(\partialE/\partialt), find \nabla x (\nabla x E) how do i take \nabla x curl E? i tried to do it by determinants, but I'm not sure which values correspond to the i, j, and k. so my next assumption is that there...

Determine Div & Curl from a given vector field V=Kyi-Kxj How do I format this? It's been a while since I've done this and every divergence and curl example I look up has the format V(x,y,z)={V1(x,y,z);V2(x,y,z);V3(x,y,z)} Should I reformat my V to be V{x,y}={V1(x,y);V2(x,y)}={Ky,-Kx}...

I want to express A as a function of B in the following equation: curl{A}=B So I need the inverse of the curl operator, but I don't know if it exist. Thanks.

"Definition" of Curl. Can anyone derive the gradient operator? Can anyone prove why this equality is true? http://en.wikipedia.org/wiki/Curl_%28mathematics%29#Definition Wikipedia says it is defined, however that's BS since the gradient operator was already defined so this needs to be proven...

untruncated version of question: As electrons move through a wire, is the direction of the curl of the magnetic field lines observed derivable from an underlying property? =========================== more detail (only to further clarify my question as useful): > looking for a mechanistic...

Given the two vector fields: \vec E and \vec B Where the first is the electric vector field and the second is the magnetic vector field, we have the following identity: curl(\vec E) = -\frac{\partial \vec B } { \partial t } and further that: curl(curl(\vec E)) =...

Curl div... Homework Statement f is a scalar field. What does div(f) curl(f) rotgrad(f) divgrad(f) stand for? I need to know if a scalar field can have the meanings of roration and diverge like a vector field

Hello, I find that one of my biggest weakness is following along with the math of waves (Wave function, SHO, Schrödinger Equation, etc). Is there a book out there that gives treatment to wave functions and PDE's in the same way that Div, Grad, Curl does for multi var/vector calculus? I do...

I know how to calculate the divergence and curl of a vector field. What I am lacking is any intuition on what these values mean. example: V= {x, y, z} ∇.V = 3 ∇xV = {0,0,0} F={-y, x, 0} ∇.F = 0 ∇xF = {0,0,2} G={0, 3y, 0} ∇.G = 3 I understand that that the divergence is a measure of how much...

could someone please help me? what would the divergence, gradient and curl of an oil spill be? I'm a bit confused. Thank you

Homework Statement 1. Evaluate \int_{S}\int curl F \cdot N dS where S is the closed surface of the solid bounded by the graphs of x = 4, z = 9 - y^2, and the coordinate planes. F(x,y,z) = (4xy + z^2)i + (2x^2 + 6y)j + 2xzk 2. Use Stokes's Theorem to evaluate \int_{C}F\cdot T dS...

Say V is a vector field. Is there a way (or rather a reasonable algorithm) to find V, given Curl V? Thanks

Homework Statement Question Attached Homework Equations The Attempt at a Solution So here I'm attempting b), I know \nabla \times \vec F\ is the curl, which in this case is defined by the matrix \left[ \begin {array}{ccc} x&y&z\\ \noalign{\medskip}{\frac {d}{{\it dx}}}&{\frac {d}{{\it...

Does Stokes's theorem imply that the flux integral of a curl of a vector field over a closed surface is always zero? (because then there is no boundary curve and thus the line integral over the boundary curve is zero) Is there an insightful way to see why this is always true? Maybe a...

Is this a theorem, that for a vector field B satisfying div B = 0 everywhere then there is a vector field A such that B = curl A? If so, is it hard to prove? Of course, the converse is obviously true.

What do Divergence and Curl of a vector function actually mean? They are nice to understand as mathematical operators and then we can work on with them, but what do they mean physically and why are they so important in our study of electromagnetism?

Homework Statement A bucket of water is rotated slowly with angular velocity w about its vertical axis..When a steady state has been reached the water rotates with a velocity field v(r) as if it were a rigid body. Calculate div(v) and interpret the result. Calculate curl (v). Can the flow be...

I have some conceptual problems with divergence... 1.divergence is supposed to be the flux per unit volume at a particular point...again,I saw on wikipedia,that they define divergence as "the derivative of net flow of of the vector field across surface of a small region relative to the volume...

Is it possible to find the vector when its curl is known?

When studying the curl, one often finds the explanation that the curl is a measure of the tendency of a vector field to circulate around a given point. But this doesn't make much sense to me, since there's no clear way to measure "tendency"? What are the units of "tendency"? Wouldn't you agree...

Hello everyone! Having a field \bf B = \nabla \times \bf A , how is it possible to get \bf A ? For constant fields, the answer is easy, but is there a general approach to find A ? Some algorithm to do it numerically would help me immensly, too. If anyone knows some book or reference...

Homework Statement find the divergence and curl of the vector field A = (x/(\sqrt{x^2 + y^2 + z^2}))i + (y/(\sqrt{x^2 + y^2 + z^2}))j + (z/(\sqrt{x^2 + y^2 + z^2}))k Homework Statement The Attempt at a Solution Im not going to go through the whole lot but i have done the whole...

I would like to demonstrate an identity with the INDICIAL NOTATION. I have attached my attempt. Please let me know where I made mistakes. Any suggestion? I am trying to understand tensors all by myself because they are the keys in continuum mechanics Thanks

Is the field of an electric dipole conservative? Initially I thought it would be, for no particular reason but that's just what my high school intuition thought. (haha I thought everything would be conservative apart from friction) But I was reading up on some vector calculus and...

I am reading the Griffiths about finding the curl of B using Biot-Savart Law. I do not understanding the step between equation (5.52) and (5.53) which finding the x components of the following: (\boldsymbol{J}\cdot\nabla^{\prime})\dfrac{\hat{\xi}}{\xi^{2}} where...

Hello. I have tried to search all day for the answer to my question but could not find anything. Is there a way to generalize the curl of a vector field to dimensions greater than three? It seems pretty straightforward to find the gradient and divergence of higher dimensional vector fields...

Homework Statement Homework Equations The Attempt at a Solution I think for solving this problem I should write \vec{r}=(x,y,z) and \vec{m}=(m_{x},m_{y},m_{z}) and then do a lot of algerbra to work it out. Is it correct? Is there a better/another solution?

Homework Statement U(x,y) = 3x2 - 7y A) Calculate the force at the coordinate point (3,3) B) Determine if the following forces are conservative and find the change in potential energy correspoinding to each for an interval 0 to x i) Fx = ax + bs2 a and b are constants ii) Fx =...

I'm studying vector calculus, and have a question about the curl and its relation to a cross product of the del operator and a vector. When doing a standard cross product as the formula I have i(det1) - j(det2) + k(det3), where det1, 2 3 are the appropriate 2x2 determinants. However for the...

Homework Statement Evaluate the divergence and curl of the following vectors. A(r) is everywhere parallel to the y-axis with a magnitude A = cx + A0 , where c and A0 are constants. Homework Equations The Attempt at a Solution I can evaluate the div and curl, but i don't know...

Homework Statement I want to calculate \nabla\times[\vec{F}(r)] and \nabla^2[\vec{F}(r)] where F if a function that depends of r, and r = \sqrt{x^2+y^2+z^2} Homework Equations 1)\nabla \times \vec A = \left|\begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ \\...

Hello all, I understand the fact that the principles LaTeX Code: F= \\nabla \\phi . LaTeX Code: \\nabla \\times F = 0 . must apply in order for a force field to be conservative however what i don't get is why showing: LaTeX Code: f_y= g_x, f_z= h_x, g_z= h_y where subscripts are what you...

like the divergence of a charged particle (whether + or -) Is there any possibility of presence of curl of elctric field of that particle?

I'm trying do derive the vorticity equation \begin{align}\frac{D\vec\omega}{Dt} &= \frac{\partial \vec \omega}{\partial t} + (\vec V \cdot \vec \nabla) \vec \omega \\ &= (\vec \omega \cdot \vec \nabla) \vec V - \vec \omega (\vec \nabla \cdot \vec V) + \frac{1}{\rho^2}\vec \nabla \rho \times...

Hi, Is it possible to have a magnetic field B which has curl B = 0 in all space? intuitivly such a field will be in a constant direction (like the electric field of an infinite charged plate ) and magnetic field "don't behave" like that, they make circles around currents, but this is not...

Find a pair of fields having equal and divergences in some region, having the same values on the boundary of that region, and yet having different curls. I really have no idea on where to start for this. Would making up 2 arbitrary fields in spherical co-ordinates work? a(theta) + b\phi +...

What is the divergence and curl at the edge of a black hole?

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?