Curl Definition and 371 Threads

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)...

Well, the reason I'm asking this is because we recently did a problem in my class where we were supposed to show some vector identity, with the conditions that both curl B = 0 and div B = 0 The problem was really about the maths, but it was phrased as if the field were a magnetic...

Homework Statement The problem is to find the value of Curl of A X B. I used the usual vector triple product formula to write as below. Δ X (A X B) = (Δ.B)A - (Δ.A)B = (div B)A - (divA)B Homework Equations But this is not the answer. Please suggest where i was wrong...

on pages 14-15, in deriving the normal vector to a surface, they use a plane to cut the surface (the plane is parallel to the xz plane) then use the curve 'c' in the xz plane (this curve being where the plane intersects the surface), draw a tangent vector 'u' and want to use the components of...

I need to prove this: u x (\nabla x u) = \frac{1}{2}\nabla(u²) - (u \cdot \nabla)u. I've came to this: uj∂iuj - uj∂jui (i think it's correct) But how this 1/2 appears?

\nabla\timesgrad(f) is always the zero vector. Can anyone in terms of physical concepts make it intuitive for me, why that is so. I get that the curl is a measure of the tendency of a vector field to rotate or something like that, but couldn't really assemble an understanding just from that.

I have a question here about Maxwell's equations: according to faraday's law at some point in space changing magnetic field with time creates the curl of electric field at that point and according to Ampere's law with Maxwell's correction changing with time electric field or electric current...

Homework Statement The curl satisfies (A) curl(f+g) = curl(f) + curl(g) (B) if h is real values, then curl(hf) = hcurl(f) + h'·f (C) if f is C2, then curl(gradf) = 0 Show that (B) holds. 2. The attempt at a solution I'm not quite sure how to interpret the "h is real valued"...

The differiantial form of faraday's law tells that at a any point in space changing with time magnetic field creates the rotor of electric field (let's say circular electric field at that point), but in the centre of the circular field there is no E vector, it's zero, there only is it's rotor...

Hi guys, I am trying to create a magnetic field from an image contour and an attempt to create an active contour model. I have a function which takes the image derivative via a Guassian: function J=ImageDerivatives2D(I,sigma,type) % Gaussian based image derivatives % %...

Hello! I'm reading up on Hamiltonian mechanics and i stumbled on the fact that the curl of the vector potential can be expressed as B_k = \sum_k \epsilon_{kij}\frac{\partial A_i}{\partial x_j} Now the text that I'm reading says that this formula can be inverted as \sum_k \epsilon_{kij}...

Homework Statement If \phi= xy^{2} A=xzi-z^{2}j+xy^{2}k B=zi+xj+yk Verify that \nabla.(\phiA)=A.\nabla\phi+\phi.\nablaA Homework Equations The Attempt at a Solution I have worked out the first two parts of the question: \phiA = (x^{2}y^{2}z, -xy^{2}z^{2},x^{2}y^{4}) div(\phiA) =...

I'm trying to figure this out. Say you have a cylinder of perfectly rotating fluid, so that it's velocity field is: F(x,y,z) = yi - xj which has curl -2k assuming there is 'infinite' fluid drag and you have an 'infinitely' light ball which you place into the fluid at any point (let's say...

I am trying to get a good grasp of the relation between the curl of a vector field and the exterior derivative of a 1-form field. In cartesian coordinates for flat R^3 the relationship is misleadingly simple. However, it still requires us to make an identification of the 2-form basis dx \wedge...

Homework Statement So this is part of a problem set in which I have to show that a vector field is divergence free but not the curl of any vector field. LetF =\frac{<x,y,z>}{(x^2 + y^2 + z^2)^{3/2}} Then F is smooth at every point of R3 except the origin, where it is not defined. (This...

Homework Statement 1.F=(x-8z)i+(x+9y+z)j+(x-8y)k find the curl of F Homework Equations curl of F= del X FThe Attempt at a Solution 1. First I took the partial with respect to y of (x-8y) and subtracted the partial with respect to z of (x+9y+z). From this I got (-8-1) Then I took the partial...

Can anyone help me proving this: http://img88.imageshack.us/img88/3730/provei.jpg And just for curiosity, is there a proof for why is the Laplace operator is defined as the divergence (∇·) of the gradient (∇ƒ)? And why it doesn't work on vetorial function. Thanks in advance, guys! Igor.

I am curious if there are any issue with commuting the curl of a vector with the partial time derivative? For example if we take Faraday's law: Curl(E)-dB/dt=0 And I take the curl of both sides: Curl(Curl(E))-Curl(dB/dt)=0 Is Curl(dB/dt)=d/dt(Curl(B)) I assume this is only...

Curl is easy to compute in 3 dimensions and if you let the third component be 0, its also easy in 2 dimensions. If you let the second and third components be 0, it is also easy in 1 dimension. My question is, is there a generalisation for curl to n dimensions and if there is, what is it and is...

Can someone help me intuitively understand why if a field has zero curl then it must be the gradient of a scalar potential? Thanks!

In a right-handed cartesian coordinate system the divergence and curl operators are respectively: \nabla \cdot A= \frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}+\frac{\partial A_{z}}{\partial z} \nabla \times \mathbf{A}= \begin{vmatrix} \widehat{x} & \widehat{y} &...

Homework Statement Suppose the length of your forearm is 34cm and its mass is 1.3kg. If your bicep inserts into the forearm 3.5cm from the pivot (the elbow), and your biceps muscle can produce a force of 800 N, how much weight can you curl? Model your forearm as a uniform rod. I have no...

Homework Statement Show that \nabla \times (a \cdot \nabla a) = a\cdot\nabla(\nabla\times a) + (\nabla \cdot a(\nabla \times a) - (\nabla \times a)\cdot \nabla aHomework Equations \nabla \times (\nabla \phi) = 0 \nabla \cdot (\nabla \times a) = 0 The Attempt at a Solution I started with...

Homework Statement Find the div v and curl v of v = (x2 + y2 + z2)-3/2(xi + yj + zk) Homework Equations div v = \nabla \cdot v and \nabla \times v The Attempt at a Solution I am just confused and drawing a blank in basic algebra Is it right to expand v like this v = x-3 + y-3 +...

How do you interpret the divergence or curl of the unit normal defined on a surface? This sometimes comes up when applying Stokes' theorem. A simple example would be Surface area = \int_{S} \hat{n} \cdot \hat{n} dA = \int_{V} \nabla \cdot \hat{n} dV where S is the closed surface that...

Homework Statement I want to prove this: \vec{r} \nabla^2 - \nabla(1+r \frac{\partial}{\partial {r}})=i \nabla \times \vec{L} Homework Equations BAC-CAB rule: \nabla \times (\vec{A} \times \vec{B}) = (\vec{B} . \nabla) \vec{A} - (\vec{A} . \nabla ) \vec{B} - \vec{B} (\nabla . \vec{A}...

Homework Statement If scalar s=x^3 + 2xy + yz^2 and vector v = (xy^3, 2y + z, z^2) find: (a) grad (s) (b) div v (c) curl v Homework Equations The Attempt at a Solution I'm entirely lost at how to do this. I think that grad s is the derivative of the scalar. I think that div is...

Hey, I've been stuck on this question for quite a while now: Homework Statement 1a. Write down an expression for the position vector r in spherical polar coordinates. 1b. Show that for any function g(r) of r only, where r = |r|, the result \nabla x [g(r)r] = 0 is true. Why does this...

I noted that if [itex]f : C \to C[\itex] is holomorphic in a subset [itex]D \in C[\itex], then [itex]\nabla \by \hat{f} = 0, \nabla \dot \hat{f} = 0[\itex]. Moreover, those two expressions are equivalent to the Cauchy-Riemann equations. I'm rewriting this in plaintext, in case latex doesn't...

For there to be curl is some vector field fxy cannot equal fyx. Where fx= P, and fy=Q. Since the (partial of Q with respect to x)-(Partial of P with respect to y) is a non zero quantity giving curl. I understand that the terms will cancel due to the right-handedness of the definition but we...

Anybody know Einstein notation for divergence and curl? What I would like to do is give each of these formulas in three forms, and then ask a fairly simple question; What is the Einstein notation for each of these formulas? The unit vectors, in matrix notation...

the divergence and the curl of a vector field "A" are specified everywhere in a volume V. The normal component of curl A is also specified on the surface S bounding V. Show that these data enable one to determine the vector field in the region

Homework Statement Use stokes theorem to find double integral curlF.dS where S is the part of the sphere x2+y2+z2=5 that lies above plane z=1. F(x,y,z)=x2yzi+yz2j+z3exyk Homework Equations stokes theorem says double integral of curlF.dS = \intC F.dr The Attempt at a Solution...

Divergence and Curl in cylindrical and spherical co are: \nabla \cdot \vec E \;=\; \frac 1 r \frac {\partial r E_r}{\partial r} + \frac 1 r \frac {\partial E_{\phi}}{\partial \phi} + \frac {\partial E_z}{\partial z} \;=\; \frac 1 {R^2} \frac {\partial R^2 E_R}{\partial R} + \frac 1 {R\;sin...

Faraday's law has an integral and a differential version: curl \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} \mbox{ and } \oint_{C} \mathbf{E} \cdot d \mathbf{l}=- \frac{d}{dt} \int_{S} \mathbf{B} \cdot d \mathbf{S} When I use the differential version I always have a constant of...

If a vector field has any component in a circular direction how can its curl be zero? If I imagine a vortex of water, it makes sense that it will be easier to go with the water in a circle than it would be to go against the water in a circle. Or more mathsy: A vector field in cylindrical...

Hi, this is a very simple question about the curl theorem. It says in my book: " If F is a vector field defined on all R3 whose component functions have continuous partial derivatives and curl F = 0 , then F is a conservative vector field" I might sound stupid, but what exactly does...

What is the direction of the curl of a vector A? If y-component of the above A is uniform can we say that the y-component of curl of a A is zero?

Homework Statement Find the curl of the vector field \mathbf{F} = <xyz,0,-x^2 y> The Attempt at a Solution I am mostly just having problems with computing the determinant. I could just go with crossing the first row and first column. But i noticed that the intermediate step...

Hello, I would like to ask a question on curl. The wikipedia page http://en.wikipedia.org/wiki/Vector_calculus_identities" gives formulas of various operations, among which: \nabla \times (A \times B) = A(\nabla \cdot B) - B(\nabla \cdot A) + (\underbrace {B \cdot \nabla...

Homework Statement Three small squares, S1, S2, and S3, each with side 0.1 and centered at the point (4,5,7), like parallel to the xy, yz, and xz planes respectively. The squares are oriented counterclockwise when viewed from the positive z, x, y axes respectively. A vector field G has...

Can anyone show me how you get the curl in polar or spherical coordinates starting from the definitions in cartesian coordianates? I haven't been able to do this.

This might be math problem, but I only see it in EM books. \nabla X (\vec A X \vec B) \;=\; (\vec B \cdot \nabla)\vec A - \vec B(\nabla \cdot \vec A) -(\vec A \cdot \nabla)\vec B + \vec A ( \nabla \cdot \vec B) . What is \vec A \cdot \nabla ?

I have a number of books which give a vector identity equation for the curl of a cross product thus: \nabla \times \left(a \times b \right) = a \left( \nabla \cdot b \right) + \left( b \cdot \nabla \right) a - b \left( \nabla \cdot a \right) - \left( a \cdot \nabla \right) b But doesn't b...

Homework Statement Hi, i am trying to find the div and curl in spherical polar coordinates for the vector field, F I have attempted both and would really appreciate it if someone could tell me if the answers look ok as I am really not sure whether i have correctly followed the method...

Homework Statement Hi, i am trying to find the div, grad and curl in cylindrical polar coordinates for the scalar field \ phi = U(R+a^2/R)cos(theta) + k*theta for cylindrical polar coordinates (R,theta,z) I have attempted all three and would really appreciate it if someone could tell me...

Hi, During the description of vector spherical harmonics, where N = curl of M , I came across the following : Laplacian of N = Laplacian of (Curl of M) = Curl of (Laplacian of M) How do we know that these operators can be interchanged ? What is the general rule for such interchanges...