Curl Definition and 359 Threads

I am trying to understand the magnetic dipole field via loop of wire. The above pictures show how this problem is typically setup and how the field lines are typically shown. The math is messy but every textbook yields the following: β = ∇xA = (m / (4⋅π⋅R3)) ⋅ (2⋅cos(θ) r + sin(θ) θ) The...

Homework Statement Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D? Homework Equations Curl of gradient f = 0 Curl of Vector D = <x,y,z>The Attempt at a Solution Curl of vector D Where vector D=<A,B,C> Cy - Bz = x Az - Cx = y Bx -...

I'm familiar with the relationship \nabla\cdot\frac{\hat{r}}{r^2}=4\pi\delta(r) in classical electromagnetism, where \hat{r} is the separation unit vector, that is, the field vector minus the source vector. This is result can be motivated by applying the divergence theorem to a single point...

Homework Statement Without explicit calculation, argue why the following expression cannot be correct: $$\nabla \times (\mathbf{c} \times \mathbf{r}) = c_{2}\mathbf{e_{1}}+c_{1}\mathbf{e_{2}}+c_{3}\mathbf{e_{3}}$$ where ##\mathbf{c}## is a constant vector and ##\mathbf{r}## is the position...

Let's assume the vector field is NOT a gradient field. Are there any restrictions on what the curl of this vector field can be? If so, how can I determine a given curl of a vector field can NEVER be a particular vector function?

Homework Statement The angular velocity vector of a rigid object rotating about the z-axis is given by ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point. a.) Assuming that ω is constant, evaluate v...

Today when I ask a professor about maxwell eqation He tells me " it seems that the unknowns exceed the number of equations. What are the missing ingredients? The answer is the boundary condition .With appropriate boundary conditions, zero divergence and zero curl will nail down a unique solution...

This is an old problem, but one that may confuse many beginners. ##\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}## Let's say that we're trying to find the electric field produced by a changing magnetic field. We could take the inverse curl of the RHS, but the curl...

Homework Statement Can I, for all purposes, say that Nabla, on index notation, is $$\partial_i e_i$$ and treat it like a vector when calculating curl, divergence or gradient? For example, saying that $$\nabla \times \vec{V} = \partial_i \hat{e}_i \times V_j \hat{e}_j = \partial_i V_j (\hat{e}_i...

Suppose we have do a curl of two 2-d vectors... we get the 3rd axis about which it is rotating. But when we do the curl of two 3-d vectors.. we get a answer like x-y plane is rotating wrt z axis, y-z plane rotating wrt to x-axis and similarly x-z plane rotating wrt to y axis. My question is...

Homework Statement ∇ . r = 3, ∇ x r = 0 Homework EquationsThe Attempt at a Solution So far I've gotten up to ∇(∇^2 r)

Homework Statement Verify the identity: ## \nabla \times ( A \times B) = (B\bullet \nabla)A - (A\bullet\nabla)B + A(\nabla \bullet B)-B(\nabla\bullet A)## My issue here is I don't understand the significance of why a term has B or A on the left of the dot product, and another has B or A on...

Ok, so I know the curl represents how much something rotates about an axis. Let's assume we have a vector field F = Fx + Fy + Fz, where x y and z are direction vectors. So the rotation about the Z axis is made possible by a change in the Y direction and a change in the X direction. But the...

Hi, I'm looking at the following graph, but there are a few things I don't get. For instance: curl should always be zero in circles where the field lines are totally straight (right-most figure) curl should always be non-zero in circles where the field lines are rotating (center figure in 2nd...

I know that the curl of electrostatic field vector is zero. I want to know what will be the curl of electrostatic field at the edge region of the finite parallel plate capacitor?

This is more of an intuitive question than anything else: the curl of a vector field \mathbf{F} , \nabla \times \mathbf{F} is defined by (\nabla \times \mathbf{F})\cdot \mathbf{\hat{n}} = \lim_{a \to 0} \frac{\int_{C} \mathbf{F}\cdot d\mathbf{s}}{a} Where the integral is taken around a...

Homework Statement Using index-comma notation only, show: \begin{equation*} \underline{\bf{v}} \times \text{curl } \underline{\bf{v}}= \frac{1}{2} \text{ grad}(\underline{\bf{v}} \cdot \underline{\bf{v}}) - (\text{grad } \underline{\bf{v}}) \underline{\bf{v}} \end{equation*} Homework Equations...

##\nabla p = \rho \nabla \phi ## My textbook says that by taking the curl we get: ## 0=\nabla \rho X \nabla \phi ## ** I don't follow. I understand the LHS is zero, by taking the curl of a divergence. But I'm unsure as to how we get it into this form, from which it is clear that the gradients...

How can a curl of 4-vector or 6-vector be writen? Let's say that we have a 4-vector A4=(a1,a2,a3,a4) how can we write in details the ∇×A4 Can we follow the same procedure for 6-vector?

Hi, I stumbled upon thinking that "Is curl operator a linear operator" ? I was reading EM Theory and studied that the electromagnetic field satisfies the curl relations of E and B. But if the operator was not linear then how can a non linear operator give rise to a linear solution. Thus it...

While investigating about the curl I have found this interesting perspective: http://mathoverflow.net/a/21908/69479 I lack the knowledge to do the derivation on my own so I would like to ask for your help. I am an undergraduate. I do not understand what a "first order differential operator"...

What test can we perform on a vector field to determine if there exist vector field(s) that describe its inverse curl?

Hello! (Wave) Is it true that for a vector field $\vec{F}$, a function $f$ such that $\vec{F}=\nabla{f}$ can exist only if $\text{ curl } \vec{F}=\nabla \times \vec{F}=\vec{0}$ ? How can we check it? (Thinking)

Homework Statement $$\bar{v}=\nabla \times \psi \hat{k}$$ The problem is much bigger, i know how a rotor or curl is calculated in cylindrical coordinates, but I'm just asking to see what would be the "determinant" rule for this specific curl. Homework Equations $$\psi$$ is in cylindrical...

In the image attached to this post, there is an equation on the top line and one on the bottom line. In the proof this image was taken from, they say this is a consequence of divergence theorem but I'm not quite understanding how it is. If anyone could explicitly explain the process to go from...

I've got ∇×(∇×R)=∇(∇.R)-∇2R [call it eq.1] However I have the identity ∇×(A×B)=A(∇.B)-B(∇⋅A)+ (B⋅∇)A-(A⋅∇)B [call it eq.2] Substituting in A=∇ and R=B into eq.2 we get ∇×(∇×R)=∇(∇.R)-R(∇⋅∇)+ (R⋅∇)∇-(∇⋅∇)R which i work out to be ∇×(∇×R)=∇(∇.R)-R(∇⋅∇)+ (R⋅∇)∇-∇2R Basically I don't understand...

Homework Statement I need a pointer to a proof of the following items: if div X =0 then X = curl Y for some field Y. if curl X = 0 then X = grad Y for some field Y. Can anyone provide a pointer to a proof? Thanks. Bob Kolker Homework EquationsThe Attempt at a Solution

My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are: \mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...

Hello I've been having trouble finding the curl of A⃗ = r^2[e][/Φ]. Could someone help me please?

Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...

I have attached the equation that I do not quite understand how is true. This is the current density and was simply used as a part of a proof. But how is this equation true? How does taking the curl of J give you this expression?

I know by definition that if T is a 2nd order tensor and v is a vector, curl(Tv)=curl(T)v but what if instead of constant vector v, I have w=grad(u), not constant but obviously an irrotational vector field. Is this still true: curl(Tw)=curl(T)w ? My guess is yes since curl(w)=0 but have no...

I not understand because why if I have a (constant) force of friction and I apply the curl, I finding that this not is equal to zero, since this force is non conservative.

In trying to get an intuition for curl and divergence, I've understood that in the case of R2, div f(x,y) = 2Re( d/dz f(z,z_)) and curl f(x,y) = 2Im( d/dz f(z,z)), where f(z,z) is just f(x,y) expressed in z and z conjugate (z). Is there any way of proving the fundamental properties of div and...

Joos asserts on page 31 https://books.google.com/books?id=btrCAgAAQBAJ&lpg=PP1&pg=PA31#v=onepage&q&f=false that $$\nabla \times \mathfrak{v} = \lim_{\Delta \tau \to 0} \frac{1}{\Delta \tau }\oint d\mathfrak{S}\times \mathfrak{v}$$ I tried to demonstrate this, and neglected to place the surface...

I have a problem. So the curl of the E field is supposed to be zero always, which tells us that it is a conservative force (path independence and scalar potential and so on). But what about the fact that the induced electric field consequent upon changes in magnetic flux is circular? Doesn't...

A modern standard way of deriving the EM wave equation from Maxwell's equations seems to be by taking the curl of curl of E and B field respectively, and use some vector identity. See for instance on wikipedia. So, I have a basic understanding of the curl of a vector field. Defined as the...

Divergence can be defined as the net outward flux per unit volume and can be explained using Gauss' theorem. (I read this in Feynman lectures Vol. 2) In the next page, He derives Stokes' theorem using small squares. The left side of equation represents the total circulation of a vector...

In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl. Curl{v}=∇×v But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the...

What is the geometrical meaning of ##\nabla\times\nabla T=0##? The gradient of T(x,y,z) gives the direction of maximum increase of T. The Curl gives information about how much T curls around a given point. So the equation says "gradient of T at a point P does not Curl around P. To know about...

Suppose [FONT=Verdana]u is a vector-valued function. Is it true that [FONT=Verdana] (∇×u)⋅( (u⋅∇)u ) = (u⋅∇)(∇×u)⋅u ? [FONT=Verdana]Please note the lack of a dot product on the first two terms of the RHS and the parenthesis around the second term of the LHS. I'm trying to understand whether...

Homework Statement The problem puts forth and identity for me to prove: or . It says that I can use "straight-forward" calculation to solve this using the definition of nabla or I can use Gauss's and Stoke's Theorum on an example in which I have a solid 3D shape nearly cut in two by a curve...

I first learned Maxwell's equations in their integral form before I was introduced to the differential form, i.e. w/curl & divergence. As I understand, in order to derive the curl form from the integral form, apply Stokes Theorem to the integral form of ∫(closed)E⋅dl=-d/dt[∫(closed)B⋅dA], and...

My understanding of the curl of a vector field is the amount of circulation per unit area with a direction normal to the area. For the vector field described as \textbf{B} =\boldsymbol{\hat\phi} \frac{\mu_{0}I}{2 \pi r} I figured the curl would be something more like this, because it points in...

Homework Statement / Homework Equations[/B] I was looking at Example 5.12 in Griffiths (http://screencast.com/t/gGrZEPBpk0) and I can't manage to work out how to verify that the curl of the vector potential, A, is equal to the magnetic field, B. I believe my problem lies in confusion about how...

Hello, new to this website, but one question that's been killing me is how can curl of a gradient of a scalar field be null vector when mixed partial derivatives are not always equal?? consider Φ(x,y,z) a scalar function consider the determinant [(i,j,k),(∂/∂x,∂/∂y,∂/∂z),(∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)]...

Hi! I have recently been independently studying vector calculus. I understand that divergence measures change in magnitude and curl is the change in direction, however, I don't understand what certain divergences and curls represent. For example, how would you describe a field with a divergence...

In what condition(s) curl of electric displacement is zero? Is it okay to say curl of electric displacement is zero in: 1) in electrostatics (curl of E is zero) then followed by the following conditions: 2) when there is no polarization (curl of P is zero) 3) in uniform polarization (which...

Hello everybody, i have some troubles with the interpretation of curl in 2D/3D space. I was looking for a better understanding of Curl, watching this video generally curl represents the 'amount' of local rotation in a vector field, point by point. If we think at the 2D vector field described...

See these equations: \left | \frac{d \vec{f}}{d \vec{r}} \right | = \left | \frac{d(f_1,f_2)}{d(x,y)} \right | = \frac{\partial f_1 \wedge \partial f_2}{\partial x \wedge \partial y} = \left | \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial...