1. ### Vector potential ##\vec A## in terms of magnetic field ##\vec B##

My solution is making an analogy of the ##\text{Relevant equations}## as shown above, starting from the equation ##\vec \omega = \frac{1}{2} \vec \nabla \times \vec v##. We have ##\vec B = \vec \nabla \times \vec A = \frac{1}{2} \vec \nabla \times 2\vec A \Rightarrow 2\vec A = \vec B \times...
2. ### How to find the curl of a vector field which points in the theta direction?

I have a vector field which is originallly written as $$\mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$ (##r## is the distance from origin, ##\phi## is azimuthal angle and...
3. ### Vector Cross Product With Its Curl

Starting with LHS: êi εijk Aj (∇xA)k êi εijk εlmk Aj (d/dxl) Am (δil δjm - δim δjl) Aj (d/dxl) Am êi δil δjm Aj (d/dxl) Am êi - δim δjl Aj (d/dxl) Am êi Aj (d/dxi) Aj êi - Aj (d/dxj) Ai êi At this point, the LHS should equal the RHS in the problem statement, but I have no clue where...
4. ### Magnetic field of vector potential

So I was able to do out the curl in the i and j direction and got 3xz/r5 and 3yz/r5 as expected. However, when I do out the last curl, I do not get 3z2-3r2. I get the following \frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}...
5. ### A Angular Moment Operator Vector Identity Question

In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere...
6. ### I Magnetic Dipole Field from a Loop of Wire

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...

12. ### Div and curl in other coordinate systems

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}...
13. ### Nonconservative Force

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.
14. ### Meaning of Curl from stokes' theorem

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...
15. ### Definition of curl

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...
16. ### Geometrical meaning of Curl(Gradient(T))=0

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...
17. ### Stoke's and Gauss's Theorum in proving div(curlA)=0

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...
18. ### Curl of the vector potential produced by a solenoid

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...
19. ### Intuitive interpretation of some vector-dif-calc identities

Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
20. ### Taking the time derivative of a curl

Is the time derivative of a curl commutative? I think I may have answered this question.... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and...
21. ### Prove that curl of a vector is a vector

Homework Statement Proove it Iam supposed to change coordinate system, and proove that the result depends on coordinate system. The Attempt at a Solution My attempt was to start from definition of cross product using levicivita. I already prooved that divergence of a vector is a scalar. But...
22. ### Finding Beltrami field in Cartesian coordinates

Homework Statement Working in Cartesian coordinates (x,y,z) and given that the function g is independent of x, find the functions f and g such that: v=coszi+f(x,y,z)j+g(y,z)k is a Beltrami field. Homework Equations From wolfram alpha a Beltrami field is defined as v x (curl v)=0 The Attempt...
23. ### Curl of a function and vector field

Hello, I'm having some difficulty with a conceptual question on a practice test I was using to study. I have the answer but not the solution unfortunately. 1. Homework Statement "For every differentiable function f = f(x,y,z) and differentiable 3-dimensional vector field F=F(x,y,z), the...