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...
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...
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...
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}}...
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...
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
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 $$\nabla \times \vec{V} = \partial_i \hat{e}_i \times V_j \hat{e}_j = \partial_i V_j (\hat{e}_i...
This is more of an intuitive question than anything else: the curl of a vector field \mathbf{F} , \nabla \times \mathbf{F} is defined by
(\nabla \times \mathbf{F})\cdot \mathbf{\hat{n}} = \lim_{a \to 0} \frac{\int_{C} \mathbf{F}\cdot d\mathbf{s}}{a}
Where the integral is taken around a...
How can a curl of 4-vector or 6-vector be writen? Let's say that we have a 4-vector A4=(a1,a2,a3,a4)
how can we write in details the ∇×A4
Can we follow the same procedure for 6-vector?
Homework Statement
$$\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...
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}...
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.
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...
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...
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...
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...
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...
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...
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...
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...