Calculating Curl With Index Notation

In summary, the conversation discusses the calculation of curl using a Levi-Civita tensor and provides a simplified formula for finding the x, y, and z components of curl. The conversation also includes a clarification on the use of implicit sums in the formula.
  • #1
poonintoon
17
0
Hi, does anyone know a link showing how to calculate curl with a Levi-Civita tensor. I can't figure it out but I am sure if I could see an actual example would be able to work out what is going on.
Thanks.
 
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  • #2
Isn't the curl of some vector, A, say [itex](\nabla \times \vec{A})_i[/tex] just [itex]\epsilon_{ijk}\partial_j A_k[/itex] ?
 
  • #3
cristo said:
Isn't the curl of some vector, A, say [itex](\nabla \times \vec{A})_i[/tex] just [itex]\epsilon_{ijk}\partial_j A_k[/itex] ?

It is, but I can't find out how to use it.
 
  • #4
cristo said:
Isn't the curl of some vector, A, say [itex](\nabla \times \vec{A})_i[/tex] just [itex]\epsilon_{ijk}\partial_j A_k[/itex] ?

poonintoon said:
It is, but I can't find out how to use it.
Have you tried just writing it out?

If Ak= <f(x,y,z), g(x,y,z),h(x,y,z)> where x= x1, y= x2, z= x3[/sub], then [itex]\epsilon_{ijk}\partial_j A_k[/itex] is:

We can simplify some of the "writing out" by noting that [itex]\epsilon_{ijk}= 0[/itex] if any of i, j, k are the same, [itex]\epsilon_{123}= \epsilon_{231}= \epsilon{312}= 1[/itex] and [itex]\epsilon_{132}= \epsilon{213}= \epsilon{321}= -1[/itex].

So for B= curl A, we have
[tex]B_x= B_1= \epsilon_{123}\partial_2\ A_3+ \epsilon{132}\partial_3 A_2[/tex]
[tex]= \partial_2 A_3- \partial_3 A_2= \frac{\partial h}{\partial z}- \frac{\partial g}{\partial y}[/tex]

[tex]B_2= \epsilon_{213}\partial_1\ A_3+ \epsilon{231}\partial_3 A_1[/tex]
[tex]= -\partial_1 A_3+ \partial_3 A_1= \frac{\partial h}{\partial x}- \frac{\partial f}{\partial z}[/tex]

[tex]B_3= \epsilon_{312}\partial_1\ A_2+ \epsilon{321}\partial_2 A_1[/tex]
[tex]= \partial_1 A_2- \partial_2 A_1= \frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}[/tex]
which are, of course, the usual formulas for curl A.
 
  • #5
So you have
[tex]\nabla\times\vec{A}=\partial_iA_j\hat{u_k}\epsilon_{ijk}[/tex].

So to get the x component of the curl, for example, plug in x for k, and then there is an implicit sum for i and j over x,y,z (but all the terms with repeated indices in the Levi-Cevita symbol go to 0)

[tex](\nabla\times\vec{A})_x = \partial_yA_z\epsilon_{yzx} + \partial_zA_y\epsilon_{zyx}=\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}[/tex]
 
  • #6
HallsofIvy said:
Have you tried just writing it out?

We can simplify some of the "writing out" by noting that [itex]\epsilon_{ijk}= 0[/itex] if any of i, j, k are the same, [itex]\epsilon_{123}= \epsilon_{231}= \epsilon{312}= 1[/itex] and [itex]\epsilon_{132}= \epsilon{213}= \epsilon{321}= -1[/itex].

So for B= curl A, we have
[tex]B_x= B_1= \epsilon_{123}\partial_2\ A_3+ \epsilon{132}\partial_3 A_2[/tex]
[tex]= \partial_2 A_3- \partial_3 A_2= \frac{\partial h}{\partial z}- \frac{\partial g}{\partial y}[/tex]

Thanks that's what I wanted, I thought once I had seen this I would be able to figure it out, unfortunately it's just not clicking for me.

Can I check I have the right thinking...
[itex]\epsilon_{ijk}\partial_j A_k[/itex]

For B1 you set i to 1. then that leaves two combinations for [itex]partial_j A_k[/itex]
[itex]partial_2 A_3[/itex] or [itex]partial_3 A_2[/itex]

Then you can have any tensor as long as i is 1 i.e [itex]epsilon_{123}\ \epsilon_{112}\ \epsilon_{111}[/itex]
but obviously any with two 1's in are zero.

I think this gives the right answer but I should be thinking more in terms of implicit sums than what combination I have left.
 

What is the definition of curl?

The curl of a vector field is a measure of the rotation of the field at a given point. It is a vector quantity that describes the direction and magnitude of the rotation.

How is curl calculated using index notation?

The curl of a vector field is calculated using the cross product of the gradient operator and the vector field. In index notation, this can be written as curl(F) = εijkjFk, where εijk is the Levi-Civita symbol and ∂j and Fk represent the partial derivatives and components of the vector field, respectively.

What is the physical significance of the curl?

The curl of a vector field is related to the angular momentum and vorticity of a system. It is also used to describe the circulation and rotation of fluid and electromagnetic fields.

Can the curl be calculated in three dimensions?

Yes, the curl can be calculated in any number of dimensions. In three dimensions, the curl is a vector with three components, representing the rotation around the x, y, and z axes.

How is the curl related to other vector operations?

The curl is related to the gradient and divergence through the fundamental theorem of vector calculus. In particular, the curl of the gradient of a scalar field is always zero, and the divergence of the curl of a vector field is also zero.

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