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poonintoon
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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.
Thanks.
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] ?
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] ?
Have you tried just writing it out?poonintoon said:It is, but I can't find out how to use it.
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]
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.
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) = εijk∂jFk, where εijk is the Levi-Civita symbol and ∂j and Fk represent the partial derivatives and components of the vector field, respectively.
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.
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.
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.