Explanation of the curl of a curl in EM theory

[amriths04]

hi people,
i have come across curl(curl(A)) many times in EM theory. i can understand that curl of a vector essentially gives the sense of rotation of a field or a surface. but what is that curl of a curl? is it somethin like a 4th dimension? also i heard from my friends that "H. M. Schey (1997). Div Grad Curl and all that: An informal text on vector calculus.", explains these issues. is that a nice book. does any one have the link for it?
and how could we ever imagine and relate them to EM fields?

thank you,

 

[las3rjock]

curl curl vector

I wouldn't worry too much about coming up with a physical interpretation of \[\nabla \times \nabla \times \vec{A}\]. The only reason it appears in EM theory is because the standard derivation of electromagnetic waves is to take the curl of either of the curl equations in Maxwell's equations, and then to apply vector identities and the other Maxwell equations in order to obtain wave equations for the electric and magnetic fields. There is not really any physical insight to be gained here other than the fact that Maxwell's equations and the identities of vector calculus allow for the possibility (and reality) of electromagnetic waves.

As for Schey's div, grad, curl, and all that, it is a nice, informal book to learn vector calculus. It's pretty cheap (~$30) and pretty readily available, even at chain bookstores (Barnes & Noble, Borders, etc.).

 

[jallem]

curl of curl of a vector

As you said the curl is the vector which gives the rotation of
another vector. curl(curl(A)) give then the rotation of the
rotation. For example, imagine a flow(current) of water in the ocean
where the rotation vector describes a closed curve, just like would
be if a tornado could close to itself, the end and the beginning touch
each other. In that way the curl always describe a rotation. Or at
least the direction of the axis of the rotation and its value(how
strong it is). You got to have a little bit of imagination to
visualizate it.
Let me try again, imagine you have three cords(or cables) and
you twist then, that give you a rotation. Now make the two ends
of the twisted cords touch each other, you have a second rotation.

 

[Bibhu166]

curl of a curl vector

Hi,
If you see maxweel eqn, curl of E gives H and vice versa.Here we cant make a eqn containg only E or H. So we need to take curl of curl so that we get a eqn in only E or H.Physciallly aslo curl of A gives B and crl of B gives E. Just thunk rotaion of E produced H and rotaion of H produces E. this process continues and field propagates.
Regards,
Bibhu

I wouldn't worry too much about coming up with a physical interpretation of \[\nabla \times \nabla \times \vec{A}\]. The only reason it appears in EM theory is because the standard derivation of electromagnetic waves is to take the curl of either of the curl equations in Maxwell's equations, and then to apply vector identities and the other Maxwell equations in order to obtain wave equations for the electric and magnetic fields. There is not really any physical insight to be gained here other than the fact that Maxwell's equations and the identities of vector calculus allow for the possibility (and reality) of electromagnetic waves.

As for Schey's div, grad, curl, and all that, it is a nice, informal book to learn vector calculus. It's pretty cheap (~$30) and pretty readily available, even at chain bookstores (Barnes & Noble, Borders, etc.).