- #1
Oxymoron
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Im currently learning some Exterior calculus which I am hoping will help me with my honours project.
The problem I am stuck at is the following.
[tex]\mbox{Show that } i_X\ast \omega = \ast(\omega \wedge X^{\flat})[/tex]
where [itex]X^{\flat}[/itex] is the one-form related to the vector field [itex]X[/itex] by the metric, and [itex]\omega[/itex] is some p-form. Also, [itex]i_X[/itex] is meant to be the interior derivative. Note: this is not a homework problem, but intended as a discussion thread on the concepts concerned with problems of this type. In effect, such a thread could help me understand more about what is going on so that I may be able to show what I have written.
But the main reason for posting here is that I would like to understand how I can use Hodge star operators, exterior derivatives, and musical isomorphisms to define an operation which is identical to all the classical vector calculus operations, and in particular curl. I believe it helps me more to discuss with other people.
Any discussion on any of the material that I have mentioned would be greatly recieved.
Cheers.
The problem I am stuck at is the following.
[tex]\mbox{Show that } i_X\ast \omega = \ast(\omega \wedge X^{\flat})[/tex]
where [itex]X^{\flat}[/itex] is the one-form related to the vector field [itex]X[/itex] by the metric, and [itex]\omega[/itex] is some p-form. Also, [itex]i_X[/itex] is meant to be the interior derivative. Note: this is not a homework problem, but intended as a discussion thread on the concepts concerned with problems of this type. In effect, such a thread could help me understand more about what is going on so that I may be able to show what I have written.
But the main reason for posting here is that I would like to understand how I can use Hodge star operators, exterior derivatives, and musical isomorphisms to define an operation which is identical to all the classical vector calculus operations, and in particular curl. I believe it helps me more to discuss with other people.
Any discussion on any of the material that I have mentioned would be greatly recieved.
Cheers.
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