Thread: differential forms and Hodge star

I am trying to figure out these relations (Hodge operator acts on the following r-forms)

this one I do not get

how do they work?

They only way I can explain it is that the wedge product work like the cross product for example

and so on
is it a correct way of thinking?

You need to be careful beucase the wedge product can be used in on any forms in the exterior algebra but the cross product only works in three dimientions.

The more general idea is

The Hodge dual is a map from



Loosely speaking If we are given an r form w we are looking for a 3-r form v
such that ()



Where J is the Jacobian matrix (the volume element of the curvilinear coordinate system.)

In rectangular coordinates |J|=1

So in your first case you have



2nd



Notice the the order is wrong we need to move dx to the front and since the wedge product is antisymmetric we get a factor of (-1) each time we move through a 1-form this gives



So the Hodge dual is



and for the last one I assume you have a typo and that you mean



So the Hodge dual is