The Hodge Star in Coordinates
It will be useful to be able to write down the Hodge star in a local coordinate system. So let’s say that we’re in an oriented coordinate patch of an oriented Riemannian manifold , which means that we have a canonical volume form that locally looks like
Now, we know that any -form on can be written out as a sum of functions times -fold wedges:
Since the star operation is linear, we just need to figure out what its value is on the -fold wedges. And for these the key condition is that for every -form we have
Since both sides of this condition are linear in , we also only need to consider values of which are -fold wedges. If is not the same wedge as , then the inner product is zero, while if then
And so must be times the -fold wedge made up of all the that do not show up in . The positive or negative sign is decided by which order gives us an even permutation of all the on the left-hand side of the above equation.