## 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.

[...] Armstrong: (Pseudo)-Riemannian Metrics, Isometries, Inner Products on 1-Forms, The Hodge Star in Coordinates, The Hodge Star on Differential Forms, Inner Products on Differential [...]

Pingback by Thirteenth Linkfest | October 8, 2011 |

I have a question for you, if I wanted to show the rate of change in a person personality due to a control stimuli, how do you think the formula should llook like.

Comment by Marcus Cox | October 9, 2011 |

[...] easiest to work this out in coordinates. If is some -fold wedge then is times the wedge of all the indices that don’t show up in . [...]

Pingback by The Hodge Star, Squared « The Unapologetic Mathematician | October 18, 2011 |

[...] implications does this have on the coordinate expression of the Hodge star? It’s pretty much the same, except for the determinant part. You can think about it yourself, [...]

Pingback by Minkowski Space « The Unapologetic Mathematician | March 7, 2012 |