You may recall that I deemed yesterday Differential Geometry Sunday and posted a small expository thing on the Hodge Star/Dual operator. Apparently in my cloudy haze of mathematical mediocrity, I concluded my post without having touched on the derivations I actually intended to touch on.
Sometimes I feel like I need a vacation from my vacation.
In any case, I’m going to take a stab at saying some of the things I’d meant to say yesterday, but in order to ensure we’re all on the same page, I’m going to recall what exactly the Hodge Star/Dual operator is. Then, after the break, I’m going to show some of the cool derivations that come about because of it.
Let be a manifold of dimension and let be a -form on for . Here, denotes a multi-index which – without loss of generality – can be assumed to be increasing. Then the hodge star operator is the operator which takes to the -form where , where is the multi-index consisting of all numbers not in , and where denotes the sign of .
So what we said is that for the most commonly-recognized example with orthogonal 1-forms and the usual metric , the Hodge Star operator sends to and vice versa, it sends to respectively, and it sends , , and . But the question then remains: Why does anybody care?