Hodge Dual part deux

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 M be a manifold of dimension \dim(M)=n and let \omega=\sum_I f_I dx_I be a k-form on M for k\leq n. Here, I 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 \omega to the (n-k)-form *\omega =\sum_I f_I(*dx_I) where *dx_I=\varepsilon_I dx_{I^C}, where I^C is the multi-index consisting of all numbers 1,\ldots,n not in I, and where \varepsilon_I=\pm 1 denotes the sign of dx_I dx_{I^C}.

So what we said is that for the most commonly-recognized example \mathbb{R}^3 with orthogonal 1-forms dx,dy,dz and the usual metric ds^2=dx^2+dy^2+dz^2, the Hodge Star operator sends 1 to dx\wedge dy\wedge dz and vice versa, it sends dx,dy,dz to dy\wedge dz, dz\wedge dx,dx\wedge dy respectively, and it sends (dx\wedge dy)\mapsto dz, (dy\wedge dz)\mapsto dx, and (dx\wedge dz)\mapsto -dy. But the question then remains: Why does anybody care?

From a big picture perspective, mathematicians are all about generality and the reason is obvious: Why should I have to resolve problems time and again using specific information when generalities will allow me to solve multiple cases at one time without having to do repeat a process indefinitely? It makes sense, as much as our brains hate to admit it, and mathematicians are all about making sense.

Differential geometry is, in many ways, an offspring of this mentality. One way to think about the advancement towards modern differential geometry is as follows: Learn one-variable calculus, use basic intuition to advance to multi-variable vector calculus, and then ask yourself, What if I want to do calculus outside the confines of simple Euclidean spaces? What if I want to do calculus on all smooth surfaces at once? The solution, of course, is differential geometry, with its use of differential forms to generalize standard Euclidean infinitesimals and its three main operations (namely \wedge, d, and *) to mimic the behavior of differentiation and integration in far more general environments.

So, long story short: The Hodge Dual * should be thought of as one of three main operations, and so its properties are studied in part because they’re intrinsically beautiful and in part because the interplay between it and its two cousin operations is really really nice.

And that’s the purpose of this post.

Consider the 1-form \alpha=\sum_{i=1}^n F_i dx_i where for the time being we’re working in \mathbb{R}^n. Clearly, then, \alpha can be thought of as a vector-valued 1-form of the form \alpha=\mathbf{F}\cdot d\mathbf{x} where \mathbf{F}=\left(F_1,\ldots,F_n\right)^T and d\mathbf{x}=\left(dx_1,\ldots,dx_n\right)^T. This yields an obvious one-to-one correspondence between 1-forms \alpha and vector fields \mathbf{F}, and this correspondence yields some particularly interesting results when hit with the Hodge Star. To see this, let f:\mathbb{R}^n\to\mathbb{R} be an arbitrary smooth function (i.e., a 0-form) with associated 1-form df=\sum_{i=1}^n (\partial f/\partial x_i)dx_i. Rewriting, we see that

df = \widehat{\mathbf{F}}\cdot d\mathbf{x} where \widehat{\mathbf{F}}=\left(\partial f/\partial x_1,\ldots,\partial f/\partial x_n\right)^T.

The kicker, here, is that the vector field \widehat{\mathbf{F}} is precisely the gradient field \text{grad}(f). Hence,

df = \text{grad}(f)\cdot d\mathbf{x}.

Next, consider a vector field \mathbf{F} and a 1-form \alpha=\mathbf{F}\cdot d\mathbf{x}=\sum_{i=1}^n F_i dx_i. As mentioned above,

*\alpha= \sum_{i=1}^n F_i(*dx_i)=\sum_{i=1}^n F_i(-1)^{i+1}dx_1dx_2\cdots\widehat{dx_i}\cdots dx_n

where \widehat{dx_i} means “omitting dx_i” and where the second equation comes from rewriting the (n-1)-form *dx_i in increasing multi-index notation. This notation can be simplified by writing *d\mathbf{x} for the vector-valued (n-1)-form of the form

*d\mathbf{x} = \left(*dx_1,\ldots,*dx_n\right)^T

and by noting then that *\alpha=\mathbf{F}\cdot *d\mathbf{x}. Taking the exterior derivative of *\alpha yields the following:

\begin{array}{rcl} d*\alpha & = & d\left(\sum_{i=1}^n F_i(-1)^{i+1}dx_1dx_2\cdots\widehat{dx_i}\cdots dx_n\right) \\[1.5em] & = & \sum_{i=1}^n (\partial F_i/\partial x_i)dx_i(-1)^{i+1}dx_1\cdots\widehat{dx_i}\cdots dx_n \\[1.5em] & = & \sum_{i=1}^n (\partial F_i/\partial x_i)dx_1\cdots dx_i\cdots dx_n\end{array}.

Recollecting the vector field \sum_{i=1}^n (\partial F_i/\partial x_i) as the so-called divergence field \text{div}(\mathbf{F}), whereby it follows that

d*\alpha = \text{div}(\mathbf{F})\cdot d\mathbf{x}.

Note, also, that this identity can be rewritten by observing that *(dx_1\wedge\cdots\wedge dx_n)=1 in \mathbb{R}^n, whereby it follows that *(d*\alpha)=*(\text{div}(\mathbf{F})\cdot d\mathbf{x})=\text{div}(\mathbf{F}). Hence,

*d*\alpha = \text{div}(\mathbf{F}).

Finally, note than in \mathbb{R}^3, evaluating d\alpha and then applying * yields that

*d\alpha = \text{curl}(\mathbf{F})\cdot d\mathbf{x}.

The details of the last identity are omitted for brevity but can be found – along with further details of the identities previously mentioned – in [1].

So there we have it: A (more-complete) (somewhat-less-)brief exposition of reasons why differential geometers love the Hodge Star operator. Of course, this last statement is an absurdly-over-simplified over-simplification, but for now, it summarizes where we stand with our knowledge of the topic at hand.

Until next time….


Resources
1. Sjamaar, Reyer, Manifolds and Differential Forms. Retrieved online from here.

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