Math needed to understand hodge star operator

Question

I'm a physics student trying to understand the hodge star operator and how to use it to derive the Laplacian. I looked into it, but the math language used is complete jibberish (notation seems insanely prohibitive). What exact field does the hodge star pertain to? Can you list the things I need to know in order to learn about this/books to read?

Answer 1

The Hodge star operator belongs to the subject of multilinear algebra, or perhaps exterior algebra. (The Wikipedia page on "exterior algebra" is probably the more helpful of the previous two.) Typically, one first encounters the Hodge star in a course on calculus on manifolds, or a course on Riemannian geometry.

If you're interested in a having a clear mathematical understanding of the Hodge star, then you'll need to know (at minimum) what the following concepts mean:

• Vector space $V$
• Dual space $V^*$ (of a vector space)
• Inner product $\langle v,w\rangle$ of vectors $v,w \in V$.
• Orientation of a vector space.
• Multilinear map $\omega \colon \underbrace{V \times \cdots \times V}_{k} \to \mathbb{R}$. By slight abuses of terminology and notation, a multilinear map (involving $k$ copies of $V$) is called an (algebraic) $k$-form, and the set of multilinear maps ($k$-forms) is often denoted by $\Lambda^k(V^*)$.
• Wedge product $\alpha \wedge \beta$ of a $k$-form $\alpha$ and an $\ell$-form $\beta$.

Now, suppose you understand the above words. If $V$ is an $n$-dimensional real vector space --- so you can think of $V$ as $\mathbb{R}^n$ --- equipped with an inner product $\langle \cdot, \cdot \rangle$ and an orientation, then one can define (but I won't define it here) the Hodge star operator $$\ast \colon \Lambda^k(V^*) \to \Lambda^{n-k}(V^*).$$ Note that if $\alpha \in \Lambda^k(V^*)$ is a $k$-form on $V = \mathbb{R}^n$, then $\ast \alpha \in \Lambda^{n-k}(V^*)$ is an $(n-k)$-form on $V = \mathbb{R}^n$.

Standard Example: Taking $V = \mathbb{R}^3$ with its standard inner product and orientation, letting $\{e_1, e_2, e_3\}$ be the standard basis, and $\{e^1, e^2, e^3\}$ be the dual basis of $V^* = (\mathbb{R}^3)^*$, then: \begin{align*} \ast e^1 & = e^2 \wedge e^3 \\ \ast e^2 & = e^3 \wedge e^1 \\ \ast e^3 & = e^1 \wedge e^2 \end{align*}

Remark: It turns out that the the gradient and divergence operators (in $\mathbb{R}^n$), as well as the curl operator (in $\mathbb{R}^3$), can all be expressed in terms of three concepts: the Hodge star operator, the exterior derivative, and the musical isomorphisms. Using those formulas, one can derive a similar formula for the Laplacian on $\mathbb{R}^n$.

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Now, if you are interested in the Laplacian in even greater generality --- e.g., on a (possibly curved) Riemannian $n$-manifold $M$ --- then of course you'll need to know more words, such as:

• Manifold
• Tangent space
• Riemannian metric
• Volume form

When one speaks of the Hodge star operator on a Riemannian manifold $M$, what one really means is the Hodge star operator $\ast$ on the tangent spaces $T_pM$ of $M$. In other words, the Hodge star operator at a point $p \in M$ is defined to be the Hodge star operator on the vector space $V = T_pM$.