HodgeStarOperator - crowlogic/arb4j GitHub Wiki

The Hodge star operator is a mathematical operation used in differential geometry and exterior calculus, particularly in the study of differential forms on a manifold. It is named after the British mathematician W. V. D. Hodge. The Hodge star operator is used to define the Hodge dual of a differential form, which is a fundamental concept in the study of differential equations, electromagnetism, and other areas of physics and mathematics.

Given a differential form $\alpha$ of degree $k$ on an $n$-dimensional manifold $M$ equipped with a Riemannian metric $g$ and an orientation, the Hodge star operator maps $\alpha$ to another differential form, denoted by $*\alpha$, of degree $(n-k)$ on the same manifold. The operation is based on the metric and orientation, and it essentially rotates the form by 90 degrees in the geometric sense.

The Hodge star operator has some useful properties, including:

  • If $\alpha$ and $\beta$ are two differential forms of the same degree, then their wedge product $(\alpha \wedge \beta)$ is a differential form of degree $2k$, and the Hodge dual of their wedge product is equal to the inner product of $\alpha$ and $\beta$ times the volume form: $*(\alpha \wedge \beta) = \langle \alpha, \beta \rangle \operatorname{vol}$.

  • Applying the Hodge star operator twice on a differential form $\alpha$ of degree $k$ results in the original form multiplied by $(-1)^{k(n-k)}$, assuming a positively oriented orthonormal basis: $**\alpha = (-1)^{k(n-k)}\alpha$.

  • The Hodge star operator can be used to define the Laplace-Beltrami operator and the codifferential, which are important concepts in the study of harmonic forms and the Hodge decomposition theorem.

The Hodge star operator plays a significant role in physics, especially in the study of electromagnetism, where it is used to relate the electric and magnetic fields in a unified framework, as well as in the formulation of the equations governing these fields, such as Maxwell's equations in the language of differential forms.