codifferential

English

Noun

codifferential (plural codifferentials)

1. (mathematics) The projected differential of an extensor field.
• 2016, Terence Tao, “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation”, in arXiv[1]:
In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as ${\displaystyle {\begin{matrix}\partial _{t}\omega +{\mathcal {L}}_{u}\omega &=0\\u&=\delta {\tilde {\eta }}^{-1}\Delta ^{-1}\omega \end{matrix}}}$ where ${\displaystyle \omega }$ is the vorticity ${\displaystyle 2}$-form, ${\displaystyle {\mathcal {L}}_{u}}$ denotes the Lie derivative with respect to the velocity field ${\displaystyle u}$, ${\displaystyle \Delta }$ is the Hodge Laplacian, ${\displaystyle \delta }$ is the codifferential (the negative of the divergence operator), and ${\displaystyle {\tilde {\eta }}^{-1}}$ is the canonical map from ${\displaystyle 2}$-forms to ${\displaystyle 2}$-vector fields induced by the Euclidean metric ${\displaystyle \eta }$.
2. (differential geometry) the formal adjoint of the exterior derivative; a differential-geometric version of the divergence operator; the exterior derivative sandwiched between two Hodge star operators with some additional factor(s) that take(s) care of the sign; the Hermitian conjugate of the exterior derivative under the inner product for k-form fields over some manifold M: ${\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge \star \beta }$, so that ${\displaystyle (\alpha ,d\beta )=(\delta \alpha ,\beta )}$.
• 2015, 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri)‎[2], K Raviteja (YouTube), spoken by Hiroshi Ooguri (Hiroshi Ooguri), 1:19:34 from the start:
One important concept is the introduction of codifferential operator