codifferential

English

Etymology

Etymology tree

Proto-Indo-European *ḱe?

Proto-Indo-European *ḱóm

Proto-Italic *kom

Proto-Italic *kom-

Latin con-

Latin co-lbor.

English co-

Proto-Indo-European *dwóh₁

▲

Proto-Indo-European *trísinflu.

Proto-Indo-European *d(w)is-

Proto-Italic *dis-

Latin dis-

Proto-Indo-European *bʰer-

Proto-Indo-European *-eti

Proto-Indo-European *bʰéreti

Proto-Italic *ferō

Latin ferō

Latin differō

Latin differēns

Proto-Indo-European *-(i)yós

Proto-Italic *-ijos

Proto-Italic *-ios

Old Latin -ios

Latin -iusnom.

▲

Ancient Greek -ίᾱ (-íā)influ.

Latin -ia

Latin differentia

Proto-Indo-European *h₂el-

Proto-Indo-European *-is

Proto-Indo-European *h₂élisder.?

Proto-Italic *-ālis

Latin -ālis

Latin differentiālislbor.

English differential

English codifferential

From co- + differential.

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 $\begin{array}{cc}{\partial }_t\omega +{ℒ}_u\omega & =0\\ u& =\delta {\tilde{\eta }}^{-1}{\mathrm{\Delta }}^{-1}\omega \end{array}$ where $\omega$ is the vorticity $2$-form, ${ℒ}_u$ denotes the Lie derivative with respect to the velocity field $u$, $\mathrm{\Delta }$ is the Hodge Laplacian, $\delta$ is the codifferential (the negative of the divergence operator), and ${\tilde{\eta }}^{-1}$ is the canonical map from $2$-forms to $2$-vector fields induced by the Euclidean metric $\eta$.

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: $(\alpha ,\beta )={\int }_M\alpha \wedge \star \beta$, so that $(\alpha ,d\beta )=(\delta \alpha ,\beta )$.
   • 2015, 1:19:34 from the start, in 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri)‎^[2], spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):

     One important concept is the introduction of codifferential operator

Further reading

• Hodge star operator on Wikipedia.Wikipedia
• Weitzenböck identity on Wikipedia.Wikipedia