Dictionary

codifferential en · NOUN

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 -ius 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.

Meanings

1. The projected differential of an extensor field.
   • In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as #92;begin#123;matrix#125;#92;partial#95;t#92;omega#43;#123;#92;mathcalL#125;#95;u#92;omegaamp;#61;0#92;#92;uamp;#61;#92;delta#92;tilde#92;eta#123;-1#125;#92;Delta#123;-1#125;#92;omega#92;end#123;matrix#125; where #92;omega is the vorticity 2-form, #123;#92;mathcalL#125;#95;u denotes the Lie derivative with respect to the velocity field u, #92;Delta is the Hodge Laplacian, #92;delta is the codifferential (the negative of the divergence operator), and #92;tilde#92;eta#123;-1#125; is the canonical map from 2-forms to 2-vector fields induced by the Euclidean metric #92;eta. 2016, Terence Tao, “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation”, in arXiv:
2. (qualifier: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: (α,β)=∫_Mα∧⋆β, so that (α,dβ)=(δα,β).
   • One important concept is the introduction of codifferential operator 2015, 1:19:34 from the start, in 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri), spoken by Hiroshi Ooguri (Hiroshi Ooguri), K Raviteja (YouTube):

Forms