codifferential

Noun

(mathematics) The projected differential of an extensor field.

Japanese Meaning

伸長体場における微分作用のうち、特定の射影によって得られる（射影された微分）概念

What is this buttons?

In differential geometry, the codifferential maps an extensor field to its projected differential, revealing subtle cohomological properties.

Google Translation / DeepL Translation

微分幾何学では、エクステンソル場の射影微分はエクステンソル場を微分の射影へ写像し、微妙なコホモロジー的性質を明らかにする。

Details

Sense（1）

(mathematics) 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 ∂ₜ𝜔+ℒᵤ𝜔=0\u=𝛿̃𝜂⁻¹𝛥⁻¹𝜔 where 𝜔 is the vorticity 2-form, ℒᵤ denotes the Lie derivative with respect to the velocity field u, 𝛥 is the Hodge Laplacian, 𝛿 is the codifferential (the negative of the divergence operator), and ̃𝜂⁻¹ is the canonical map from 2-forms to 2-vector fields induced by the Euclidean metric 𝜂.

Google Translation / DeepL Translation

Details

Sense（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: (𝛼,𝛽)=∫_M𝛼∧ star 𝛽, so that (𝛼,d𝛽)=(𝛿𝛼,𝛽).