codifferential

English

Etymology

co- +‎ differential

Noun

codifferential (plural codifferentials)

(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:
  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 ${\\displaystyle \\eta }$ .
(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: ${\\displaystyle (\\alpha ,\\beta )=\\int _{M}\\alpha \\wedge \\star \\beta }$ $\text{[math]}$ , so that ${\\displaystyle (\\alpha ,d\\beta )=(\\delta \\alpha ,\\beta )}$ $\text{[math]}$ .
- 2015, Hiroshi Ooguri as Hiroshi Ooguri, 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri)‎, K Raviteja (YouTube), 1:19:34 from the start:
  One important concept is the introduction of codifferential operator

单词	codifferential
释义	codifferential English Etymology co- +‎ differential Noun codifferential (plural codifferentials) (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: 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 ${\\displaystyle \\eta }$ . (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, Hiroshi Ooguri as Hiroshi Ooguri, 2 Tangent Space, Differential Forms, Metric (Mathematics for Theoretical Physicists - Hirosi Ooguri)‎, K Raviteja (YouTube), 1:19:34 from the start: One important concept is the introduction of codifferential operator Further reading Hodge star operator on Wikipedia.Wikipedia Weitzenböck identity on Wikipedia.Wikipedia
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codifferential

English

Etymology

Noun

Further reading