Codifferential

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.

One important concept is the introduction of codifferential operator