lemmaroutine

Hessian cofactor is column-divergence-free

Shows Hessian cofactor

C(H(u))

has divergence-free columns

enables divergence-form integration by parts of the determinant

Lemma·lem:cofactor-divergence·§3.3 Column divergence and the boundary term

Statement (paper)

For

k=3

4

and a smooth real

u

, the Hessian cofactor satisfies

\sum_{a,\alpha}\bar e_\alpha\,\partial_{a\alpha} C(H(u))_{ab}=0 \quad(1\le b\le m).

In plain English

For

k=3

4

and smooth real

u

, the cofactor matrix

C(H(u))

of the quaternionic Hessian is divergence-free in each column:

\sum_{a,\alpha}\bar e_\alpha\,\partial_{a\alpha} C(H(u))_{ab}=0

for

1\le b\le m

Why it's here

This is the quaternionic analogue of the classical fact that the cofactor matrix of a real Hessian is divergence-free. It lets

\det H

be written in divergence form after integrating by parts against

C(H(u))

, which is what the energy balance and the determinant-energy computations on fibers (the Green identity with

M=C(H(\cdot))

) require. The proof rests on the cofactor being a polynomial derivative of the Moore determinant (the quaternionic matrix lemma) and on the Hermitian structure of

H(u)

(the Hessian calculus lemma).

Classified routine:

Quaternionic analogue of the classical divergence-free property of the cofactor matrix of a real (or complex) Hessian, as used for real and complex Monge–Ampère operators. It follows from the Moore determinant cofactor properties and the Hermitian structure of the quaternionic Hessian established in the earlier lemmas.

Uses

lemMoore determinant and cofactor propertieslemHessian calculus in quaternionic coordinates

Used by

propEnergy flux identity with full wall normalclaimLaplacian of segment energy equals full determinantclaimDeterminant energy and balance on pencil fibers

Sources

PaperOpen

Section: §3.3 Column divergence and the boundary term
Page: —
Label: lem:cofactor-divergence
Line: sections/calculus.tex:188

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