A concise formula for the Hessian determinant of a function parameterising a quadratic hypersurface

TL;DR

This paper derives a concise formula for the Hessian determinant of smooth functions whose graphs lie on quadratic hypersurfaces, using matrix algebra techniques.

Contribution

It provides a new, simplified formula for the Hessian determinant in the specific geometric context of quadratic hypersurfaces.

Findings

Derived a concise formula for the Hessian determinant.

Utilized matrix algebra for the proof.

Applicable to functions on quadratic hypersurfaces.

Abstract

We will give a concise formula for the Hessian determinant of a smooth function $y:\mathbb R^n\supseteq\Omega\to\mathbb R$ such that its graph is contained in a quadratic hypersurface. The proof will make heavy use of matrix algebra.