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The Hessian covariant of a form {{harvtxt|Hilbert|1993|loc=p.88}} is the determinant of the [[Hessian matrix]]
:<math>H(f) = \begin{bmatrix}
\frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x\,\partial y} \\
\frac{\partial^2 f}{\partial y\,\partial x} & \frac{\partial^2 f}{\partial y^2}
\end{bmatrix}.</math>
It is a covariant of order 2''n''− 4 and degree 2.
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The [[Jacobian matrix and determinant|Jacobian]]
:<math> \det \begin{bmatrix}
\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\
\frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}
\end{bmatrix}.</math>
is a simultaneous invariant of two forms ''f'', ''g''.
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