The Hessian (or Hessian matrix) of
is the Jacobian of the derivatives, , ..., of a function with respect to , , ..., ,
i.e.,
The Hessian matrix is also called the Hesse matrix (Dutta et al. 2023). As in the case of the Jacobian, the term "Hessian"
unfortunately appears to be used both to refer to this matrix
and to the determinant of this matrix (Gradshteyn and
Ryzhik 2000, p. 1069).
of 
is the Jacobian of the derivatives 
, 
, ..., 
of a function 
with respect to 
, 
, ..., 
,
i.e.,
![Hf(x_1,x_2,...,x_n)=[(partial^2f)/(partialx_1^2) (partial^2f)/(partialx_1partialx_2) (partial^2f)/(partialx_1partialx_3) ... (partial^2f)/(partialx_1partialx_n); (partial^2f)/(partialx_2partialx_1) (partial^2f)/(partialx_2^2) (partial^2f)/(partialx_2partialx_3) ... (partial^2f)/(partialx_2partialx_n); | | | ... |; (partial^2f)/(partialx_npartialx_1) (partial^2f)/(partialx_npartialx_2) (partial^2f)/(partialx_npartialx_3) ... (partial^2f)/(partialx_n^2)].](/images/equations/Hessian/NumberedEquation1.svg)
, the discriminant 
is given by
