The Jacobian of the derivatives 
, 
, ..., 
of a function 
with respect to 
, 
,
..., 
is called the Hessian (or Hessian
matrix) 
of 
, 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) |
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).
In the second derivative test for determining extrema of a function 
,
the discriminant 
is given by
 |
The Hessian can be implemented in the Wolfram Language as
HessianH[f_, x_List?VectorQ] := D[f, {x, 2}]
