Hessian matrix

The Jacobian of the derivatives df/dx_1, df/dx_2, ..., df/dx_n of a function f(x_1, x_2, ..., x_n) with respect to x_1, x_2, ..., x_n is called the Hessian (or Hessian matrix) H of f, i.e., H f(x_1, x_2, ..., x_n) = [(d^2 f)/(dx_1^2) | (d^2 f)/(dx_1 dx_2) | (d^2 f)/(dx_1 dx_3) | ... | (d^2 f)/(dx_1 dx_n) (d^2 f)/(dx_2 dx_1) | (d^2 f)/(dx_2^2) | (d^2 f)/(dx_2 dx_3) | ... | (d^2 f)/(dx_2 dx_n) ⋮ | ⋮ | ⋮ | ⋱ | ⋮ (d^2 f)/(dx_n dx_1) | (d^2 f)/(dx_n dx_2) | (d^2 f)/(dx_n dx_3) | ... | (d^2 f)/(dx_n^2).] 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.

Jacobian | second derivative test