Sei \(f:R^n \longrightarrow R\) (für Gradient und Hesse-Matrix)

Gradient: (Vektor aus \(R^n\))

 \(\nabla f(x) = \begin{pmatrix}\frac{\partial f}{\partial x_1}(x) \\
                    \frac{\partial f}{\partial x_2}(x) \\ \vdots \\
                    \frac{\partial f}{\partial x_n}(x) \end{pmatrix}\)

 mit \(x=(x_1,...,x_n)\)

Hesse-Matrix (\(n\times n)\)

\(H_f(x) := \begin{pmatrix}\frac{\partial^2 f}{\partial x_1\,\partial x_1}(x) &
\frac{\partial^2 f}{\partial x_1\,\partial x_2}(x) & & \cdots &
\frac{\partial^2 f}{\partial x_1\,\partial x_n}(x) \\[1mm]
\frac{\partial^2 f}{\partial x_2\,\partial x_1}(x) &
\frac{\partial^2 f}{\partial x_2\,\partial x_2}(x) & & \cdots &
\frac{\partial^2 f}{\partial x_2\,\partial x_n}(x)   \vdots  & \vdots  & \hfill & \hfill &\vdots  \\[1mm]
\frac{\partial^2 f}{\partial x_n\,\partial x_1}(x) & \frac{\partial^2 f}{\partial x_n\,\partial x_2}(x) & & \cdots &
\frac{\partial^2 f}{\partial x_n\,\partial x_n}(x) \end{pmatrix}\)

mit \(x=(x_1,...,x_n)\)

Sei nun \(f:R^n \longrightarrow R^m\).

Jacobi-Matrix (\(m \times n)\):

\(Df(x) :=
\begin{pmatrix} \frac{\partial f_1}{\partial x_1}(x) & \frac{\partial f_1}{\partial
x_2}(x) & \cdots & \frac{\partial f_1}{\partial x_n}(x)\\[1mm]
\frac{\partial f_2}{\partial x_1}(x) & \frac{\partial f_2}{\partial x_2}(x) & \cdots & \frac{\partial f_2}{\partial x_n}(x)\\
                           \vdots  & \vdots  & \hfill & \vdots  \\
\frac{\partial f_m}{\partial x_1}(x) & \frac{\partial f_m}{\partial x_2}(x) & \cdots & \frac{\partial f_m}{\partial x_n}(x)
\end{pmatrix}\)

mit \(x=(x_1,...,x_n)\)