Zum Begriff des Differentials

Ein Differential erster Ordnung ist der lineare Anteil des Zuwachses von $f$ .

f (x_{0} + h) = f (x_{0}) + \underset{f^{'} (x_{0}) h}{\underset{︸}{(d f) (x_{0}; h)}} + o (∥ h ∥) h \to 0

h \to d x = (d x_{1}, \dots, d x_{n})

d f = f^{'} (x_{0}) d x = \frac{\partial f}{\partial x_{1}} d x_{1} + \dots + \frac{\partial f}{\partial x_{n}} d x_{n}

x = x (u) \Leftrightarrow \begin{matrix} x_{1} = x_{1} (u_{1}, \dots, u_{n}) \\ ⋮ \\ x_{n} = x_{n} (u_{1}, \dots, u_{n}) \end{matrix}

\begin{matrix} d x_{1} = \frac{\partial x_{1}}{\partial u_{1}} d u_{1} + \dots + \frac{\partial x_{1}}{\partial u_{n}} d u_{n} \\ ⋮ \\ d x_{n} = \frac{\partial x_{n}}{\partial u_{1}} d u_{1} + \dots + \frac{\partial x_{n}}{\partial u_{n}} d u_{n} \end{matrix}

d x = J d u = \frac{D x}{D u} d u