Definition: Totally Differentiable Functions, Total Derivative

Let \(V\subseteq \mathbb R^n\) and \(W\subseteq \mathbb R^m\) be finitely dimensional vector spaces over the field \(\mathbb R\), let \(G\subseteq V\) be an open set and let \(\varphi \colon G\rightarrow W\) be a function. We call \(\varphi \) differentiable (or totally differentiable) at a point \(x\in G\), if there exists a linear function \(A\colon V\rightarrow W\) with the property

\[\varphi (x+\xi)=\varphi (x)+A\xi+r(\xi)\,,\]

where \(r\) is a function defined in a neighborhood of \(0\) with values in \(W\), i.e. \(r:B\left(0,\delta \right)\rightarrow W\) and fulfilling the property

\[\lim_{\xi\to 0}\frac{r(\xi)}{\mid \mid \!\xi\!\mid \mid}=0\]

for all \(\xi\in V\) with \(x+\xi\in U\left(x,\delta \right)\subseteq G\).

The linear function \(A\), if it exists, is called the (total) derivative of \(\varphi \) at the point \(x\) and is also denoted by \[\left(D\varphi \right)_{x}.\]

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück