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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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of OsnabrÃ¼ck