Let \(U\subseteq\RR^n\) be open, let \(F\colon U\to\RR^m\), and let \(\vector p\in U\). The map \(F\) is real-differentiable at \(\vector p\) if there exists a linear map \(A\colon\RR^n\to\RR^m\) such that \[ \lim_{\vector h\to0}\frac{\norm{F(\vector p+\vector h)-F(\vector p)-A(\vector h)}}{\norm{\vector h}}=0. \]
Equivalently, there is a function \(R\), defined for \(\vector h\) with \(\vector p+\vector h\in U\), such that \(\norm{R(\vector h)}\to0\) as \(\vector h\to0\) and \[ F(\vector p+\vector h)=F(\vector p)+A(\vector h)+\norm{\vector h}\,R(\vector h). \] In little-\(o\) notation, \(F(\vector p+\vector h)=F(\vector p)+A(\vector h)+o(\norm{\vector h})\) as \(\vector h\to0\).