Let $\;f \in C^3(\mathbb R^m;\mathbb R)\;$ and $\;g,h:\mathbb R^n \to \mathbb R^m\;$ be two smooth functions.
If $\;I(h)=f(g+h)-f(g)-Df(g)\cdot h-\frac{1}{2}D^2(f(g))h \cdot h\;$ then it follows $\;I(h)=\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} p^2 l D^3(f(g+klph))(h,h,h)\;dp\;dl\;dk\;$
NOTE: $\;h \cdot h\;$ is the $\;L^2-$inner product.
However I 'm having a really hard time understanding what exactly $\;(h,h,h)\;$ represents here
Any help here would be valuable! Thanks in advance
1) Actually for $D^{3}$, it is a map that $D^{3}:{\bf{R}}^{n}\rightarrow L({\bf{R}}^{n},L({\bf{R}}^{n},L({{\bf{R}}^{n}},{\bf{R}}^{m})))\cong L({\bf{R}}^{n}\times{\bf{R}}^{n}\times{\bf{R}}^{n},{\bf{R}}^{m})$, so we can input three $h\in{\bf{R}}^{n}$ to $D^{3}(a)$, that is, $D^{3}(a)(h,h,h)$ makes sense.