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This is from Woll's "Functions of Several Variables," but there's no proof.

If $g$ is of class $C^k$ ($k \ge 2$) on a convex open set $U$ about $p$ in $\mathbb{R}^d$, then for each $q \in U$,

$ g(q) = g(p) + \sum_{i=1}^d \frac{\partial g}{\partial r_i} \bigg|_p (r_i(q) - r_i(p)) + \sum_{i,j} (r_i(q) - r_i(p)) (r_j(q) - r_j(p)) \int_0^1 (1-t) \frac{\partial^2g}{\partial r_i \partial r_j} \bigg|_{p + t(q - p)} dt. $

It looks like Taylor's or mean value theorem. I especially don't understand the integral part.

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Consider function $$ h(t)=g(p+t(q-p)) $$ and its Taylor series with the remainder in the integral form $$ h(1)=h(0)+h'(0)+\frac{1}{2}\int\limits_{0}^{1}(1-t)h''(t)dt $$ Now note that $$ h(0)=g(p) $$ $$ h'(0)=\sum\limits_{i=1}^d\frac{\partial g}{\partial r_i}\biggl|_p(r_i(p)-r_i(q)) $$ $$ h''(t)=\sum\limits_{i=1}^d\sum\limits_{j=1}^d\frac{\partial^2 g}{\partial r_i\partial r_j}\biggl|_{p+t(q-p)}(r_i(p)-r_i(q))(r_j(p)-r_j(q)) $$

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Thanks! $ $ $ $ –  Steven Li Jun 17 '12 at 4:56

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