14.2.3: Taylor's Theorem

Suppose $f:\mathcal{X}\rightarrow\mathbb{R}$ is $p+1$ times continuously differentiable in the open set $\mathcal{X}\subseteq\mathbb{R}^n$. Define, for all $0\leq s\leq p$, $$h_s(x,y)\mathop{=}\limits^{\Delta}\frac{1}{s!}\langle\mathcal{D}^sf(y),(x-y)^{s}\rangle,$$ as the inner product of the $s$-dimensional array of partial derivatives $\mathcal{D}^sf(y)$ and the $s$-dimensional outer power of $x-y.$ Both arrays are super-symmetric, and have dimension $n^s.$ By convention $h_0(x,y)=f(y)$.

Also define the Taylor Polynomials $$g_p(x,y)\mathop{=}\limits^{\Delta}\sum_{s=0}^ph_s(x,y)$$ and the remainder $$r_p(x,y)\mathop{=}\limits^{\Delta}f(x)-g_p(x,y).$$ Assume $\mathcal{X}$ contains the line segment with endpoints $x$ and $y$. Then Lagrange's form of the remainder says there is a $0\leq\lambda\leq 1$ such that $$r_p(x,y)=\frac{1}{(p+1)!}\langle\mathcal{D}^{p+1}f(x+\lambda(y-x)),(x-y)^{p+1}\rangle,$$ and the integral form of the remainder says $$r_p(x,y)=\frac{1}{p!}\int_0^1(1-\lambda)^p\langle\mathcal{D}^{p+1}f(x+\lambda(y-x)),(x-y)^p\rangle d\lambda.$$