偏微分
{\displaystyle \operatorname {H} _{f}=\left({\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}\right)={\begin{pmatrix}{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{1}}}&\dots &{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}&\dots &{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{n}}}\end{pmatrix}}.}
高次元版のテイラーの公式: k-回連続的微分可能函数 f: U → R は点 a = (a1, ..., an) ∈ U の近傍でテイラー多項式を用いて f ( a + h ) = ∑ s = 0 k ∑ j 1 + ⋯ + j n = s 1 j 1 ! ⋯ j n !