Higher Order Partial Derivatives of Functions from Rn to Rm
Definition: Let $S \subseteq \mathbb{R}^n$ be open, $\mathbb{c} \in S$, and $\mathbf{f} : S \to \mathbb{R}^m$. Let $D_1 \mathbf{f} (\mathbf{c})$, $D_2 \mathbf{f} (\mathbf{c})$, …, $D_n \mathbf{f} (\mathbf{c})$ be the partial derivatives of $\mathbf{f}$ at $\mathbf{c}$. The Second Order Partial Derivatives of $\mathbf{f}$ at $\mathbf{c}$ are the partial derivatives of $D_1 \mathbf{f}$, $D_2 \mathbf{f}$, …, $D_n \mathbf{f}$, at $\mathbf{c}$. |
For example, consider the partial derivative of $\mathbf{f}$ with respect to the $k^{\mathrm{th}}$ variable at $\mathbf{c}$, $D_k \mathbf{f} (\mathbf{c})$. Then $D_k \mathbf{f} : S \to \mathbb{R}^n$ and we can take partial derivatives of $D_k \mathbf{f}$ (provided that they exist). For each $r \in \{ 1, 2, ..., n \}$, we have $D_r (D_k \mathbf{f} (\mathbf{c}))$ is a second order partial derivative of $\mathbf{f}$ at $\mathbf{c}$. Other notations include "$D_{r, k} \mathbf{f}$" and "$\displaystyle{\frac{\partial^2 \mathbf{f}}{\partial x_r \partial x_k} = \frac{\partial}{\partial x_r} \left ( \frac{\partial \mathbf{f}}{\partial x_k} \right )}$".
For example, consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined for all $(x, y) \in \mathbb{R}^2$ by:
(1)Then the first order partial derivatives of $f$ are:
(2)The second order partial derivatives of $f$ are:
(3)For a more complicated example, consider the function $f : \mathbb{R}^3 \to \mathbb{R}$ defined by:
(4)Then the first order partial derivatives of $f$ are:
(5)And the second order partial derivatives of $f$ are
(6)We can even define higher order partial derivatives of functions from $\mathbb{R}^n$ to $\mathbb{R}^m$.
Definition: Let $S \subseteq \mathbb{R}^n$ be open, $\mathbb{c} \in S$, and $\mathbf{f} : S \to \mathbb{R}^m$. Let $\{ D_k \mathbf{f} (\mathbf{c}) : k \in \{1, 2, ..., n \} \}$ be the set of all first order partial derivatives of $\mathbf{f}$ at $\mathbf{c}$ and let $\{ D_{r, k} \mathbf{f} (\mathbf{c}) : r, k \in \{ 1, 2, ..., n \} \}$ be the set of all second order partial derivatives of $\mathbf{f}$ at $\mathbf{c}$. Then the set of all Third Order Partial Derivatives of $\mathbf{f}$ at $\mathbf{c}$ is $\{ D_{p, r, k} \mathbf{f} (\mathbf{c}) : p, r, k \in \{ 1, 2, ..., n \} \}$ provided that they exist. Higher order partial derivatives of $\mathbf{f}$ at $\mathbf{c}$ are defined analogously. |