Higher Order Partial Derivatives of Functions from Rn to Rm

# 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)
\begin{align} \quad f(x, y) = e^x (\cos y + \sin y) \end{align}

Then the first order partial derivatives of $f$ are:

(2)
\begin{align} \quad D_1 f(x, y) = e^x(\cos y + \sin y) \quad \mathrm{and} \quad D_2f(x, y) = e^x(-\sin y + \cos y) \end{align}

The second order partial derivatives of $f$ are:

(3)
\begin{align} \quad D_{1, 1} f(x, y) & = e^x (\cos y + \sin y) \quad D_{2, 1} f(x, y) & = e^x(-\sin y + \cos y) \\ \quad D_{1, 2} f(x, y) & = e^x(-\sin y + \cos y) \quad D_{2, 2} f(x, y) & = -e^x (\cos y + \sin y) \end{align}

For a more complicated example, consider the function $f : \mathbb{R}^3 \to \mathbb{R}$ defined by:

(4)
\begin{align} \quad f(x, y, z) = xy^2z^3 \end{align}

Then the first order partial derivatives of $f$ are:

(5)
And the second order partial derivatives of $f$ are
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.