Functions of Class Cᵏ: Notation

(i) Base. This setting extends Higher Derivatives on Open Subsets of ℝⁿ: Notation: its conventions and notation are in force.

(ii) Classes $C^{k}$ and smooth functions. For $k\in\mathbb{N}_{0}$, $\mathcal{C}^{k}(U)$ is the set of Functions of Class Cᵏ on an Open Subset of ℝⁿ (i); a function $f:U\to\mathbb{R}$ or $f:U\to\mathbb{R}^{m}$ is of class $C^{k}$ on $U$ as in Functions of Class Cᵏ on an Open Subset of ℝⁿ (ii) and Functions of Class Cᵏ on an Open Subset of ℝⁿ (iii), and smooth on $U$ as in Smooth Functions on an Open Subset of ℝⁿ (i) and Smooth Functions on an Open Subset of ℝⁿ (ii). The same applies on an open subset $V$ of $\mathbb{R}^{m}$, as in Higher Derivatives on Open Subsets of ℝⁿ: Notation (iii).

(iii) Second partial derivatives. Let $f:U\to\mathbb{R}$ and $i,j\in[n]$ be such that $\partial_{j}f(x)$ exists for every $x\in U$ and the function $\partial_{j}f$ has an $i$-th partial derivative at every point of $U$. Then $\partial_{i}\partial_{j}f$ denotes the partial derivative function

$$\partial_{i}\partial_{j}f=\partial_{i}(\partial_{j}f):U\to\mathbb{R},$$

also written $\dfrac{\partial^{2}f}{\partial x_{i}\,\partial x_{j}}$.

(iv) Local extrema. For $f:U\to\mathbb{R}$ and $a\in U$, a local maximum, strict local maximum, local minimum and strict local minimum of $f$ at $a$ are those of Local Maximum of a Function Relative to a Subset of a Metric Space (i), Local Maximum of a Function Relative to a Subset of a Metric Space (ii), Local Minimum of a Function Relative to a Subset of a Metric Space (i) and Local Minimum of a Function Relative to a Subset of a Metric Space (ii), relative to $U$ in $(\mathbb{R}^{n},d_{E})$.

(v) Pointwise operations. Sums, scalar multiples and products of functions on $U$ are the pointwise ones of Pointwise Operations on Real-Valued Maps (i), formed in the same way for $\mathbb{R}^{m}$-valued functions. For $f,g:U\to\mathbb{R}$ with $g(x)\neq0$ for every $x\in U$, $1/g$ and $f/g$ are the functions $x\mapsto1/g(x)$ and $x\mapsto f(x)/g(x)$ on $U$. The same applies on an open subset $V$ of $\mathbb{R}^{m}$.

(vi) Split coordinates. For $n,m\in\mathbb{N}$, the point $\iota(x,y)$ of $\mathbb{R}^{n+m}$, where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$ and $\iota$ is the concatenation bijection of Euclidean Space and Real Matrices: Standing Notation (viii), is written $(x,y)$; for $A\subseteq\mathbb{R}^{n}$ and $B\subseteq\mathbb{R}^{m}$, $A\times B$ denotes the set of the points $(x,y)$ with $x\in A$ and $y\in B$. Let $O$ be an open subset of $\mathbb{R}^{n+m}$, with functions on $O$ treated as in Higher Derivatives on Open Subsets of ℝⁿ: Notation (iii), and let $(x,y)\in O$. For $F:O\to\mathbb{R}^{m}$ whose Jacobian matrix $DF(x,y)\in\mathcal{M}_{m\times(n+m)}(\mathbb{R})$ is defined, $D_{x}F(x,y)\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $D_{y}F(x,y)\in\mathcal{M}_{m}(\mathbb{R})$ are the blocks of its first $n$ and its last $m$ columns:

$$\bigl(D_{x}F(x,y)\bigr)_{ij}=\bigl(DF(x,y)\bigr)_{ij},\qquad\bigl(D_{y}F(x,y)\bigr)_{il}=\bigl(DF(x,y)\bigr)_{i,\,n+l}\qquad(i,l\in[m],\ j\in[n]).$$

Likewise, for $F:O\to\mathbb{R}$ whose gradient $\nabla F(x,y)$ is defined, $\nabla_{x}F(x,y)\in\mathbb{R}^{n}$ and $\nabla_{y}F(x,y)\in\mathbb{R}^{m}$ consist of its first $n$ and its last $m$ components: $\bigl(\nabla_{x}F(x,y)\bigr)_{j}=\partial_{j}F(x,y)$ and $\bigl(\nabla_{y}F(x,y)\bigr)_{l}=\partial_{n+l}F(x,y)$ for $j\in[n]$ and $l\in[m]$.