Multivariable Calculus: Notation

(i) Base. This setting extends Vector- and Matrix-Valued Functions on a Compact Interval: Standing Conventions and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: their conventions and notation are in force.

(ii) Domain, components and balls. Unless stated otherwise, $n$ and $m$ denote natural numbers and $U$ denotes an open subset of $\mathbb{R}^{n}$, open for $d_{E}$ as in Euclidean Space and Real Matrices: Standing Notation (ii). For a function $f:U\to\mathbb{R}^{m}$ and $k\in[m]$, its $k$-th component map is $f_{k}:U\to\mathbb{R}$, $f_{k}(x)=f(x)_{k}$, as in Component Maps and Entry Maps (i). $e_{1},\dots,e_{n}$ are the standard basis vectors of $\mathbb{R}^{n}$. For $a\in\mathbb{R}^{n}$ and a real number $r>0$, $B(a,r)=B_{d_{E}}(a,r)$ is the open ball in $(\mathbb{R}^{n},d_{E})$ with centre $a$ and radius $r$.

(iii) Continuity. Continuity of a function defined on a subset of $\mathbb{R}^{n}$ with values in $\mathbb{R}$, in $\mathbb{R}^{m}$ or in $\mathcal{M}_{p\times q}(\mathbb{R})$ ($p,q\in\mathbb{N}$) is understood for $d_{E}$ on the domain and for the metrics of Vector- and Matrix-Valued Functions on a Compact Interval: Standing Conventions (iii) on the values.