Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$

Let $U\subseteq {\mathbb{R}}^m$, $a\in U$, $v\in {\mathbb{R}}^m$, and $f:U\to {\mathbb{R}}^n$. If the line map $t\mapsto f(a+tv)$ is defined near $0$, its derivative at $0$ is the directional derivative

$D_{v}f(a):={\lim }_{t\to 0}\frac{f(a+tv)-f(a)}{t}\mathrm{.}$

For a standard basis vector $e_j$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$), $D_{e_j}f(a)$ is the $j$th partial derivative, written ${\mathrm{\partial }}_{j}f(a)$. For $n\ge 1$, these are ordinary vector-valued one-variable derivatives in the sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral.

For $n=0$, use the zero-space case directly, without invoking that supplier's positive-codomain standing hypothesis. The space ${\mathbb{R}}^0$ has only its zero vector and carries the zero Euclidean norm and its induced metric (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$ clause 5, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). Whenever the line map is defined near $0$, every difference quotient for real $t\mathrm{\ne }0$ is this zero vector; define $D_{v}f(a)=0$. This is the unique possible derivative value, and its normwise difference-quotient error is exactly zero for all such $t$, so the real parameter still approaches an actual limit point. Every existing coordinate direction therefore has ${\mathrm{\partial }}_{j}f(a)=0$; these zero-valued partial maps are continuous in the induced norm metrics. If the source dimension is zero, there are no partial indices and the sole direction is $v=0$; its constant line map is defined near $0$.

The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms