Here's a decent reason to have the thing we call the directional derivative.
And not require the reference vector to be of unit magnitude.
Start with the case of $f:\mathbb{R}\to\mathbb{R}$. Write the directional
derivative with respect to some arbitrary constant $h\ne0$, and apply
first principles
$D_{h}f\left[a\right]=\lim_{t\to0}\frac{f[a+th]-f[a]}{t}$
$=h\lim_{t\to0}\frac{f[a+th]-f[a]}{th}$
$=hf^{\prime}\left[a\right]$.
Neither surprising nor particularly interesting. It's just another
way of saying what we already knew.
$x\left[t\right]=a+th$
$g\left[t\right]=f\left[x\left[t\right]\right]$
$g^{\prime}\left[0\right]=f^{\prime}\left[a\right]x^{\prime}=hf^{\prime}\left[a\right].$
But it gets a bit more interesting if we use it to build up a Taylor
series for a real-valued function of a real argument. I'll leave the
notation to speak for itself.
$D_{h}D_{h}f\left[a\right]=D_{h}^{2}f\left[a\right]=h^{2}f^{\prime\prime}\left[a\right]$.
$D_{h}^{r}f\left[a\right]=h^{r}f^{(r)}\left[a\right]$.
$D_{h}^{0}f\left[a\right]=f\left[a\right]$.
$f(a+h)=\sum_{r=0}^{k}\frac{D_{h}^{r}f\left[a\right]}{r!}+\frac{D_{h}^{k+1}f\left[\zeta\right]}{(k+1)!}$.
I think that's kind of pretty.
Now lets call $f:\mathbb{R}^{n}\to\mathbb{R}$. I use this non-standard
notation as a shorthand:
$\{\mathbb{J}\}=\left\{ \left\{ j_{i}\geq0\right\} _{n}|\sum_{i=1}^{n}j_{i}=k\right\} $.
Think of the $\mathbb{J}^{'s}$ as multi-indices. Each $\mathbb{J}$
is a set of non-negative integers $\{j_{1},\dots,j_{n}\}$ such that
$j_{1}+\dots+j_{n}=k$.
Now use those indecies to write the multinomial expansion of a sum
of n terms:
$\left(x_{1}+\dots+x_{n}\right)^{k}=\sum_{\mathbb{J}}\begin{pmatrix}k\\
j_{1}\dots j_{n}
\end{pmatrix}x_{1}^{j_{1}}\dots x_{n}^{j_{n}}$.
Here the coefficients are the multinomial generalization of the binomial
coefficients:
$\begin{pmatrix}k\\
j_{1}\dots j_{n}
\end{pmatrix}=\frac{k!}{j_{1}!\dots j_{n}!}=\begin{pmatrix}k\\
\mathbb{J}
\end{pmatrix}$.
Now write the Taylor series as before, but using $\mathfrak{h}\in\mathbb{R}^{n}$.
$f(\mathfrak{a}+\mathfrak{h})=\sum_{r=0}^{k}\frac{D_{\mathfrak{h}}^{r}f\left[\mathfrak{a}\right]}{r!}+\frac{D_{\mathfrak{h}}^{k+1}f\left[\vec{\zeta}\right]}{(k+1)!}$
$=P_{k}\left[\mathfrak{h}\right]+R_{k}\left[\mathfrak{h}\right].$
The directional derivative is:
$D_{\mathfrak{h}}f\left[\mathfrak{a}\right]=\mathfrak{h}\cdot\nabla\left[f\left[\mathfrak{a}\right]\right]=\left(h_{1}D_{1}+\dots+h_{n}D_{n}\right)\left[f\left[\mathfrak{a}\right]\right].$
So:
$D_{\mathfrak{h}}^{r}f\left[\mathfrak{a}\right]=\left(h_{1}D_{1}+\dots+h_{n}D_{n}\right)^{r}\left[f\left[\mathfrak{a}\right]\right].$
So the Taylor polynomial of degree $k$ is:
$P_{k}\left[\mathfrak{h}\right]=\sum_{r=0}^{k}\sum_{\mathbb{J}}\begin{pmatrix}r\\
j_{1}\dots j_{n}
\end{pmatrix}\left(h_{1}D_{1}\right)^{j_{1}}\dots\left(h_{n}D_{n}\right)^{j_{n}}\left[f\right]$.
The $\left[f\right]$ indicates that the beast to the left is an opperator
being applied to $f$.
