Physicist here not a mathematician. I am trying to understand the notation for the Finsler metric in Chern and Shen's book.
The equation is
$$\textbf{g}_y(u,v):=\frac{1}{2} \frac{\partial^2}{\partial s \partial t} \left[ F^2( y + su + tv) \right]_{s=t=0} ~.$$
Normally I would have written something like
$$ \textbf{g}(u,v)= \frac{1}{2}\frac{ \partial^2 F^2}{\partial u\partial v} u v~. $$
If I make substitutions $a=su$ and $b=tv$, evaluation of the top equation gives me
$$ \frac{1}{2}\frac{ \partial^2 F^2(y)}{\partial a\partial b} uv ~.$$
Which is not the middle equation. Thanks for the help.
The logic of notation in the book is that $u,v $ are vectors and $s,t$ are scalars. So the composition $F^2(y+su+tv)$ is a function of two scalar variables $s,t$. We take second mixed partial of that function.
What you wrote in the second equation may be familiar to you, but I do not recognize "partial derivative with respect to a vector" as a precise mathematical concept.