I have a question about the monotonicity definition when it's applied to high-dimensional functions $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$.
In the answer of my other question: Monotonicity of Convex/Concave Function's Gradient Function in High Dimension, URL (version: 2020-11-13), user daw gives a definition with
$$ ( f(x) - f(y) ) ^T(x-y) \ge 0 $$ for all $x,y$.
Is this a general definition about monotonicity in $f:\mathbb{R}^d\rightarrow \mathbb{R}^d$? For example, suppose $g:\mathbb{R}^d\rightarrow \mathbb{R}$ is a concave function, let $f=\nabla g:\mathbb{R}^d\rightarrow \mathbb{R}^d$. I believe we have $$ (f(x)-f(y))^T(x-y)\leq 0 \,\, \,\, \big(\text{i.e. }( \nabla g(x) - \nabla g(y) ) ^T(x-y) \le 0\big) $$ for any concave function $g$ (can be proved with concavity definition $g(y)\leq g(x) + \nabla g(x)^T(y-x)$).
If so, then I am really confused. Does this mean the gradient of multi-variable concave function is not monotone? Or the monotonicity definition of $( f(x) - f(y) ) ^T(x-y) \ge 0$ on multivariable function $f:\mathbb{R}^d \rightarrow \mathbb{R}^d$ is not formal?