I am preparing a note, and I am looking for a nice example of a monotone nonlinear operator $f:\mathbb{R}^2 \to \mathbb{R}^2$, i.e., for all $a,b\in\mathbb{R}^2$ $$(f(a)-f(b))^\top (a-b) \ge 0.$$ However, I failed to find any simple illustrative example; all textbook examples are either linear or rather weird and not really pedagogical. Could you please help with that?
UPD: To make the example more pedagogical, I am looking for $$f(x) = \begin{bmatrix}f_1(x_1,x_2)\\f_2(x_1,x_2)\end{bmatrix}$$ rather than $$f(x) = \begin{bmatrix}f_1(x_1)\\f_2(x_2)\end{bmatrix}.$$
You can use
$$f(x,y) = \big(1+\ln(x/y), -x/y \big)$$ for $x>0$ and $y>0$.
Note: This is the gradient of the Kullback-Leibler divergence $x\ln(x/y)$ which is convex in $(x,y)$ so its gradient is monotone.
If you really need full domain, you can use $$f(x,y) = (x^3-y,y^3+x)$$ which was obtained by adding the rotator by $\pi/2$ to the gradient of the convex function $\frac{1}{4}(x^4+y^4)$.