Consider a function $f:[a,b]^2\rightarrow[a,b]$.
What does the expression $\nabla f\cdot (t,s)$ mean? I read that it indicates "the rate of change of $f$ in the direction of $(s,t)$". I would appreciate if someone could elaborate this.
Moreover, suppose that $\nabla f\cdot (t,s) \le 0$. What is the intuitive meaning of this inequality?
Dot product of the gradient with a vector is not that clear to me, so any clarifications would be great.