The rate of change of $f$ in the direction of unit vector $\vecu = \langle a, b \rangle$: $$D_\vecuf = f_x a + f_y b$$
The Calculus III course notes are structured into several major chapters that cover the transition from 2D to 3D mathematics:
Used to optimize $f(x,y)$ subject to constraint $g(x,y) = k$.
