Directional Derivative

$$\nabla_{\bf u}f \equiv\nabla f\cdot {{\bf u}\over\vert{\bf u... ... \propto\lim_{h\to 0} {f({\bf x}+h{\bf u})-f({\bf x})\over h}.$$ (1)

$\nabla_{\bf u} f(x_0,y_0,z_0)$ is the rate at which the function $w = f(x,y,z)$ changes at $(x_0,y_0,z_0)$ in the direction ${\bf u}$. Let ${\bf u}$ be a Unit Vector in Cartesian Coordinates, so

$$\vert{\bf u}\vert = \sqrt{{u_x}^2+{u_y}^2+{u_z}^2} = 1,$$ (2)

then

$$\nabla_{\bf u}f={\partial f\over\partial x}u_x+{\partial f\over\partial y}u_y+{\partial f\over\partial z}u_z.$$ (3)

The directional derivative is often written in the notation

$${d\over ds} \equiv \hat {\bf s}\cdot\nabla = s_x{\partial\o... ...+s_y{\partial\over\partial y} + s_z {\partial\over\partial z}.$$ (4)