Directional derivative

The directional derivative of a differentiable multivariable function $f$ in the direction of a vector $v$ is the instantaneous rate of change of the function while moving in the direction of $v$ . In physical terms, this can be thought of as the rate of change of the function while moving with velocity $v$ . In many cases, only the directional derivative with respect to space (not time) is desired, making velocity irrelevant; in such a case, we would use the unit vector $u$ . In two dimensions, the directional derivative is equal to the dot product of the gradient and directional vector.

$D_{\vec v}f(x,y)=\nabla f\cdot\vec v=\frac{\part f}{\part x}v_x+\frac{\part f}{\part y} v_y$

Generalized to $h$ dimensions, this becomes

$D_{\vec v}f(x_1,\ldots,x_n)=\nabla f\cdot\vec v=\frac{\part f}{\part x_1}v_{x1}+\cdots+ \frac{\part f}{\part x_n} v_{xn}$