Translations:The directional derivative/14/en

where $\alpha$ is the angle between the gradient and the directional vector. The maximum value of the directional derivative is obtained when the direction vector points in the same direction as the gradient, that is, when $\alpha {\rm {=0}}$. This maximum value is equal to the magnitude of the gradient. In other words, the gradient gives the direction of maximum slope. Suppose the directional vector points along a contour line. Because the contour line is level, the directional derivative in the direction of a contour line must be zero. Thus, ${\rm {\ cos\ }}\alpha$ must be zero, so $\alpha$ is ${\rm {9}}0^{\rm {o}}$. Therefore, the gradient vector is orthogonal (or perpendicular) to the contour line.