Directional Derivatives and Gradient

Abstract

Let a real-valued function f of two real variables of class C¹ be given. A vector, called the gradient of the function, whose components are the partial derivatives is defined. The directional derivative of f with respect to the oriented direction of the line r is the scalar product of the unit vector of r and the gradient. The gradient of f at a point P is a vector orthogonal to the tangent line to the level curve k = f(P). The above properties of gradient lead to a procedure of intuitive appeal, for finding relative extrema or critical points, known as the steepest descent algorithm.