An Extension to High Order for Directional Derivative of Multivariate Function

The directional derivative concept of the multivariate function is expanded from first order to high order in this paper. After getting the definition of second order directional derivative and the calculation formula,the high order directional derivative has been given. Applications of the high order directional derivative are proposed as following: 1) A general way to expanding simple variable function characteristics to multivariate function is presented. 2) The necessary conditions,necessary and sufficient conditions of extreme value criterion of function are easily obtained. 3) The geometrical meaning of semi-positive and semi-negative definite is explained according to second-order directional derivatives. 4) It is revealed that there is extreme value problem of the function when the matrix of the linear equations is positive or negative definite. 5) Taylor's expansion formula of the multivariate function is easily deduced.