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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 7 First and Second Order Conditions for Multivariate Objective Functions We now characterize first and second order conditions by using gradient vectors and Hessian matrices of objective functions. Outline A. Partial Derivatives, Gradient Vectors and Jacobian Matrices B. Second-Order Partial Derivatives and Hessian Matrices A. Partial Derivatives, Gradient Vectors and Jacobian Matrices Definition. Let f : < n < . When the limit lim h f ( x 1 ,x 2 ,...,x i + h,...,x n )- f ( x 1 ,x 2 ,...,x i ,...,x n ) h exists, it is called the partial derivative of f with respect to x i and is denoted by f i ( x ) , f x i ( x ) , f ( x ) x i . Definition. Suppose that f : X < is differentiable at x . Then the n- dimensional column vector f ( x ) x 1 f ( x ) x 2 . . . f ( x ) x n 1 is called the gradient of f and denoted by f ( x ). Chain Rule. Suppose X is an open set in < p , f : X < q is differentiable at x X , Y is an open set containing f ( X ), and g : Y < r is differentiable at y = f ( x ). Then h = g f is differentiable at x and Dh ( x ) = Dg ( f ( x )) Df ( x ) ....
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- Fall '07