Question

NAME: Complete the following statements of the theorems relating the total derivative at a and partial derivatives near a. Theorem. Let f: ACR" R™ be differentiable at the interior point a € A. Then for all i el } and all je (D,S.) (a) Moreover, the matrir representation of the total derivative Df, is equal to the matrir with ith and jth equal to (D, f.) a) Theorem. For all i e{ let (D, f.) (s) satisfy _}and all je (1) AND AND Then

Answer 1

W Theorem; det f. ACRW, R be differentiable at the interion point a e A. Then for all it ? 1,2,...ms and for all je 21,2... og Dif; (a) exists Moreover, the matrix representation ofa is equal to the matrix with ith equal to Bit, (a). of the row total derivative and jth column Theorem: for all ie 2 1, 2,....mb and all je? 2.2,.. ng tet (D; f; ) (2) Satisfy (2) Dj f; (a) exists AND (2) D; f; (21) exists in a neighborhood of a AND (3) D; f; (2) are continuous at a Thette Dia excrete -is equal to f: ACIRM, RM is differentiable Then at a