Multivariable Advanced Calculus

6.7. C K FUNCTIONS 131I propose to iterate this observation, starting with f and then going to Df and thenD 2 f and so forth. Hopefully it will yield a rational way to understand higher orderderivatives in the same way that matrices can be used to understand linear transformations.Thus beginning with the derivative,Df (x) = ∑ ij 1D vj1 f i (x) w i v j1 .Then letting w i v j1 play the role of w i in 6.8,D 2 f (x) = ∑ ij 1 j 2D vj2(Dvj1 f i)(x) wi v j1 v j2≡∑ ij 1j 2D vj1 v j2f i (x) w i v j1 v j2Then letting w i v j1 v j2 play the role of w i in 6.8,D 3 f (x) = ∑≡etc. In general, the notation,( )D vj3 Dvj1 v j2f i (x) wi v j1 v j2 v j3ij ∑1j 2j 3ij 1j 2j 3D vj1 v j2 v j3f i (x) w i v j1 v j2 v j3w i v j1 v j2 · · · v jndefines an appropriate linear transformation given byw i v j1 v j2 · · · v jn (v k ) = w i v j1 v j2 · · · v jn−1 δ kjn .The following theorem is important.Theorem 6.7.1 The function x → D k f (x) exists and is continuous for k ≤ pif and only if there exists a basis for X, {v 1 , · · · , v n } and a basis for Y, {w 1 , · · · , w m }such that forf (x) ≡ ∑ f i (x) w i ,iit follows that for each i = 1, 2, · · · , m all Gateaux derivatives,for any choice of v j1 v j2 · · · v jkD vj1 v j2 ···v jkf i (x)and for any k ≤ p exist and are continuous.Proof: This follows from a repeated application of Theorems 6.5.1 and 6.5.4 at eachnew differentiation. Definition 6.7.2 Let X, Y be finite dimensional normed vector spaces and letU be an open set in X and f : U → Y be a function,f (x) = ∑ if i (x) w iwhere {w 1 , · · · , w m } is a basis for Y. Then f is said to be a C n (U) function if for everyk ≤ n, D k f (x) exists for all x ∈ U and is continuous. This is equivalent to the othercondition which states that for each i = 1, 2, · · · , m all Gateaux derivatives,D vj1 v j2 ···v jkf i (x)for any choice of v j1 v j2 · · · v jkexist and are continuous.where {v 1 , · · · , v n } is a basis for X and for any k ≤ n