Multivariable Advanced Calculus

124 THE DERIVATIVEIt follows≡limt→0 π j( )f (x + tvk ) − f (x)tf j (x + tv k ) − f j (x)lim≡ D vk f j (x)t→0 t= π j( ∑iJ ik (x) w i)= J jk (x)Thus J ik (x) = D vk f i (x).In the case where X = R n and Y = R m and v is a unit vector, D v f i (x) is thefamiliar directional derivative in the direction v of the function, f i .Of course the case where X = F n and f : U ⊆ F n → F m , is differentiable and thebasis vectors are the usual basis vectors is the case most commonly encountered. Whatis the matrix of Df (x) taken with respect to the usual basis vectors? Let e i denote thevector of F n which has a one in the i th entry and zeroes elsewhere. This is the standardbasis for F n . Denote by J ij (x) the matrix with respect to these basis vectors. ThusDf (x) = ∑ ijJ ij (x) e i e j .Then from what was just shown,J ik (x) =f i (x + te k ) − f i (x)D ek f i (x) ≡ limt→0 t≡∂f i(x) ≡ f i,xk (x) ≡ f i,k (x)∂x kwhere the last several symbols are just the usual notations for the partial derivative ofthe function, f i with respect to the k th variable wherem∑f (x) ≡ f i (x) e i .i=1In other words, the matrix of Df (x) is nothing more than the matrix of partial derivatives.The k th column of the matrix (J ij ) is∂ff (x + te k ) − f (x)(x) = lim≡ D ek f (x) .∂x k t→0 tThus the matrix of Df (x) with respect to the usual basis vectors is the matrix ofthe form⎛⎞f 1,x1 (x) f 1,x2 (x) · · · f 1,xn (x)⎜⎟⎝ . .. ⎠f m,x1 (x) f m,x2 (x) · · · f m,xn (x)where the notation g ,xk denotes the k th partial derivative given by the limit,g (x + te k ) − g (x)lim≡ ∂g .t→0 t∂x kThe above discussion is summarized in the following theorem.Theorem 6.3.1 Let f : F n → F m and suppose f is differentiable at x. Then allthe partial derivatives ∂f i(x)∂x jexist and if Jf (x) is the matrix of the linear transformation,Df (x) with respect to the standard basis vectors, then the ij th entry is given by ∂fi∂x j(x)also denoted as f i,j or f i,xj .