1978/03/01 by L. E. Fraenkel · 1 citation
Mathematics · #Algebraic and Geometric Analysis #Mathematical and Theoretical Analysis #Fixed Point Theorems Analysis #Mathematics #Scalar (mathematics) #Euclidean space #Banach space #Order (exchange) #Function (biology) #Euclidean geometry #Space (punctuation) #Pure mathematics #Function space #Combinatorics #Composite number #Mathematical analysis #Geometry #Computer science #Algorithm
paper · doi:10.1017/s0305004100054402
openalex publication_date 1978/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/20
This note concerns a question of elementary calculus. Given a smooth composite function u = g o f [with values u ( x ) = g ( f ( x ))], we write explicit formulae for its derivatives, of arbitrary order, in terms of derivatives of f and g . We consider (A) the general case, in which E, F and G are Banach spaces, and U, V are open sets; (B) the finite-dimensional case E = ℝ M and F = ℝ N , where ℝ M denotes real M -dimensional Euclidean space; and (C) the particular case of (B) (due to restricting g to part of an M -dimensional surface in ℝ M + 1 ) in which N = M + 1 and u ( x )= g ( x , φ ( x )), so that φ denotes a real-valued (scalar-valued) function of x = ( x 1 , …, x M ).