Multivariable Analysis

Description

This book provides a rigorous treatment of multivariable differential and integral calculus. Inverse and implicit function theorems based on total derivatives are given and the connection with solving systems of equations is included. There is an extensive treatment of extrema, including constrained extrema and Lagrange multipliers, covering both first order necessary conditions and second order sufficient conditions. The material on Riemann integration in n dimensions, being delicate by its very nature, is discussed in detail. Differential forms and the general Stokes' Theorem are explained in the last chapter. With a focus on clarity rather than brevity, this text gives clear motivation, definitions and examples with transparent proofs. Some of the material included is difficult to find in most texts, for example, double sequences in Chapter 2, Schwarz’ Theorem in Chapter 3 and sufficient conditions for constrained extrema in Chapter 5. A wide selection of problems, ranging from simple to challenging, is included with carefully written solutions. Ideal as a classroom text or a self study resource for students, this book will appeal to higher level undergraduates in Mathematics. Front Matter....Pages i-ix Preliminaries....Pages 1-21 Functions Between Euclidean Spaces....Pages 23-75 Differentiation....Pages 77-115 Inverse and Implicit Function Theorems....Pages 117-150 Extrema....Pages 151-176 Riemann Integration in Euclidean Space....Pages 177-216 Transformation of Integrals....Pages 217-248 The General Stokes Theorem....Pages 249-301 Solutions....Pages 303-386 Back Matter....Pages 387-394