Basic Analysis II A Modern Calculus in Many Variables
Basic Analysis II: A Modern Calculus in Many Variables focuses on differentiation in Rn and important concepts about mappings from Rn to Rm, such as the inverse and implicit function theorem and change of variable formulae for multidimensional integration. These topics converge nicely with many other important applied and theoretical areas which are no longer covered in mathematical science curricula. Although it follows on from the preceding volume, this is a self-contained book, accessible to undergraduates with a minimal grounding in analysis.
- Can be used as a traditional textbook as well as for self-study
- Suitable for undergraduates in mathematics and associated disciplines
- Emphasises learning how to understand the consequences of assumptions using a variety of tools to provide the proofs of propositions
1. Beginning Remarks 2.Preliminaries 3.Vector Spaces 4.Linear Transformations 5.Symmetric Matrices 6.Continuity and Topology 7.Abstract Symmetric Matrices 8.Rotations and Orbital Mechanics 9.Determinants and Matrix Manipulations 10.Differentiability 11.Multivariable Extremal Theory 12.The Inverse and Implicit Function Theorems 13.Linear Approximation Applications 14.Integration in Multiple Dimensions 15.Change of Variables and Fubini’s Theorem 16.Line Integrals 17.Differential Forms 18.The Exponential Matrix 19.Nonlinear Parametric Optimization Theory 20.Summing It All Up. References. Index
"Mathematics is fortunate to be populated by bright practitioners. Nonetheless, amongst these we are fortunate to have rare individuals who are wise. Professor Peterson is a member of this distinguished group. His works clearly demonstrate the importance of a long career of research and teaching where he combines the two perspectives of: (1) clearly understanding the needs of diverse readers for clear exposition that scaffolds their exposure to complex material with a transparency about both where they are going and what the utility is of what they are currently reading; and, (2) the benefits of having used the mathematics under consideration in so many diverse applications. The masterly synthesis of so much complex material by a single individual is a superb achievement which will reward serious readers with insight, surprise, and breadth as well as depth."
– Professor John R. Jungck, University of Delaware
"Analysis is the bedrock of rigorous mathematical thinking and abstraction. Prof. Peterson's book does a fascinating job by taking a critical approach - highly recommended."
– Professor Nithin Nagaraj, National Institute of Advanced Studies
"Dr. Peterson's thoughtful and detailed explanations reflect his insights to a very fundamental but complex subject in Mathematics. The treatment in the book does justice to recent trends in Mathematical Analysis while staying true to the classical spirit of the subject. A thoroughly enjoyable read."
– Professor Snehanshu Saha, BITS PIlani (K K Birla Goa Campus)