Multivariable Analysis

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 foundation of linear systems theory goes back to Newton and has been followed over the years by many improvements such as linear operator theory, Laplace Transformation etc. After the World War II, feedback control theory has shown a rapid development, and standard elegant analysis and synthesis techniques have been discovered by control system workers, such as root-locus (Evans) and frequency response methods (Nyquist, Bode). These permitted a fast and efficient analysis of simple-loop control systems, but in their original "paper-and-pencil" form were not appropriate for multiple­ loop high-order systems. The advent of fast digital computers, together with the development of multivariable multi-loop system techniques, have eliminated these difficulties. Multivariable control theory has followed two main avenues; the optimal control approach, and the algebraic and frequency-domain control approach. An important key concept in the whole multivariable system theory is "ob­ servability and controllability" which revealed the exact relationships between transfer functions and the state variable representations.

This self-contained textbook gives a thorough exposition of multivariable calculus. It can be viewed as a sequel to the one-variable calculus text, A Course in Calculus and Real Analysis, published in the same series.

This text is appropriate for a one-semester course in what is usually called ad­ vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func­ tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop­ erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de­ fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat­ ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.

This study guide is intended to aid your understanding of multivariable calculus. We have organized it into chapters and sections corresponding to Basic Multivariable Calculus by Jerrold E. Marsden, Anthony ]. Tromba, and Alan Weinstein (Springer-Verlag/W.H. Freeman, 1993). Each section contains Goals, Study Hints, and (most important) Solutions to Every Other Odd Exercise. In addition, we have written chapter tests at the end of each review section.