An Introduction to Multivariable Mathematics

The text is designed for use in a 40 lecture introductory course covering linear algebra, multivariable differential calculus, and an introduction to real analysis. The core material of the book is arranged to allow for the main introductory material on linear algebra, including basic vector space theory in Euclidean space and the initial theory of matrices and linear systems, to be covered in the first 10 or 11 lectures, followed by a similar number of lectures on basic multivariable analysis, including first theorems on differentiable functions on domains in Euclidean space and a brief introduction to submanifolds. The book then concludes with further essential linear algebra, including the theory of determinants, eigenvalues, and the spectral theorem for real symmetric matrices, and further multivariable analysis, including the contraction mapping principle and the inverse and implicit function theorems. There is also an appendix which provides a 9 lecture introduction to real analysis. There are various ways in which the additional material in the appendix could be integrated into a course—for example in the Stanford Mathematics honors program, run as a 4 lecture per week program in the Autumn Quarter each year, the first 6 lectures of the 9 lecture appendix are presented at the rate of one lecture a week in weeks 2–7 of the quarter, with the remaining 3 lectures per week during those weeks being devoted to the main chapters of the text. It is hoped that the text would be suitable for a 1 quarter or 1 semester course for students who have scored well in the BC Calculus advanced placement examination (or equivalent), particularly those who are considering a possible major in mathematics. The author has attempted to make the presentation rigorous and complete, with the clarity and simplicity needed to make it accessible to an appropriately large group of students.