An introduction to multivariable mathematics

Leon Simon1598298011, 9781598298017

The text is designed for use in a forty-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 ten or eleven 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 nine-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 four-lecture per week program in the Autumn Quarter each year, the first six lectures of the nine-lecture appendix are presented at the rate of one lecture per week in weeks two through seven of the quarter, with the remaining three 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 quarter or 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. Table of Contents: Linear Algebra / Analysis in R / More Linear Algebra / More Analysis in R / Appendix: Introductory Lectures on Real Analysis

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Table of contents :
Preface……Page 6
Vectors in Rn……Page 12
Dot product and angle between vectors in Rn……Page 13
Subspaces and linear dependence of vectors……Page 16
Gaussian Elimination and the Linear Dependence Lemma……Page 18
The Basis Theorem……Page 22
Matrices……Page 23
Rank and the Rank-Nullity Theorem……Page 26
Orthogonal complements and orthogonal projection……Page 29
Row Echelon Form of a Matrix……Page 33
Inhomogeneous systems……Page 38
Analysis in Rn……Page 41
Open and closed sets in Euclidean Space……Page 42
Bolzano-Weierstrass, Limits and Continuity in Rn……Page 44
Differentiability……Page 46
Directional Derivatives, Partial Derivatives, and Gradient……Page 48
Chain Rule……Page 52
Higher-order partial derivatives……Page 53
Second derivative test for extrema of multivariable function……Page 55
Curves in Rn……Page 59
Submanifolds of Rn and tangential gradients……Page 64
More Linear Algebra……Page 71
Permutations……Page 72
Determinants……Page 75
Inverse of a Square Matrix……Page 80
Computing the Inverse……Page 83
Orthonormal Basis and Gram-Schmidt……Page 84
Matrix Representations of Linear Transformations……Page 86
Eigenvalues and the Spectral Theorem……Page 87
More Analysis in Rn……Page 91
Contraction Mapping Principle……Page 92
Inverse Function Theorem……Page 93
Implicit Function Theorem……Page 95
Lecture 1: The Real Numbers……Page 98
Lecture 2: Sequences of Real Numbers and the Bolzano-Weierstrass Theorem……Page 102
Lecture 3: Continuous Functions……Page 107
Lecture 4: Series of Real Numbers……Page 111
Lecture 5: Power Series……Page 116
Lecture 6: Taylor Series Representations……Page 119
Lecture 7: Complex Series, Products of Series, and Complex Exponential Series……Page 124
Lecture 8: Fourier Series……Page 127
Lecture 9: Pointwise Convergence of Trigonometric Fourier Series……Page 132
Index……Page 136