An Introduction to Multivariable Mathematics

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Morgan & Claypool Publishers, 2008 - Mathematics - 132 pages
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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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Contents

Linear Algebra
1
2 Dot product and angle between vectors in Rᴺ
3
3 Subspaces and linear dependence of vectors
5
4 Gaussian Elimination and the Linear Dependence Lemma
7
5 The Basis Theorem
11
6 Matrices
13
7 Rank and the RankNullity Theorem
15
8 Orthogonal complements and orthogonal projection
18
2 Determinants
64
3 Inverse of a Square Matrix
69
4 Computing the Universe
72
5 Orthonormal Basis and GramSchmidt
73
6 Matrix Representations of Linear Transformations
75
7 Eigenvalues and the Spectral Theorem
76
More Analysis in Rᴺ
81
2 Inverse Function Theorem
82

9 Row Echelon Form of a Matrix
22
10 Inhomogeneous systems
27
Analysis in Rᴺ
31
2 BolzanoWeierstrass Limits and Continuity in Rᴺ
33
3 Differentiability
35
4 Directional Derivatives Partial Derivatives and Gradient
37
5 Chain Rule
41
6 Higherorder partial derivatives
42
7 Second derivative test for extrema of multivariable function
44
8 Curves in Rᴺ
48
9 Submanifolds of Rᴺ and tangential gradients
53
More Linear Algebra
61
3 Implicit Function Theorem
84
Introductory Lectures on Real Analysis
87
Sequences of Real Numbers and the BolzanoWeierstrass Theorem
91
Continuous Functions
96
Series of Real Numbers
100
Power Series
105
Taylor Series Representations
108
Complex Series Products of Series and Complex Exponential Series
113
Fourier Series
116
Pointwise Convergence of Trigonometric Fourier Series
121
Index
127
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