## An Introduction to Multivariable MathematicsThe 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

1 | |

3 | |

5 | |

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 |

127 | |

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### Common terms and phrases

1-variable absolutely convergent assume augmented matrix Bolzano-Weierstrass Theorem Cauchy-Schwarz inequality chain rule column space continuous deﬁned deﬁnition denoted detA directional derivative elementary row operations equations Exercise exists f is differentiable f(a+ fact ﬁnd ﬁrst Fourier series function f Function Theorem Gaussian elimination given hence Hint identity induction inequality integer j-th column l.i. vectors Lecture Linear Dependence Lemma local maximum m x n matrix maximum minimum n x n nonempty nontrivial subspace nonzero notation Notice null space Observe open sets orthogonal projection orthonormal basis partial derivatives partial sums particular permutation positive integers power series properties Py(x radius of convergence real numbers row echelon form rrefA says sequence solution span span{v1 submanifold subset subspace Suppose terminology transpositions trivial vectors in Rn zero