An introduction to multivariable mathematics

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