Page 2 - Vector Analysis

Contents

1 Linear Algebra 1

1.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The linear independence of vectors . . . . . . . . . . . . . . . . . . . 2

1.1.2 The dimension of a vector space . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Bases of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Inner Products and Inner Product Spaces . . . . . . . . . . . . . . . . . . . 3

1.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Elementary Row Operations and Elementary Matrices . . . . . . . . 13

1.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 Variations of determinants . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6 Bounded Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6.1 Matrix norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.7 Representation of Linear Transformations . . . . . . . . . . . . . . . . . . . 28

1.8 Matrix Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.9 The Einstein Summation Convention . . . . . . . . . . . . . . . . . . . . . . 30

2 Differentiation of Functions of Several Variables 32

2.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Definition of Derivatives and the Matrix Representation of Derivatives . . . 37

2.4 Conditions for Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Properties of Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . 47

2.5.1 Continuity of Differentiable Functions . . . . . . . . . . . . . . . . . 47

2.5.2 The Product Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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