Lectures on Vector Calculus
                                Paul Renteln
                              Department of Physics
                             California State University
                             San Bernardino, CA 92407
                           March, 2009; Revised March, 2011
                c
                
Paul Renteln, 2009, 2011
                          ii
                         Contents
                         1 Vector Algebra and Index Notation                                                1
                             1.1   Orthonormality and the Kronecker Delta . . . . . . . . . . . .            1
                             1.2   Vector Components and Dummy Indices . . . . . . . . . . . .               4
                             1.3   Vector Algebra I: Dot Product . . . . . . . . . . . . . . . . . .         8
                             1.4   The Einstein Summation Convention . . . . . . . . . . . . . .           10
                             1.5   Dot Products and Lengths . . . . . . . . . . . . . . . . . . . .        11
                             1.6   Dot Products and Angles . . . . . . . . . . . . . . . . . . . . .       12
                             1.7   Angles, Rotations, and Matrices . . . . . . . . . . . . . . . . .       13
                             1.8   Vector Algebra II: Cross Products and the Levi Civita Symbol            18
                             1.9   Products of Epsilon Symbols . . . . . . . . . . . . . . . . . . .       23
                             1.10 Determinants and Epsilon Symbols . . . . . . . . . . . . . . .           27
                             1.11 Vector Algebra III: Tensor Product . . . . . . . . . . . . . . .         28
                             1.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     31
                         2 Vector Calculus I                                                               32
                             2.1   Fields   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  33
                             2.2   The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . .      34
                             2.3   Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . .      37
                             2.4   The Divergence . . . . . . . . . . . . . . . . . . . . . . . . . .      41
                             2.5   The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . .     41
                             2.6   The Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    42
                             2.7   Vector Calculus with Indices . . . . . . . . . . . . . . . . . . .      43
                             2.8   Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    47
                         3 Vector Calculus II: Other Coordinate Systems                                    48
                             3.1   Change of Variables from Cartesian to Spherical Polar          . . . .  48
                                                                  iii
                             3.2   Vector Fields and Derivations . . . . . . . . . . . . . . . . . .       49
                             3.3   Derivatives of Unit Vectors . . . . . . . . . . . . . . . . . . . .     53
                             3.4   Vector Components in a Non-Cartesian Basis           . . . . . . . . .  54
                             3.5   Vector Operators in Spherical Coordinates . . . . . . . . . . .         54
                             3.6   Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    57
                         4 Vector Calculus III: Integration                                                57
                             4.1   Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .    57
                             4.2   Surface Integrals    . . . . . . . . . . . . . . . . . . . . . . . . .  64
                             4.3   Volume Integrals . . . . . . . . . . . . . . . . . . . . . . . . .      67
                             4.4   Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    69
                         5 Integral Theorems                                                               70
                             5.1   Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . .       71
                             5.2   Stokes’ Theorem      . . . . . . . . . . . . . . . . . . . . . . . . .  73
                             5.3   Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .      74
                             5.4   The Generalized Stokes’ Theorem . . . . . . . . . . . . . . . .         74
                             5.5   Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    75
                         A Permutations                                                                    76
                         B Determinants                                                                    77
                             B.1 The Determinant as a Multilinear Map . . . . . . . . . . . . .            79
                             B.2 Cofactors and the Adjugate . . . . . . . . . . . . . . . . . . .          82
                             B.3 The Determinant as Multiplicative Homomorphism . . . . . . 86
                             B.4 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . .       89
                                                                  iv