Multivariable and vector calculus

Preface......Page 4 Points and Vectors on the Plane......Page 5 Scalar Product on the Plane......Page 12 Linear Independence......Page 16 Geometric Transformations in two dimensions......Page 18 Determinants in two dimensions......Page 24 Parametric Curves on the Plane......Page 28 Vectors in Space......Page 35 Cross Product......Page 43 Matrices in three dimensions......Page 48 Determinants in three dimensions......Page 51 Some Solid Geometry......Page 54 Cavalieri, and the Pappus-Guldin Rules......Page 56 Dihedral Angles and Platonic Solids......Page 58 Spherical Trigonometry......Page 61 Canonical Surfaces......Page 65 Parametric Curves in Space......Page 70 Multidimensional Vectors......Page 73 Some Topology......Page 79 Multivariable Functions......Page 80 Limits......Page 81 Definition of the Derivative......Page 86 The Jacobi Matrix......Page 88 Gradients and Directional Derivatives......Page 95 Levi-Civitta and Einstein......Page 99 Extrema......Page 102 Lagrange Multipliers......Page 105 Differential Forms......Page 109 Zero-Manifolds......Page 111 One-Manifolds......Page 112 Closed and Exact Forms......Page 115 Two-Manifolds......Page 119 Change of Variables......Page 126 Change to Polar Coordinates......Page 132 Three-Manifolds......Page 136 Change of Variables......Page 140 Surface Integrals......Page 143 Green's, Stokes', and Gauss' Theorems......Page 146 Answers and Hints......Page 152 4. MODIFICATIONS......Page 202 10. FUTURE REVISIONS OF THIS LICENSE......Page 203