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