Differential Forms: A Heuristic Introduction

1 Partial Differentiation.- 1.1 Partial Derivatives.- 1.2 Differentiability, Chain Rule.- 1.3 Taylor's Theorem.- 2 Differential Forms.- 2.1 Line Integrals.- 2.2 One Forms.- 2.3 Wedge Product.- 2.4 Change of Coordinates.- 3 Integration in Higher Dimensions.- 3.1 Jacobians.- 3.2 Implicit Function Theorem.- 3.3 Manifolds.- 3.4 Integration on Manifolds.- 4 Exterior Differentiation.- 4.2 Exterior Derivative.- 4.2 Fundamental Theorem of Calculus.- 4.3 Closed Forms.- 4.4 Exact Forms.- 5 Vector Operations in ?3.- 5.1 Nabla.- 5.2 Higher Derivatives.- 5.3 Integral Formulae.- 6 Extremals.- 6.1 Generic Extremals.- 6.2 Extremals with Constraints.- 7 Integral Geometry.- 7.1 Measure of Points and Lines.- 7.2 Kinematic Measure.- 7.3 Formulae of Poincare and Blaschke.- 1. The Volume Element on a Manifold.- 2. The Algebra of Forms.- 3. A Remark on Curl Curl.