A geometric approach to differential forms

Description

For the Student......Page 3 1. So what is a Differential Form?......Page 9 2. Generalizing the Integral......Page 10 4. What went wrong?......Page 11 5. What about surfaces?......Page 14 1. Coordinates for vectors......Page 17 2. 1-forms......Page 19 3. Multiplying 1-forms......Page 22 4. 2-forms on TpR3 (optional)......Page 27 5. n-forms......Page 29 1. Families of forms......Page 33 2. Integrating Differential 2-Forms......Page 35 3. Orientations......Page 42 4. Integrating n-forms on Rm......Page 45 5. Integrating n-forms on parameterized subsets of Rn......Page 48 6. Summary: How to Integrate a Differential Form......Page 52 1. The derivative of a differential 1-form......Page 57 2. Derivatives of n-forms......Page 60 3. Interlude: 0-forms......Page 61 4. Algebraic computation of derivatives......Page 63 1. Cells and Chains......Page 65 2. Pull-backs......Page 67 3. Stokes' Theorem......Page 70 4. Vector calculus and the many faces of Stokes' Theorem......Page 74 1. Maxwell's Equations......Page 81 2. Foliations and Contact Structures......Page 82 3. How not to visualize a differential 1-form......Page 86 1. Forms on subsets of Rn......Page 91 2. Forms on Parameterized Subsets......Page 92 3. Forms on quotients of Rn (optional)......Page 93 4. Defining Manifolds......Page 96 5. Differential Forms on Manifolds......Page 97 6. Application: DeRham cohomology......Page 99 1. Surface area and arc length......Page 103