1. §1.3.2 The fundamental theorem of differentials [FTD]
Christopher Crawford
PHY 416

2. Outline
Regions – what you integrate over Boundary operator : boundaries vs. cycles Boundary of a boundary and converse
Geometric representation of fields – flux and flow Derivatives as boundaries – coboundary Duality – two different ways of looking at things
Poincaré lemma – analog of : exact vs. closed Vector identities stemming from and converse
Generalized Stokes’ theorem – a geometric duality [next class] Pictures of FTVC, Stokes’ Gauss’ theorems, proof by induction

3. Regions and boundaries

4. Flux/flow representation of fields

5. Differential as a boundary
Small change in [source of] potential / flow / flux equals equipotential / flux / subst. at the boundary
One higher dimension (extra `d’)
d2=0 (boundary of a boundary)
What about the converse?

6. Fundamental Theorem of Differentials
Given a star-like [spherical] coordinate system,

7. Poincaré lemma and converse
Differentials = everything after the integral sign – type of vector
Pictoral representation of vector/scalar fields – integration by eye
Exact sequence – mathematical structure