§1.3.2 The fundamental theorem of differentials [FTD] Christopher Crawford PHY 416 2014-09-19

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

Regions and boundaries

Flux/flow representation of fields

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?

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

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