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Christopher Crawford PHY

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Presentation on theme: "Christopher Crawford PHY"— Presentation transcript:

1 Christopher Crawford PHY 416 2014-09-19
§ Boundary Theorems Christopher Crawford PHY 416

2 Outline Review of FTD – multivariate analog of FTC Antiderivative – conservative/solenoidal fields, potentials Derivative chains – Definite integral – fundamental boundary theorems Gradient (FTVC), Curl (Stokes’), Divergence (Gauss’) theorems Picture – geometric duality Duality – integration as a contraction (inner product)

3 Review: FTD & Poincaré lemmas
Fundamental theorem of differentials Everything “is a boundary” or “has a boundary” Which category is larger?

4 Derivative [boundary] chains
Differentials = everything after the integral sign – type of vector Pictoral representation of vector/scalar fields – integration by eye Exact sequence of fields – gauge – potential – field – source – dynamics

5 Dual chain

6 Fundamental Boundary Theorems

7 Fundamental Boundary Theorems
Integrate out to the boundary

8 Pictures of Gauss’, Stokes’ theorems
Integral = intersection of geometric objects Geometric duality of two boundaries Adjoint (transpose) operators

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