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§1.3 Integrals Flux, Flow, Subst

§1.3 Integrals Flux, Flow, Subst. Christopher Crawford PHY 311 2014-01-27. Outline. Integration Classification of integrals – let the notation guide you! Calculation: 1) parameterize, 2) pull-back

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§1.3 Integrals Flux, Flow, Subst

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  1. §1.3IntegralsFlux, Flow, Subst Christopher Crawford PHY 311 2014-01-27

  2. Outline • IntegrationClassification of integrals – let the notation guide you!Calculation: 1) parameterize, 2) pull-back • vs. Natural derivativesGradient, Curl, Divergence – differentials in 1d, 2d, 3dSet stage for fundamental theorems of vector calculus • Natural integralsFlow, Flux, Substance – canonical 1d, 2d, 3d integralsGeometric interpretation • NEXT CLASS: BOUNDARY operator ` ‘ (opposite of `d’)Derivative, boundary chains: dd=0, =0 ; (and converse)Gradient, curl, divergence -> generalized Stokes’ theorem

  3. Classification of integrals • Scalar/vector - fields/differentials – 14 combinations (3 natural) • 0-dim (2) • 1-dim (5) • 2-dim (5) • 3-dim (2) • ALWAYS boils down to • Follow the notation! • Differential form – everything after the integral sign • Contains a line element: – often hidden • Charge element: • Current element: • Region of integration: – contraction of region and differential • Arbitrary region : (open region) • Boundary of region : (closed region)

  4. Recipe for Integration • Parameterize the region • Parametric vs. relational description • Parameters are just coordinates • Boundaries correspond to endpoints • Pull-back the parameters • x,y,z -> s,t,u • dx,dy,dz -> ds,dt,du • Chain rule + Jacobian • Integrate • Using single-variable calculus techniques

  5. Example – verify Stokes’ theorem • Vector field Surface • Parameterization • Line integral • Surface integral

  6. Example – verify Stokes’ theorem • Vector field Surface • Parameterization • Line integral • Surface integral

  7. Unification of vector derivatives • Three rules: a) d2=0, b) dx2 =0, c) dx dy = - dy dx • Differential (line, area, volume) elements as transformations

  8. … in generalized coordinates • Same differential d as before; hi comes from unit vectors

  9. Example redux – using differential • Vector field Surface • Parameterization • Line integral • Surface integral

  10. Natural Integrals • Flow, Flux, Substance – related to differentials by TFVC • Graphical interpretation of fundamental theorems

  11. Summary of differentials / integrals

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