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Comprehensive Guide to Vector Calculus Integrals and Derivatives

Dive deep into the classification and calculation of vector integrals, derivatives, and fundamental theorems in 1D, 2D, and 3D spaces. Learn the geometry and application of flow, flux, and substance integrals with clear examples and techniques.

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Comprehensive Guide to Vector Calculus Integrals and Derivatives

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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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