1 3 Integrals Flux Flow Subst Christopher Crawford

§ 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 • vs. Natural derivatives Gradient, Curl, Divergence – differentials in 1 d, 2 d, 3 d Set stage for fundamental theorems of vector calculus • Natural integrals Flow, Flux, Substance – canonical 1 d, 2 d, 3 d integrals Geometric interpretation • NEXT CLASS: BOUNDARY operator ` ‘ (opposite of `d’) Derivative, boundary chains: dd=0, =0 ; (and converse) Gradient, curl, divergence -> generalized Stokes’ theorem 2

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: – Charge element: – Current element: – often hidden • Region of integration: – contraction of region and differential – Arbitrary region : – Boundary of region : (open region) (closed region) 3

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

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

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

Unification of vector derivatives • Three rules: a) d 2=0, b) dx 2 =0, c) dx dy = - dy dx • Differential (line, area, volume) elements as transformations 7

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

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

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

Summary of differentials / integrals 11