§1.3 Integrals Flux, Flow, Subst Christopher Crawford PHY

Outline Integration Classification of integrals – let the notation guide you! Calculation: 1) parameterize, 2) pull-back vs. Natural derivatives Gradient, Curl, Divergence – differentials in 1d, 2d, 3d Set stage for fundamental theorems of vector calculus Natural integrals Flow, Flux, Substance – canonical 1d, 2d, 3d 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:– often hidden – Charge element: – Current element: Region of integration: – contraction of region and differential – Arbitrary region :(open region) – Boundary of 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 – Using single-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; h i 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