1. §1.3 Integrals Flux, Flow, Subst
Christopher Crawford
PHY 311

2. Outline
Natural integrals vs. natural derivatives Classification of integrals – let the notation guide you! Flow, Flux, Substance – canonical 1d, 2d, 3d integrals Gradient, Curl, Divergence – canonical 1d, 2d, 3d differentials Water analogy – the velocity field Geometric interpretation – hint of fundamental theorems
Calculation of integrals Steps: 1) parameterize, 2) pull-back Example: verification of Stokes’ theorem
NEXT CLASS: BOUNDARY operator ` ‘ (adjoint of `d’) Potential theory: dd=0, =0 and converse (Poincaré) Fundamental theorems of vector calculus

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. Natural Integrals Flow surfaces, Flux tubes, Substance boxes
Graphical interpretation of fundamental theorems
FTVC: # of equipotentials crossed = change in potential at bounds
Stokes’ theorem: # of flow surfaces pierced by closed path = # of flux tubes punctured through bounded disk
Gauss’ theorem: # of flux tubes punctured through closed surface = # of substance boxes in enclosed volume

5. Summary of differentials / integrals

6. Velocity field: flux, flow, [and fish]

7. 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

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

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