Zen and the Art of Motorcycle Maintenance Robert Pirsig

1. Zen and the Art of Motorcycle MaintenanceRobert Pirsig • The state of “stuckness” is to be treasured. It is the moment that precedes enlightenment.

2. The complex exponential function REVIEW

3. Hyperbolic functions

4. Newton’s 2nd Law for Small Oscillations =0 ~0

5. Differential eigenvalue problems

6. Partial derivatives Increment: x part y part

7. Total derivatives x part y part t part

8. Isocontours

9. Multivariate Calculus 2:partial integrationseparation of variablesFourier methods

10. Partial differential equations Algebraicequation: involves functions; solutions are numbers. Ordinary differential equation (ODE): involves total derivatives; solutions are univariate functions. Partial differential equation (PDE): involves partial derivatives; solutions are multivariate functions.

11. Notation

12. Classification

13. Order =order of highest derivative with respect to any variable.

14. Partial integration Instead of constant, add function of other variable(s)

15. Partial integration

16. Partial integration

17. Partial integration

18. Partial integration

19. Solution by separation of variables Try to reduce PDE to two (or more) ODEs.

20. Is it possible for functions of two different variables to be equal?

21. Is it possible for functions of two different variables to be equal?

22. Is it possible for functions of two different variables to be equal?

23. The Plan

24. Example 1

25. Example 1

26. Example 1

27. Example 1

28. Thermal diffusion in a 1D bar Boundary conditions: Initial condition:

29. Applications • Diffusion of: • Heat • Salt • Chemicals, e.g. O2, CO2, pollutants • Critters, e.g. phytoplankton • Diseases • Galactic civilizations • Money (negative diffusion)

30. Thermal diffusion in a 1D bar Boundary conditions: Initial condition:

31. Solution by separation of variables

32. First try

33. Second try

35. Characteristics of time dependence: • T 0 as t  ∞ , i.e. temperature equalizes to the temperature of the endpoints. • Higher  (diffusivity) leads to faster diffusion. • Higher n (faster spatial variation) leads to faster diffusion.

36. In fact: Time scale is proportional to (length scale)2.

37. In fact: E.G. Water 1/2 as deep takes 1/4 as long to boil!

38. Sharp gradients diffuse rapidly.  This observation is surprising.

39. Now what about the initial condition? Boundary conditions: Initial condition: ?

40. Initial condition: ?

44. Fourier’s Theorem

45. To find the constants:

46. Problem solved Boundary conditions: Initial condition:

47. Homework clarification • 3.2.1 partial integration • 3.2.2 separation of variables • 3.2.3 Fourier series solution • 1. Solve for a single Fourier mode • Separate • Choose sign of separation constant • Satisfy boundary conditions and initial condition ht=0 • Write down the general solution satisfying h(x,0)=h0(x), • but don’t derive the Fourier series. • 2. Interpret • Time dependence: exponential or …? • Describe in physical terms. • How might you create the modes n=1, n=2, …? • 3. Time scale • Is the time scale for a mode proportional to the length scale squared? • If not, what?