1. Order different from syllabus: Univariate calculus Multivariate calculus Linear algebra Linear systems Vector calculus (Order of lecture notes is correct)

2. Differential equations Algebraic equation: involves functions; solutions are numbers. Differential equation: involves derivatives; solutions are functions. REVIEW

3. Classification of ODEs Linearity: Homogeneity: Order:

4. Superposition (linear, homogeneous equations) Can build a complex solution from the sum of two or more simpler solutions.

5. Properties of the exponential function Sum rule: Power rule: Taylor series: Derivative Indefinite integral

6. Tuesday Sept 15th: Univariate Calculus 3 Exponential, trigonometric, hyperbolic functions Differential eigenvalue problems F=ma for small oscillations

7. Complex numbers The complex plane

8. The complex exponential function

9. Also:

10. Hyperbolic functions

11. Application: initial condition for turbulent layer model

12. Oscillations Simple pendulum Waves in water Seismic waves Iceberg or buoy LC circuits Milankovich cycles Gyrotactic swimming current gravity Swimming direction

13. Newton’s 2 nd Law for Small Oscillations

16. =0 Small if x is small Expand force about equilibrium point:

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

18. Newton’s 2 nd Law for Small Oscillations =0 ~0 OR: Simple pendulum Waves in water Seismic waves Iceberg or buoy LC circuits Milankovich cycles Gyrotactic swimming

19. Example: lake fishing Why positive and negative?

20. Example: lake fishing

21. Inhomogeneous fishery example

22. Classify?

23. Differential eigenvalue problems

26. Multivariate Calculus 1: multivariate functions, partial derivatives

27. Partial derivatives Increment: x part y part

28. Partial derivatives Could also be changing in time:

29. Total derivatives x part y part t part

30. Isocontours

31. Isocontour examples

32. Pacific watermasses

33. Homework Section 2.9, #4: Derive the first two nonzero terms in the Taylor expanson for tan … Section 2.10, Density stratification and the buoyancy frequency. Section 2.11, Modes Section 3.1, Partial derivatives