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Order different from syllabus: Univariate calculus Multivariate calculus Linear algebra Linear systems Vector calculus

Order different from syllabus: Univariate calculus Multivariate calculus Linear algebra Linear systems Vector calculus (Order of lecture notes is correct). Differential equations. REVIEW. Algebraic equation : involves functions ; solutions are numbers.

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Order different from syllabus: Univariate calculus Multivariate calculus Linear algebra Linear systems Vector calculus

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

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

  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 Taylor series: Sum rule: Power rule: 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 forturbulent layer model

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

  13. Newton’s 2nd Law for Small Oscillations

  14. Newton’s 2nd Law for Small Oscillations

  15. Newton’s 2nd Law for Small Oscillations

  16. Newton’s 2nd Law for Small Oscillations Expand force about equilibrium point: =0 Small if x is small

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

  18. Newton’s 2nd 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 Why positive and negative?

  21. Inhomogeneous fishery example

  22. Inhomogeneous fishery example Classify?

  23. Differential eigenvalue problems

  24. Differential eigenvalue problems

  25. 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

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