340 likes | 644 Views
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.
E N D
Order different from syllabus: • Univariate calculus • Multivariate calculus • Linear algebra • Linear systems • Vector calculus • (Order of lecture notes is correct)
Differential equations REVIEW Algebraicequation: involves functions; solutions are numbers. Differential equation: involves derivatives; solutions are functions.
Classification of ODEs Linearity: Homogeneity: Order:
Superposition(linear, homogeneous equations) Can build a complex solution from the sum of two or more simpler solutions.
Properties of the exponential function Taylor series: Sum rule: Power rule: Derivative Indefinite integral
Tuesday Sept 15th: Univariate Calculus 3 Exponential, trigonometric, hyperbolic functions Differential eigenvalue problems F=ma for small oscillations
Complex numbers The complex plane
Oscillations • Simple pendulum • Waves in water • Seismic waves • Iceberg or buoy • LC circuits • Milankovich cycles • Gyrotactic swimming current Swimming direction gravity
Newton’s 2nd Law for Small Oscillations Expand force about equilibrium point: =0 Small if x is small
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
Example: lake fishing Why positive and negative?
Example: lake fishing Why positive and negative?
Inhomogeneous fishery example Classify?
Multivariate Calculus 1:multivariate functions,partial derivatives
Partial derivatives Increment: x part y part
Partial derivatives Could also be changing in time:
Total derivatives x part y part t part
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