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

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Order different from syllabus: Univariate calculus Multivariate calculus Linear algebra Linear systems Vector calculus - PowerPoint PPT Presentation

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
• (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

=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?

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