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


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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. Algebraic equation : involves functions ; solutions are numbers.

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slide1

Order different from syllabus:

  • Univariate calculus
  • Multivariate calculus
  • Linear algebra
  • Linear systems
  • Vector calculus
  • (Order of lecture notes is correct)
differential equations
Differential equations

REVIEW

Algebraicequation: involves functions; solutions are numbers.

Differential equation: involves derivatives; solutions are functions.

classification of odes
Classification of ODEs

Linearity:

Homogeneity:

Order:

superposition linear homogeneous equations
Superposition(linear, homogeneous equations)

Can build a complex solution from the sum of two or more simpler solutions.

properties of the exponential function
Properties of the exponential function

Taylor series:

Sum rule:

Power rule:

Derivative

Indefinite integral

tuesday sept 15th univariate calculus 3

Tuesday Sept 15th: Univariate Calculus 3

Exponential, trigonometric, hyperbolic functions

Differential eigenvalue problems

F=ma for small oscillations

complex numbers
Complex numbers

The complex plane

oscillations
Oscillations
  • Simple pendulum
  • Waves in water
  • Seismic waves
  • Iceberg or buoy
  • LC circuits
  • Milankovich cycles
  • Gyrotactic swimming

current

Swimming

direction

gravity

newton s 2 nd law for small oscillations3
Newton’s 2nd Law for Small Oscillations

Expand force about equilibrium point:

=0

Small if x is small

newton s 2 nd law for small oscillations5
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
Example: lake fishing

Why positive and negative?

example lake fishing1
Example: lake fishing

Why positive and negative?

partial derivatives
Partial derivatives

Increment:

x part

y part

partial derivatives1
Partial derivatives

Could also be changing in time:

total derivatives
Total derivatives

x part

y part

t part

homework
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