Zen and the art of motorcycle maintenance robert pirsig
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Zen and the Art of Motorcycle Maintenance Robert Pirsig. The state of “ stuckness ” is to be treasured. It is the moment that precedes enlightenment. Differential equations. REVIEW. Algebraic equation : involves functions ; solutions are numbers.

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Zen and the art of motorcycle maintenance robert pirsig

Zen and the Art of Motorcycle MaintenanceRobert Pirsig

  • The state of “stuckness” is to be treasured. It is the moment that precedes enlightenment.


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


Wednesday sept 15th univariate calculus 3

Wednesday Sept 15th: Univariate Calculus 3

Exponential, trigonometric, hyperbolic functions

Differential eigenvalue problems

F=ma for small oscillations


Complex numbers

Complex numbers

The complex plane


The complex exponential function

The complex exponential function


Zen and the art of motorcycle maintenance robert pirsig

Also:


Hyperbolic functions

Hyperbolic functions


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 oscillations

Newton’s 2nd Law for Small Oscillations


Newton s 2 nd law for small oscillations1

Newton’s 2nd Law for Small Oscillations


Newton s 2 nd law for small oscillations2

Newton’s 2nd Law for Small Oscillations


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 oscillations4

Newton’s 2nd Law for Small Oscillations

=0

~0


Newton s 2 nd law for small oscillations5

Newton’s 2nd Law for Small Oscillations

=0

~0


Angular frequency

Angular frequency


Pendulum

Pendulum


Pendulum1

Pendulum


Pendulum2

Pendulum


All that matters

All that matters:


Differential eigenvalue problems

Differential eigenvalue problems


Differential eigenvalue problems1

Differential eigenvalue problems


Differential eigenvalue problems2

Differential eigenvalue problems


Differential eigenvalue problems3

Differential eigenvalue problems

Zero crossings


Differential eigenvalue problems4

Differential eigenvalue problems

Zero crossings


Multivariate calculus 1 multivariate functions partial derivatives

Multivariate Calculus 1:multivariate functions,partial derivatives


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


Isocontours

Isocontours


Isocontour examples

Isocontour examples


Pacific watermasses

Pacific watermasses


Partial differential equations

Partial differential equations

Algebraicequation: involves functions; solutions are numbers.

Ordinary differential equation (ODE):

involves total derivatives; solutions are univariate functions.

Partial differential equation (PDE):

involves partial derivatives; solutions are multivariate functions.


Notation

Notation


Classification

Classification


Order

Order

=order of highest derivative with respect to any variable.


Partial integration

Partial integration

Instead of constant,

add function of other variable(s)


Partial integration1

Partial integration


Homework

Homework

Section 2.10, Density stratification and the buoyancy frequency.

Section 2.11, Small oscilations

Section 2:12, Modes

Section 3.1, Partial derivatives (typo in 4e)


Application initial condition for turbulent layer model

Application: initial condition forturbulent layer model


Lake f ishing

Lake Fishing


Lake fishing

Lake Fishing

Why positive and negative?


Inhomogeneous fishing example

Inhomogeneous fishing example


Inhomogeneous fishing example1

Inhomogeneous fishing example

Classify?


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