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 MaintenanceRobert Pirsig

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


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


Wednesday Sept 15th: Univariate Calculus 3

Exponential, trigonometric, hyperbolic functions

Differential eigenvalue problems

F=ma for small oscillations


Complex numbers

The complex plane


The complex exponential function


Also:


Hyperbolic functions


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


Newton’s 2nd Law for Small Oscillations


Newton’s 2nd Law for Small Oscillations


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


Newton’s 2nd Law for Small Oscillations

=0

~0


Angular frequency


Pendulum


Pendulum


Pendulum


All that matters:


Differential eigenvalue problems


Differential eigenvalue problems


Differential eigenvalue problems


Differential eigenvalue problems

Zero crossings


Differential eigenvalue problems

Zero crossings


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


Isocontours


Isocontour examples


Pacific watermasses


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


Classification


Order

=order of highest derivative with respect to any variable.


Partial integration

Instead of constant,

add function of other variable(s)


Partial integration


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 forturbulent layer model


Lake Fishing


Lake Fishing

Why positive and negative?


Inhomogeneous fishing example


Inhomogeneous fishing example

Classify?


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