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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





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







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



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.




Order
Order

=order of highest derivative with respect to any variable.


Partial integration
Partial integration

Instead of constant,

add function of other variable(s)



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)



Lake f ishing
Lake Fishing


Lake fishing
Lake Fishing

Why positive and negative?