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 equationsAlgebraic equation: involves functions; solutions are numbers.

Differential equation: involves derivatives; solutions are functions.

REVIEW

Classification of ODEs

2''' 3  0 linear''' 3  0 nonlinear' ''  0 nonlinear' 2 1  / mondo  nonlinear!f

f ff ff f ff f

2''' 3  0 homogeneous''' 3  0 homogeneous' '' 1  nonhomogeneous

f ff ff f f

2'  0 1st  order

''' 3  0 3rd  order' ''  0 2nd  order' 2 1  / 1st  orderf

f gf ff f ff f

Linearity:

Homogeneity:

Order:

Superposition(linear, homogeneous equations)

( ), ( ) solutions

( ) ( ) solution

f x g x

af x bg x

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

Properties of the exponential function

1

2 31 12! 3!1 , 2.71828

,

( ) , with special case  1/ ,

, 

.

x

x y yx

x x x x

x x

x x

e x x x e

e e e

e e e e

d e edx

e dx e c

Sum rule:

Power rule:

Taylor series:

Derivative

Indefinite integral

All implicit in this: '( ) ( ); (0) 1E x E x E

Wednesday Sept 15th: Univariate Calculus 3

•Exponential, trigonometric, hyperbolic functions•Differential eigenvalue problems•F=ma for small oscillations

Complex numbers

*

*

*

Add and divide by 2: .2

Subtract and divide by

1

real part; imaginary part

Co

2 :

mplex conjugate:

.2

r i

r i

r i

r

i

z z iz

z z

z z iz

z z z

z z

i

zi i

iz

rz

z

The complex plane

*z

The complex exponential function2 3 4 5

2 3 42 3 4 5 5

2 3 4 5

2 4 3 5

1 1 1 1( ) 1 ( ) ( ) ( ) ( )2! 3! 4! 5!1 1 1 1 1 2! 3! 4! 5!

1 1 1 1 =1 2! 3! 4! 5!1 1 1 1 2! 4! 3!

15!

i iE x x x x x x

x x x x x

x x x x x

x x x x

i i i i

i i i i i

i i i

i x

( ) C( ) ( )

OR

. (Euler)cos sinix

E ix x S xi

e x i x

23 24 2 25

1

1,

ii i i ii i ii i

( )C x ( )S x

Also:

ADD:

SUBTRACT:

2

2

cos sin

cos sin

2cos

cos

2 si

n

n

si

ix

ix

ix i

ix ix

ix

x

ix ix

ixi

e ex

e x i x

e x i x

e e x

e e i

e ex

x

Hyperbolic functionssinh( ) ; cosh( ) .2 2

sinh( ) 1 2tanh( ) ; sech( ) .cosh( )  cosh( ) 

x x x x

x xx x x x

e e e ex x

x e ex xe e e ex x

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

current

gravity

Swimmingdirection

Newton’s 2nd Law for Small Oscillations

22 ( )d xm F x

dt

0x

m

x

Newton’s 2nd Law for Small Oscillations

22 ( )d xm F x

dt m

x

F

0x

Newton’s 2nd Law for Small Oscillations

22 ( )d xm F x

dt m

F

x

0x

Newton’s 2nd Law for Small Oscillations

(3) ( )22

32 1 1 1''(0) (0) (0)2! 3! ! = (0) '(0) n nF x F x F xnd xd

Fm F xt

=0Small if x is small

22 ( )d xm F x

dt m

x

equilibrium point: 0F

0x

Expand force about equilibrium point:

Newton’s 2nd Law for Small Oscillations(3) ( )2

232 1 1 1''(0) (0) (0)2! 3! ! = (0) '(0) n nF x F x F xn

d xd

Fm F xt

=0~0

22 = '(0) '(0) 0 oscillationd xm F x F

dt

Newton’s 2nd Law for Small Oscillations(3) ( )2

232 1 1 1''(0) (0) (0)2! 3! ! = (0) '(0) n nF x F x F xn

d xd

Fm F xt

=0~0

22 = '(0) '(0) 0 oscillationd xm F x F

dt

0

22

e.g. Hooke's law: '(0) where spring constant

=

cos

F kk

d x xdt

x x

km

k tm

Angular frequency

0 0 0

0

cos cos 2 cos

where amplitude and

2 angular frequency

In this case = .

x x x xk t f t tm

x

f

km

Pendulum

s

mg

m

Pendulum

s

mg

22

22

sin

sin

Newton:ma F

d sm mgdt

dm mgdt

F

m

Pendulum

s

mg

22

22

sin

sin

Newton:ma F

d sm mgdt

dm mgdt

2 3 52 sin ...3! 5!

provided is small

g gddt

g

Oscillation with angular frequency g

F

m

All that matters:function of position only

Displacements small

Force =

Differential eigenvalue problems

2( ) ( ) 0;

(0) 0; ( ) 0

sin( ) cos( )

f x f x

f f

f A x B x

Differential eigenvalue problems

2''( ) ( ) 0;

(0) 0; ( ) 0

sin( ) cos( )

(0) 0 0

( ) 0 0 sin( ) sin( ) 0 0, 1, 2, 3,

sin( ),sin(2 ),sin(3 ),

f x f x

f f

f A x B x

f B

f A

f x x x

Differential eigenvalue problems

2''( ) ( ) 0;

(0) 0; ( ) 0

sin( ) cos( )

(0) 0 0

( ) 0 0 sin( ) sin( ) 0 0, 1, 2, 3, eigenvalues

sin( ),sin(2 ),sin(3 ), eigenfunctions

f x f x

f f

f A x B x

f B

f A

f x x x

eigenmodes

modesoror

Differential eigenvalue problems

2''( ) ( ) 0;

(0) 0; ( ) 0

sin( ) cos( )

(0) 0 0

( ) 0 0 sin( ) sin( ) 0 0, 1, 2, 3, eigenvalues

sin( ),sin(2 ),sin(3 ), eigenfunctions

f x f x

f f

f A x B x

f B

f A

f x x x

eigenmodes

modesoror

0 1 2 Zero crossings

Differential eigenvalue problems

2''( ) ( ) 0;

(0) 0; ( ) 0

sin( ) cos( )

(0) 0 0

( ) 0 0 sin( ) sin( ) 0 0, 1, 2, 3, eigenvalues

sin( ),sin(2 ),sin(3 ), eigenfunctions

f x f x

f f

f A x B x

f B

f A

f x x x

eigenmodes

modesoror

0 1 2 Zero crossings

Multivariate Calculus 1:

multivariate functions,partial derivatives

x

y

( , )T x y

Partial derivatives

x

y

( , )T x y

0

0

( , ) ( , )( , ) lim

( , ) ( , )( , ) lim .

x

y

T x x y T x yT x y xx

T x y y T x yT x y yy

TT x T yx y

Increment:

x part y part

"di" = partial derivative

Partial derivatives

x

y

( , , )T x y tTTT x y tx y t

T

Could also be changing in time:

Total derivatives

x

y

( , , )T x y t

TTT x y tx y tT

yT T xt t tx y t

T T

0limt

dyT dT T dxt dt x dt

T Ty dt t

x part y part t part

Isocontours

x

y

( , )T x y

0

/ isocontour slope/

TT x yx y

Ty xy x

y T xx T y

T

T

Isocontour examples

Stonewall bank: ( , )x z

Pacific Ocean: ( , )T T z

50S 0 50N

//

xdzdx z

//

Tdzd T z

Pacific watermasses

( , )T z

( , )S z

50S 0 50N

Partial differential equationsAlgebraic equation: 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

2 32 2

subscript notatio

"di" = partial der

n:

; ;

iv

,

ative

x tt xttf f ff f fx t x t

Classification

2

2

2

3  0 linear3  0 nonlinear

3  0 homogeneous3 1  nonhomogeneous

xxt

x t

x t

x t

f f ff f f

f f ff f f

If ( , ) and ( , ) are solutions of a PDE, then any ( ,

Superposition:

linear, homogeneouslinear combinati ) ( , ),

where andon

al are

so con

a sstants,

is .olution

f x t g x taf x t bg x t

a b

Order

2 0 1st order

3  0 3rd orderx t

xxx t

f f gf f f

=order of highest derivative with respect to any variable.

Partial integration

21( ,

. .

) )

,

(

( )

2

x

f x y

e g

f x y x

x c y

Instead of constant,add function of other variable(s)

Partial integration

2

2

boundary condition

(0, ) 1

(0, ) 0 ( ) 1

add :

( , ) ;

1( , ) ( )2

1complete solution: ( , 12

( ) 1

)

x f y y

f y c y y

c

f x y x c y

f x y x

f y x

y

y y

x

x

y ( , )f x y

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

3tanh , 1027 tanhkgz zU U h hm

Lake Fishing

2

2

( ) fish( ) fishermen

f tF t

dF fdtdf Fdt

d f dFdtdt

2 2

2 2

( ) fish( ) fishermen

cos( ); sin( )

f tF t

dF fdtdf Fdt

d f d fdF f fdtdt dt

f t F t

Why positive and negative?

Lake Fishing

Inhomogeneous fishing example( ) fish( ) fishermen

f tF t

dF fdtdf F sdt

Inhomogeneous fishing example

2 2

2 2

2 2 2

2 2 2

2 22 2

( ) fish( ) fishermen

Let

cos( ); sin( )

( )

f tF t

dF fdtdf F sdt

d f d fdF f fdtdt dt

dfd F d F d FF s F s F sdtdt dt dt

u F s

d d udt dt

f t F t s

F s u

Classify?