Order different from syllabus: Univariate calculus Multivariate calculus Linear algebra Linear...

Order different from syllabus:•Univariate calculus•Multivariate calculus•Linear algebra•Linear systems•Vector calculus

(Order of lecture notes is correct)

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

Tuesday 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

zii

iz

rz

z

The complex plane

*z

The complex exponential function

2 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 12! 3! 4! 5!

1 1 1 1 =12! 3! 4! 5!

1 1 1 12! 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 iS x

e x i x

2

3 2

4 2 2

5

1

1,

ii i i ii i ii i

cos( )x sin( )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

ix

i

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

Application: initial condition forturbulent layer model

3tanh , 1027 tanhkgz zU U

h hm

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

2

2( )d xm F x

dt

0x

m

x

Newton’s 2nd Law for Small Oscillations

2

2( )d xm F x

dt m

x

F

0x

Newton’s 2nd Law for Small Oscillations

2

2( )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 x

nd xd

Fm F xt

=0Small if x is small

2

2( )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

23

2 1 1 1''(0) (0) (0)2! 3! !

= (0) '(0) n nF x F x F xn

d xd

Fm F xt

=0~0

2

2 = '(0) '(0) 0 oscillationd xm F x Fdt

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

23

2 1 1 1''(0) (0) (0)2! 3! !

= (0) '(0) n nF x F x F xn

d xd

Fm F xt

=0~0

2

2= '(0) '(0) 0 oscillationd xm F x F

dt

OR:

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

0

2

2

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

=

cos

F kk

d x xdt

x x

km

k tm

Example: lake fishing

2

2

( ) fish( ) fishermen

f tF t

dF fdtdf Fdt

d f dFdtdt

Why positive and negative?

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?

Example: lake fishing

Inhomogeneous fishery example( ) fish( ) fishermen

f tF t

dF fdtdf F sdt

Inhomogeneous fishery example

2 2

2 2

2 2 2

2 2 2

2 2

2 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 sd d u

dt dt

f t F t s

F s u

Classify?

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

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 yxx

T x y y T x yT x yyy

TT x T yx y

Increment:

x part y part

Partial derivatives

x

y

( , , )T x y tTTT x y t

x y tT

Could also be changing in time:

Total derivatives

x

y

( , , )T x y t

TTT x y tx y t

T

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

Pacific watermasses

( , )T z

( , )S z

50S 0 50N

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