order different from syllabus: univariate calculus multivariate calculus linear algebra linear...
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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