REVIEW: Separation of Variables. Fourier’s Theorem.

REVIEW:

Separation of Variables

Manipulate PDE into the form ( ) ( ).

Result is 2 ODES: ( ) ; ( ) .

f x g y

f x c g y c

Fourier’s Theorem

0x x L

0 0

1

0

0

ANY FUNCTION ( ) that obeys the boundary conditions (0) 0 and ( ) 0

can be represented as a Fo

(

urier series:

2The coefficients are given by:

) sin

n

nn

xT x A nL

T x T T L

A TL

00

0

( )sin

"recipe" or ) (f

L xx n dxL

T x

Friday Sept 17th: Linear Algebra

•Vector/matrix operations•Index notation•Determinants•Cartesian position vectors•Eigenvalues/eigenvectors

Scalars and vectors• A scalar a is just a single number.• A vector is, in the simplest definition, just a list of numbers:

1

21 2Column vector , Row vector , , .

Index notation: s the 'th component of , for 1,2,3, , i

N

N

i

vv

v v v v v

v

v i v i N

.

1

, or , for 1,2,3, ,

Scalar multiplicati

Vec

on: , or .

Dot (scalar, inner) product: ·

tor sum: i i i

i i

N

i ii

w u v w u v i N

u av u av

u v u v

u

v

.

2

Einstein summation notation:

Summation over the repeated (dummy) index is implied.

•Magnitude or length of a vector :

·

| | ·

The dot product can be written a

s · |

i i

i i i

u v u v

v v v v v v

u v u

|| |cos .

Orthogonal vectors: 0 · 0 (right angles)

Unit vector: / | | ; | | 1 ; is parallel to .ˆ ˆ ˆ

Component of in the direction of : · .| |

v

cos u v

e v v e e v

vu v uv

· | |u vv

11 12

21 22

First subscript = ROW, Second =

A matrix is an array of numbers, e.g.

Matrix transpose:

Symmetric matrix: , or

COLUMN.

.

An

ij

Tij ji

ji iT

jA

A AA

A

A

A

A A

A A

11

22

tisymmetric matrix: , or .

Diagonal matrix: =0

0

unle

ss , e.g.

0

ji ij

ij

ij

T A A

A

A A

i

AA

A

j

Matrices

11 12 11 12 11 11 12 21 11 12 12 22

21 2221 22 21 11 22 21 21 12 22 22

Matrix addition: .

Matrix multiplication: .

For 2x2 case:

ij ij ij

A A B

C A B

C

B A B A B A B A BB B

A

A A A B A B A A

B

B B

or (sum on middle index).

ij ik kjC A B

AB BA

In , dummy index, , = free indices.

Free indices on LHS and RHS must correspond.

Identity matrix:

1 if 1 0 e.g. 0 if 0 1

Matrix inverse: If

ij ik kj

ij

C A B k i j

i ji j

AB

1 1

1

and , then and .

Orthogonal matrix: .T

BA B A A B

A A

2 2: x xAx x

Determinant of a matrix

2 2:

3 3:

x xAx x

x x xA x x x

x x x


2 2:

3 3:

x xAx x

x x xA x x x

x x x


.

Singular matrix: | | 0. Has no inverse.

a b cd e f aei bfg cdh gec hfa idbg h i

ORa b c

d ed fe fd e f a b cg i g hh ig h i

A

Matrix-vector multiplication, linear systems

Right multiplication: or . (Vector on right; sum over 2nd, or right, index.)

Left multiplication: or . (Vector on left; sum over 1st, or left, index.)

, o

i ij j

j i ij

u Av u A v

u vA u v A

u Av

1

r is a system of linear equations that we can solve for :

(provided | | 0!).

Homogeneous system: 0, has solution ONLY IF | | 0.

Av u v

v A u A

Av A

Cross product

1 2 3 2 3 2 3 1 3 1 3 1 2 1 2

1 2 3

( ) ( ) ( )

| || |sin

Identities:( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( )( )

i j ku v u u u i u v v u j u v v u k u v v u

v v v

u v u v

u v w u v wu v w u w v u v wu v w x u w v x u x v w

ocean

beach

60% speldfar, 40% schwartz

A linear system

?

Crimea R.

Rollinonthe R.

ocean

beach




C

R

A linear system

0.6 0.9 0.40.4 0.1 0.6

0.9 0.4 0.6OR 0.1 0.6 0.4

Invert 2 2 matrix: rearrange, divide by determinant.

0.6 0.410.9 0.6 0.1 0.4 0.1 0.

C R

CR

CR

0.6 0.4

9 0.4 0.6

0.9 0.4

0.1 0.61

Det

Beach sand is 40% from Crimea R., 60% from Rollinonthe R.

i

j

k p

Cartesian position vectors

ˆˆ ˆp x y zi j k

unit vectors

i

j

k p

Cartesian position vectors

ˆˆ ˆp x y zi j k

unit vectors

Right-handed

Left-handed

Matrices as transformations

1 0 0Ex. 1: 0 1 0 Note: | | 1.

0 0 1A v v A


1 0 0Ex. 1: 0 1 0 Note: | | 1.

0 0 1A v v A

2 0 0Ex. 2: 0 2 0 2 | | 8

0 0 2

Note: Transformation is rever .sible

A v v A


1 2

2 1

3 3

0 1 0Ex. 3: 1 0 0

0 0 1

v vA v v

v v

j

i

k

reversible


j

i

k

reversible

1 2

2 1

3 3

0 1 0Ex. 3: 1 0 0

0 0 1

v vA v v

v v

Matrices as transformations1 1

2 2

3

1 0 0Ex. 4: 0 1 0

0 0 0 0

v vA v v

v

j

i

k

Matrices as transformations1 1

2 2

3

1 0 0Ex. 4: 0 1 0

0 0 0 0

v vA v v

v

j

i

k

irreversible!| | 0A

Eigenvalues and eigenvectors

j

i

k

2 0 0Ex. 2: 0 2 0 ' 2

0 0 2

Note: Direction is unchanged.

A Av v v

Typo on p. 50

Eigenvalues and eigenvectors

2 0 0Ex. 2: 0 2 0 ' 2

0 0 2

Note: Direction is unchanged.

A Av v v

j

i

k

(special matrix)

Eigenvalues and eigenvectorsAlways true for some vectors:

'

eigenvalue

eigenvector

Av v v

v

( ) ( ) ( )

= matrix, eigval-eigvec pairs.

(no sum on )

1,2, ,

n n nAv v

N

A N N

n

N

n

Finding eigvals and eigvecs

( ) ( ) ( ) ( ) ( )

( )

th

( ) 0

homogeneous | | 0

order polynomial equation for , solutions.

n n n n n

n

Av v A v

A

N N

Example: eigvals

2

1 0 00 1 00 0 0

1 0 01 0 0 1 0 0| | 0 1 0 0 1 0 0 1 0 (1 ) 0

0 0 0 0 0 1 0 0

0,1,1

A

A

3rd order polynomial equation

Example: eigvecs

0,1,1 0 00 1 00 0

10

A

1 1 1 1

2 2 2 2

3 3 3

1 0 00 1 0 , ,0 0

0

10

v v v vv v v vv v v

=

3 0v j

i

k

1 1Av v v

Example: eigvecs

0,1,1 0 00 1 00 0

10

A

1 1 1

2 2 2

3 3

1 0 0 00 1 0 , 0 0 ,0 0 0 00

v v vv v vv v

=

1 20, 0v v j

i

k

0 00Av v

1 20 1

1 20 1

1 22 2

1 22 1

1 22 0

2x2 examples:

eigvals only

eigvals only

Homework

., ,

2

(a) Find expressions for the two eigenvalues of

in terms of the quantities and

(b) Show that both eigenvalues are real if

Plus ex

and both eige

tra problem:

n

,

A B C D

A DAD

B

BC

AC D

values are complex otherwise.

Section 4.1, Section 4.2, Section 4.3, omit #1, Section 4.4, plus:

REVIEW: Separation of Variables. Fourier’s Theorem.

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Transcript of REVIEW: Separation of Variables. Fourier’s Theorem.