REVIEW: Separation of Variables. Fourier’s Theorem.
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Transcript of 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
Determinant of a matrix
2 2:
3 3:
x xAx x
x x xA x x x
x x x
Determinant of a matrix
2 2:
3 3:
x xAx x
x x xA x x x
x x x
Determinant of a matrix
.
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
90% speldfar, 10% schwartz
40% speldfar, 60% schwartz
60% speldfar, 40% schwartz
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
Matrices as transformations
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
Matrices as transformations
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
Matrices as transformations
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
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
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: