∞
f (t) = ∑ an cos nω t + bn sin nω t
n =0
What can you predict about the a’s and b’s for this f(t)?
f(t)
A)  All terms are non-zero
B) The a’s are all zero
C) The b’s are all zero
D) a’s are all 0, except a0
E) More than one of the above (or none, or ???)
2- 1
When you finish P. 3 of the Tutorial, click in:
∞
f (t) = ∑ an cos nω t + bn sin nω t
n =0
What can you say about the a’s and b’s for this f(t)?
f(t)
t
A)  All terms are non-zero
B) The a’s are all zero
C) The b’s are all zero
D) a’s are all 0, except a0
E) More than one of the above, or, not enough info...
2- 2
1
∞
f (t) = ∑ an cos nω t + bn sin nω t
n =0
What can you say about the a’s and b’s for this f(t)?
f(t)
t
A)  All terms are non-zero
B) The a’s are all zero
C) The b’s are all zero
D) a’s are all 0, except a0
E) More than one of the above!
2- 3
∞
f (t) = ∑ bn sin(nω t)
n =1
2- 4
2
Vectors, in terms of a set of basis vectors:
 3 ˆ
v = ∑ vi ei
i=1
Inner product, or “dot product”:
  3
c ⋅ d = ∑ c i di
i=1
To find one numerical component of v:

v i = v ⋅ eˆ i
2- 5
 3 ˆ
v = ∑ vi ei
i=1
∞
f (t) = ∑ bn sin(nω t)
n =1
2- 6
3
 3 ˆ
v = ∑ vi ei
i=1

v i = v ⋅ eˆ i
∞
f (t) = ∑ bn sin(nω t)
n =1
2- 7
Inner product, or “dot product” of vectors:
  n
c ⋅ d = ∑ c i di
i=1
If you had to make an intuitive stab at what might be
the analogous inner product of functions, c(t) and d(t),
what might you try? (Think about the large n limit?)
2- 8
4
Inner product, or “dot product” of vectors:
  n
c ⋅ d = ∑ c i di
i=1
If you had to make an intuitive stab at what might be
the analogous inner product of functions, c(t) and d(t),
what might you try? (Think about the large n limit?)
How about:
∫ c(t)d(t)dt
??
2- 9
Inner product, or “dot product” of vectors:
   n
a b ≡ a ⋅ b = ∑ ai bi
i=1
If you had to make an intuitive stab at what might be
the analogous inner product of functions, a(t) and b(t),
what might you try?
2- 10
5
1.0
0.5
2
4
sin(ωt)
6
8
- 0.5
10
12
- 1.0
1.0
0.5
2
4
6
sin(2ωt)
8
10
12
- 0.5
- 1.0
T
What can you say about
∫ sin(ωt)sin(2ωt)dt
0
A)  0
B) positive C) negative D) depends
E) I would really need to compute it...
1.0
2- 11
sin(ωt)
0.5
2
4
6
8
10
12
- 0.5
- 1.0
1.0
sin(mw t)
0.5
2
4
6
8
- 0.5
- 1.0
10
12
If m>1, what can you guess about
T
∫ sin(ωt)sin(mωt)dt
0
A)  always 0
B) sometimes 0
C)???
2- 12
6
Summary (not proven by previous questions,
but easy enough to just do the integral and show this!)
T
∫ sin(nωt)sin(mωt)dt = 0
if n ≠ m
0
2- 13
Orthogonality of basis vectors:
eˆ i ⋅ eˆ j = 0
(if i ≠ j)
What does ...
T
∫ sin(nωt)sin(mωt)dt = 0
(if n ≠ m)
0
suggest to you, then?
2- 14
7
Orthonormality of basis vectors:
⎧0 (if i ≠ j)
ˆe i ⋅ eˆ j = ⎨
⎩1 (if i = j)
≡ δ i,j
T
⎧0 (if n ≠ m)
2
≡ δ n,m
sin(n
ω
t)sin(m
ω
t)dt
=
⎨
∫
T0
⎩1 (if n = m)
2- 15
Vectors, in terms of a set of basis vectors:
 n ˆ
v = ∑ vi ei
i=1
To find one numerical component:

v i = v ⋅ eˆ i
2- 16
8
Vectors, in terms of a set of basis vectors:
 n ˆ
v = ∑ vi ei
i=1
To find one numerical component:

v i = v ⋅ eˆ i
Functions, in terms of basis functions
∞
f (t) = ∑ bn sin(nω t)
n =1
To find one numerical component:
T
2
bn = ∫ f (t)sin(nωt)dt (??)
T0
2- 17
Vectors, in terms of a set of basis vectors:
 n ˆ
v = ∑ vi ei
i=1
To find one numerical component: Fourier’s trick
n

eˆ j ⋅ v = eˆ j ⋅ ∑ v i eˆ i
i=1
n
= ∑ v i eˆ j ⋅ eˆ i
i=1
n
= ∑ v i δ i, j
2- 18
i=1
9
n
∑v δ
i
=?
i, j
i=1
n
A)
∑v
i
i=1
B) v i
C) v j
D) v n
E) Other/none of these?
2- 19
n

eˆ j ⋅ v = eˆ j ⋅ ∑ v i eˆ i
n
i=1
= ∑ v i eˆ j ⋅ eˆ i
i=1
n
= ∑ v i δ i, j
i=1
= vj
D’oh!
2- 20
10
∞
f (t) = ∑ bn sin(nω t)
n =1
To find one component: Fourier’s trick
“Dot” both sides with a “basis vector”
2
2 ∞
f (t)sin(mω t)dt = ∫ ∑ bn sin(nω t)sin(mω t)dt
T ∫0
T 0 n=1
T
T
∞
T
2
= ∑ ∫ bn sin(nω t)sin(mω t)dt
n=1 T 0
2- 21
∞
2
2
f
(t)sin(m
ω
t)
=
bn sin(nω t)sin(mω t)
∑
∫
T ∫0
T
n =1
0
T
T
∞
= ∑ bnδ n,m
n =1
2- 22
11
∞
∑b δ
n
n =1
n,m
=?
∞
A)
∑b
n =1
n
B) bn
C) bm
D) Other/none of these?
2- 23
∞
2
2
f
(t)sin(m
ω
t)
=
bn sin(nω t)sin(mω t)
∑
∫
T ∫0
T
n =1
0
T
T
∞
= ∑ bnδ n,m
n =1
= bm
2- 24
12
2- 25
13