2M
A
3
6
6
0
0
F
I
N
A
L
R
E
V
I
E
W
such that
i. Each
g
k
(
t
) is continuous on
I
. (There are
m
functions to consider.)
ii. Each
p
ij
(
t
) is continuous on
I
. (There are
m
·
n
functions to consider.)
iii.
t
0
∈
I
.
Then there exists a unique solution on the interval
I
. This is known as the
Existence and Uniqueness
Theorem for Linear Systems
.
•
Consider a
LRC Circuit
i.e., an electrical circuit which contains three objects:
–
An Inductor (which behaves like a mass)
V
inductor
∝
dI
inductor
dt
=
⇒
V
inductor
=
L
·
dI
inductor
dt
.
The constant
L
is called the
inductance
.
–
A Resistor (which behaves like friction):
V
resistor
∝
I
resistor
=
⇒
V
resistor
=
R
·
I
resistor
.
The constant
R
is called the
resistance
. (This is called
Ohm’s Law
.)
–
A Capacitor (which behaves like a spring):
dV
capacitor
dt
∝
I
capacitor
=
⇒
dV
capacitor
dt
=
1
C
·
I
capacitor
.
The constant
C
is called the
capacitance
.
Let
I
=
I
(
t
) and
V
=
V
(
t
) denote the current and the voltage.
KirchhoF’s Laws of Circuits
state
1. The net Fow of current
I
through a node is zero.
2. The net voltage
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 Spring '09
 EdrayGoins
 Linear Algebra, Inductor, Invertible matrix, Quantification, Hilbert space

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