Fundamental Theorems
In your study of mathematics, you have
already
come across two “Fundamental
Theorems”
and another will be introduced in this course
. The first was the
Fundamental
Theorem of Algebra
,
the second, the
Fundamental Theorem of Calculus
, an
d this
course will introduce the
Fundamental Theorem of Linear Systems
.
If you study a
topic and it has a “fundamental theorem”, this is the single most important
idea/concept/relationship in the subject area, and therefore should be among the
ideas/concep
ts/relationships remembered forever.
We will use the review of the first two
Fundamental Theorems as an opportunity to
review some other sub topics that will be of use in our study of control engineering.
Fundamental Theorem of Algebra and other
related
topics
.
There are six
operations associated with arithmetic
and algebra
: addition, subtraction,
multiplication, division
, raising a number to a power,
and finding roots.
Addition is the most fundame
ntal of these operations because
subtraction is the p
rocess of undoing addition (i.e. adding the additive inverse)
multiplication is over and over addition
o
3*4=4+4+4=3+3+3+3
division is the process of undoing multiplication (i.e. multiplication by the
multiplicative inverse)
o
12/3 answers the question: How ma
ny times can 3 be subtracted from 12?
12

3=9, 9

3=6, 6

3=3, 3

3=0, hence 12/3=4.
raising a number to a power is defined using multiplication:
For example
o
3
0
=1
o
3
1
=3*3
0
o
3
2
=3*3
1
o
3
n
=3*3
n

1
Finding roots is the process of undoing the operation of raising a numb
er to a
power, that is
More advanced topics in arithmetic include number theory, factoring techniques, prime
numbers, etc.
The arithmetic/algebraic operations satisfy certain rules. The most basic of these rules are
the famili
ar “laws”. For numbers a, b, c
Commutative law:
a + b = b +a
Associative law:
a + (b + c) = (a + b) + c
Distributive laws:
a(b+c) = ab + ac
(a+b)c = ac +bc
Additive identity law:
There is a special number, 0, such that for any and all a,
a + 0 =
0 + a =
a
Additive inverse
law
:
For any a, there exists (

a) such that a + (

a) = 0
Multiplicative identity law:
There is a special number, 1, such that for any and all a,
a1=1a=1
Multiplicative inverse
law
:
For any and all non zero a, there exists (a

1
) such
that aa

1
=a

1
a =1.
In addition, the operations satisfy the “order of operations”
remembered
mnemonically as
“PEMDAS”. The letters mean
that the algebraic operations
are performed
in the
following order
:
o
Parenthesis.
Perform all operations within
p
arenthesis and other
grouping symbols
first.
This includes all
other
forms of “grouping symbols” such as
the division bar, {},
[], etc.
o
Exponentials.
Evaluate exponentials and any other functions next.
o
Mult
iplication/Division.
Next perform multiplications
and divisions
as they occur
f
rom left to right
.
o
Addition/Subtraction.
Finally, perform additions and subtractions
as they occur
fr
om
left to right
.
From Arithmetic to Algebra
.
The transition from arithmetic to algebra involves replacing the numbers used
in
arithmetic by symbols (
these symbols are
called variables) that either represent numbers
or re
present more complicated combinations
of variables. Hence
,
algebra is arithmetic
applied to symbols rather than numbers.
For example, an arithmetic expression
might be
3+4 = 7
In algebra we encounter expressions more like
3+x = 7
where x might represent a number (i.e. 4) or a more complex expression like
x=(y+5)
2
which means that (y+5)
2
=4; y+5=(+/

)2; or y =

5+/

(2) and finally y =

3 or

7.
Advanced topics in
algebra include polynomials, techniques for factoring polynomials,
prime polynomials, etc.
One topic in algebra is
solving equations
. Remember that all equations contain an equals
sign. The rules for solving equations are simple:
Whatever you do to one
side of the equals sign you must do to the other side
Never divide by zero (or by a variable that is equal to zero, or by a combination of
variables that is equal to zero).
T
he Fundamental Theorem
of Algebra concerns polynomials and their roots. The
root
of
a polynomial (or of any function) is a value of the variable that caused the polynomial to
evaluate
(or become equal)
to zero.
A high school level web site that discusses the Fundamental Theorem of Algebra is:
http://webpages.charter.net/thejacowskis/chapter6/section7.html
A college level web site that discus
ses the Fundamental Theorem of A
lgebra is
:
http://ccrma.stanford.edu/~jos/mdft/Fundamental_Theorem_Algebra.html
The Fundamental Theorem of Algebra
. An n
th
order p
olynomial
has n roots
(some may
be repeated)
.
If, as in the important special case considered in control engineering, ALL
of the
coefficients of the polynomial are real then any complex root is accompanied by its
complex conjugate.
Connection between roots and factors.
Suppose we have found the n roots of an n
th
order polynomial p(s), and these n roots are r
k
, k = 1, … n. T
hen p(s)=K(s

r
1
)(s

r
2
)
...
(s

r
n
).
It is easy to see that the r
k
’s are roots of the factored form. Also, it is easy to see that if all
of the complex r
k
’s are accompanied by their complex conjugates, then the coefficients of
the expanded polynomial are all
real.
Exercises.
1.
Construct the expanded polynomial from the given set of roots
, assume K=1
.
An
important partial answer is the polynomial in factored form.
(Definition of
expanded polynomial
: In the final polynomial, each power of s must not occur
more t
han once; no j’s can appear; the powers of s must decrease from left to
right
: the coefficients are as simple as possible
.)
a.
{1, 1, 3,

3}
b.
{j, 1,

j, 5}
c.
{3}
d.
{1+j, 1

j, j,

j, 0}
e.
{a}
f.
2.
Given the following polynomials, find
all of
th
e roots
(including multiplicity)
, and
put the polynomial in factored form.
a.
p(s)
=
4s+2
b.
p(s)
=
3s
2
+2s+3
c.
p(s)
=
5s
4
+
3
d.
p(s) = as+b
e.
p(s) = as
2
+bs+c
3.
Commit to memory:
a.
Fundamental Theorem of Algebra
i.
Including the special case for real coefficients
b.
Definition of
a root
c.
Relationship between roots of p(s) and factored form of p(s)
4.
Using the information in (3 above) be able to
a.
Find the roots of a polynomial
b.
Put a polynomial in factored form
c.
Given roots, construct the factored form of the polynomial
d.
Beginning with th
e factored form of a polynomial, be able to produce the
expanded form of the polynomial.
Fundamental Theorem of Calculus and other related topics
.
As was pointed out earlier, the operations of arithmetic/algebra were addition,
subtraction, multiplication
, division, raising to powers, and taking roots.
These operations
come in pairs, an operation and one that undoes it.
In arithmetic these operations are
applied to numbers. In algebra they are applied to expressions that involve one or more
variables, thes
e expressions were called functions.
Calculus introduces three more
operations that a
re applied to functions:
limits; differentiation; and integration.
Differentiation and integration are defined in terms of limits, i.e.
The
re
are two common forms of the
Fundamental Theorem of Calculus
. Each
ties the
operations of integration and differentiation together.
The Fundamental Theorem of Calculus
(1)
. If
(i.e. F is the
antiderivative of f)
and some other
technical assumptions that almost all functions of
engineering interest satisfy,
then
The Fundamental Theorem of Calculus
(2)
. If some technical assumptions that almost
all functions of engineering interest satisfy, then
Both versions show that integration and differentiation are (nearly) inverse operations,
i.e. one undoes the other similar to subtraction
undoing
addition and division
undoing
multiplication
and vice versa
.
The Fundamental Theorem
of Calculus (1)
is the basic technique for computing
definite
(as opposed to indefinite)
integrals. It says that the integral (i.e. the area under the
curve between the lower and upper limits of integration) is found by first finding the anti

derivative (i
.e. a function whose derivative is the integrand) of the integrand, second
evaluating this anti

derivative at the upper limit of integration, third evaluating it at the
lower limit of integration and finally subtracting the second value from the first.
Th
e Fundamental Theorem of Calculus (2)
says that if the integral is used to define a
function of time by letting the upper limit be the time variable, then differentiating this
function produces the integrand.
You should not
e that care was taken to use
as the dummy variable of integration.
Further, all integrals have a dummy variable of integration and that this dummy variable
NEVER shows up in the result of integration. I know you can find books, especially
engineering books, th
at violate this. However, these books are WRONG and should not
be followed
in this practice
.
Limits
. The idea of a limit is an investigation of what happens when some variable gets
very small
(
or
sometimes
very large
)
. We
discuss
the idea of a limit as a
variable gets
small.
Consider the problem of determining how much the area of a square increases
when the length of a side is increased by a small amount. The area of
a
square is A=L
2
.
Now if the length is increased from L to L+dL, the area increases from
A to A+dA =
(L+dL)
2
= L
2
+2LdL+dL
2
= A + 2LdL +dL
2
. Hence dA = 2LdL + dL
2
.
The derivative is
dA/dL = lim
dL
0
[2L + dL]. In the limit as dL approaches 0, we ignore all terms that go to
0, hence,
dA = 2LdL approximately.
See picture.
Some d
erivative formulas
.
a, b, n
constants
(real or complex)
Integration by parts
.
Integration by parts is an integration technique that
is
perhaps one
of the most important.
Antiderivative
formulas
. See table for derivative formulas.
Fundamental Theorem of Linear Systems
.
Covered in this course.
Consider a stable
linear time

invariant system described by the transfer function G(s) where
. The steady

state output
due to a sinusoidal input
, is
.
Limits
:
Used in:
Root locus; straight line Bode plots; error constants.
Fundamental Theorem of Calculus
:
Used in:
Laplace transforms; solving state
equations; starting
point for review of derivatives, integrals, limits, etc.
Roots of polynomials
:
Used in:
stability; root locus;
Fundamental Theorem of Linear Systems
:
Used in:
Foundation for Bode plots;
foundation for phasors; foundation for experimental determination of
transfer functions;
Extra credit
opportunity for students with a D or F: Turn in these exercises worked out
and I will add up to 10 points to the total points earned on the tests, quizzes, final exam,
etc. Due at time of first test.
There is a very sign
ificant difference between
a definition and a theorem. Recall the
definition of the derivative
We can use it to prove the “product rule of differentiation”
Proof:
This the
orem, along with the Fundamental Theorem of Calculus can be used to develop
the Integration

by

Parts theorem.
Proof:
In short, definitions are arbitrary, whereas theorems, once the definitions are
established
are pre

determined and are logical consequences of the definitions and earlier postulates.
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