Use of point-wise convergence for a sequence of approximating curves,
Use of concept of continuity for curves.
Length of a curve.
How does a curve "fill" space?
In
respnse to a question about how to make sense of curves being close and
sequence of curves having limits, we introduced the concept of a metric
space. - a set where closeness can be measured using non-negative real
numbers for the measurements.
In Linear algebra, an inner product such as one given for
poynomials where `<f,g> = int_a^b f(x) g(x) dx` [ integration of the product over an
interval] gives rise to a norm, `||f|| = sqrt{<f,f>}`, and then a metric,
`m(f,g) = ||f-g||`.
9-4
Discussion of the mean value theorem, its proofs, and counterexamples related to its hypotheses;
(i) continuous on [a, b] and
(ii) differentiable on
(a, b), then
there exists a number c in (a, b)
such that f'(c) = [ f (b) - f (a)]/[b-a].
Counter examples to the MVT:
(1) without Continuity: f (x) = x for x in (0, 1] and f(0) = 0 on the interval [0,1]
and
(2) without differentiability: f (x) = 1/2 -| x - 1/2| for x in [0, 1].
Also f (x) = (1/2)^(1/3) -|( x - 1/2)|^(1/3)for x in [0, 1].
Theorem:
Rolle's Theorem [proof based on Extreme Value Theorem and critical point analysis]: If f is
(i) continuous on [a, b] and
(ii) differentiable on
(a, b),
(iii) f (a) = f (b)
then
there exists a number c in (a, b)
such that f'(c) = [ f (b) - f (a)]/[b-a].
Comment: the proof of Rolle's Theorem broke into three cases-
one of which actually found the number c = (a+b)/2. This case actually
did construct the desired number - and demonstrated it was a number
with the required properties using the fact that the real numbers are a
field with an order relation! This was connected to the first
Motivtional question on the nature of the real numbers.
For Monday find some theorems that relay on the Mean Value
Theorem for their proof. Suggestion- proofs about
increasing/decreasing/ what if f'(x) = 0 for an interval/ The
fundamental theorem of calculus. [See Interactive real Analysis]and What is the point of the mean value
theorem?
Motivational Question I: What is a number?
Real numbers: Sensible Calculus Chapter 0 on numbers
See the following if you want to see a "simple" approach to defining real numbers: [This is a main theme for the course!] Vector
Calculus, Linear Algebra, and Differential
Forms: A Unified Approach by John
Hubbard and Barbara Burke Hubbard Appendix.A.1 Arithmetic of Real numbers.pdf
9-8
algebraic numbers = {z: z is a complex
number which is the root of a polynomial with integer coefficients}.
Functions:
A few more properties of Fields:
Suppose F is a field.
If a and b are in F and ab = 0 then either a
=0 or b=0.
Proof: Case 1. If a=0 then we are done.
Case 2. If a is not 0, then a has an inverse... c where
ca=1. Then b=1b= (ca)b= c(ab)
= c0 = 0.
Thus either a =0 or b=0. IRMC [I rest
my case.]
If a, b and c are in F, a+b = a+c
impliesb=c.
Proof: Let k be the element of the field where k + a
= 0 Then
b = 0 + b = (k +a)+b =k +(a+b)=
k +(a+c) =(k +a)+c = 0 + c =
c.
If a, b and c are in F with a not equal
to 0, ab = ac impliesb=c.
Proof: Similar to the last result.
Let n·1 stand for 1 + 1 + 1 + ... + 1 with n summands.
Either (i) for all n, n·1 is not 0 in which case we say the
field has "characteristic 0" , or (ii) for some n, n·1=0. In
the casen·1=0, there is a smallest n for which n·1=0,
in which case we say the field has "characteristic n". For example: R,
Q, and C all have characteristic 0, while Z_{2}, Z_{p},
where p is a prime number, and any finite field such as F_{4},
all have a non zero characteristic.
If the characteristic of F is not zero, then it is a prime number.
Proof: If n is not a prime, n = rs with 1<r,s<n. Let a =
1+1+...+1 r times and b= 1+1+...+1 s times. Then ab=n·1=0,
so either a = 0 or b = 0, contradicting the fact the n was
supposed to be the SMALLEST natural number for which n·1=0. IRMC.
9-22 25 AND 26 We covered
definitions and proofs of many results about limits of sequences and
continuity. In particular the Bolzano Weierstass Theorem, and the
intermediate value theorem. 10-2 Begin with proof of the extreme value theorem. The boundedness theorems
Discuss: Composition of continuous is continuous.
Definition of derivative.
Differentiable implies continuous.
Proof of product and chain rules.
Critical point Theorem. [this completes first proof of MVT!]
Examples of Some horrible functions for limits and continuity, including the mathematician's sine. 10-6 and 13 Begin discussion of integration and connection to continuity.
Motivation from Calc I definitions of the definite integral.
Various defintions and an example of a function that is not integrable.
10-20
Recall: Definition of an open set. U is open if for any `x in U` there is a `delta > 0` where N(x,`delta`) `sub` U.
Prop: (i) `O/` and `R`are open sets. (ii) If `a in R` then `(a, oo )` and `(- oo, a)` are open sets.
Fact: {a} is not an open set.
Prop: (i) If U and V are open sets, then `U nnn V` is an open set.
(ii) If O is a family of open sets, then `uuu O` = { `x in
R : x in U` for some `U in O`} is an open set.
Cor. : If `a < b` , then `(a,b)` is an open set.
The family of all open sets of R is sometimes called the topology of R.
Sidenote: A topological space is a set X together with a family of subsets `O` that satisfies the three properties:
(i) `O/ in O` and `X in O`. (ii) If `U , V in O` , then `U nnn V in O`. (iii)
If {`U_{alpha}`} is a family of sets with each `U_{alpha} in O`,
then `uuu U_{alpha}` = { `x in X : x in U_{alpha}` for some
`U_{alpha}` in the family} is `in O`.
Definition of a closed set. A set C is closed if (and only if) `R - C` is open.
Prop: (i) `O/` and `R` are closed sets. (ii) If `a in R` then `[a, oo )` and `(- oo, a]` are open sets.
Fact: {a} is a closed set.
Prop: (i) If U and V are closed sets, then `U uuu V` is a closed set. (ii)
If K is a family of closed sets, then `nnn K` = { `x in R : x
in C` for some `C in K`} is a closed set.
Cor. : If `a < b` , then `[a,b]` is a closed set.
Review definitions: Given f : R `->` R, and `U sub R`
The image of U under f , f (U) ={ y `in R` : y = f (x) for some `x in R`}and
the preimage of U under f, `f ^{-1}(U)` = {x ` in R` : f (x) `in U`}. Continuity and open sets
Theorem: Given f : R `->` R, f is continuous if and only if whenever U is an open subset of R, `f ^{-1}(U)` is also an open set.
10-23 (revised 10-24)
Connectedness , Continuity, and Topological Proof of the Intermediate Value Theorem. Def'n: A subset I of the R is an interval: If a, b `in` I and a < x< b, then x`in`I. Def'n: A subset S of (a metric space) R is disconnected if there are open sets U and V where
(i) U `uuu` V `sup` S; (ii) U `nnn` S `!=` `O/` , V `nnn` S `!=` `O/` , AND V `nnn` S `nnn`U `=` `O/` Def'n: A subset S of (a metric space) R is connected if it is not disconnected.
Thus If S is connected with open sets U and V where
(i) U `uuu` V `sup` S; and (ii) U `nnn` S `!=` `O/` , V `nnn` S `!=` `O/` , then V `nnn` S `nnn`U `!=` `O/`. Theorem: A subset of R is connected if and only if it is an interval. Connected and disconnected sets
If f : I `->` R is continuous and I is a connected then f (I) is connected.
Proof: (Corrected 10-24) Suppose f (I) is not connected. Then there exist U and V in R where
(i) U `uuu` V = f (I); and (ii) U `nnn` f (I)`!=` `O/` , V `nnn` f (I) `!=` `O/` , AND V `nnn` f (I) `nnn`U `=` `O/` . We will show this implies that I is not a connected set.
Since f is continuous for the domain I, there exists open sets U1 and V1 so that U1 `nnn` I = `f ^{-1}(U)` and V1 `nnn` I = `f ^{-1}(V)`.
Now
(i) U1 `uuu` V1 `sup` I; and (ii) U1 `nnn` I `!=` `O/` , V1 `nnn` I `!=` `O/` , AND U1 `nnn` I `nnn`V1 `=` `O/`. [why?]
But this contradicts the assumption that I was a connected set.
Heine Borel Theorem (simplest version): `[a,b]` is a compact subset of R.
Proof: (outline). Suppose O is a family of open sets where `uu O sup [a,b]`. Let G= {x `in` `[a,b]` where `[a,x]` is covered by a finite subset of the members of O.}. Claim ( "easy"): G `!=` `O/` and G is bounded by `a`. Then let `alpha` = lub of G.
Show `alpha in ` G and then `alpha = b`.
Theorem: A closed subset of a compact set is compact.
Corollary: If K is a closed and bounded subset of R, then K is compact.
Proof: K is bounded- so K `sub [a,b]` for some `a` and `b`. But by HB,
`[a,b]` is compact, so K is a closed subset of a compact set thus K is
compact.
Theorem: If K is a compact subset of R, then K is closed and bounded.
Proof: (i) K is bounded: Consider the family of open sets, O = {B(0,n)
for n = 1,2,3.... }. Certainly this family covers any subset of R, so
it covers K. But a finite subcover will be bounded by the largest n of
the sets in that subcover. Thus K is bounded.
(ii) K is closed. Let U = the complement of K = K^{C}. We show that U is open. Suppose p `in` U and x `in ` K. Then let r(x) = |x-p|/3 so r(x) > 0 and B(p,r(x)) `nnn`B(x,r(x)) = `O/`. Let O be the family of open sets = {B(x, r(x)) : x `in` K}. Then O covers K and because K is compact there a finite number of points x1, ... , xm where the family of open sets{B(xk, r(xk)) : k = 1,2,..., m} covers K. Then let r = min{r(xk) : k = 1,2,..., m} and we have that B(p,r) `nnn`B(xk,r(xk)) = `O/` for all k, and hence B(p,r) `nnn`K = `O/` Thus B(p,r) `sub` U and U is an open set. EOP.
Compactness and continuity. Theorem:If f : I `->` R is continuous and I is a compact then f (I) is compact.
Proof: To show f (I) is compact, suppose O is an open cover of f (I). Because f is continuous, for each U `in` O, there is an open set U* where U*`nnn` I = `f ^{-1}` (U).
Let O* = { U* : U `in` O }. Note that O* is open cover of I , and
since I is compact, O* has a finite subcover of I. For these
(finite) open sets we have corresponding U so that U*`nnn` I = `f ^{-1}` (U). Thus .... these U are a finite subcover of the family O.
Topological proof of the
Extreme Value Theorem.
BREAK! I have fallen behind in keeping these notes up to date.
Since the discussion of campactness we have been looking at integration.
We discussed both forms of the Fundamental Theorem of Calculus and proofs.
These proofs led us to a discussion of more basic theorems about the definite integral.
We went on an excursion into the concept of uniform continuity, proving
that a continuous function on a compact set of real numbers was
uniformly continuous. this allowed us to prove that any continuous
function on [a,b] was integrable.
We also demonstrated the additive property and monotonicity for the definite integral.
The sum and scalar multiple properties will be done in class by students in the near future. 11-13
Today we started an investigation of functions and comparing them to
numbers- in particular we introduced two distinct norms on functions:
the sup norm and the L2 norm.
These will be discussed later.
We turned our attention to sequences and series
reviewing some of the properties for convergence related primarily to
numbers. We noted how geometric series could be thought of as a
sequence of functions.
We reviewed and proved:
A nowhere differentiable function Metric spaces: definition and examples Convergence in a metric space Convergence in infinite dimensional spaces Properties of uniform convergence The Weierstrass approximation theorem 12-1
Given the integers as an ordered integral domain ("commutative ring
with unity and no zero divisors") we construct the rational numbers by
first considering P= {(a,b): a,b integers and b not 0} and the
equivalence relation on P where (a,b) ~ (c,d) [in P] if and only if ad=
bc.
We proved that as a relation on P, ~ is symmetric, reflexive, and
transitive. we let [a,b] = { (r,s) in P : (a,b) ~ (r,s) } and showd
this partitioned P, i.e., (i) every (a,b) in P is in [a,b], and (ii) if
[a,b] amd [c,d] have some element in common, then [a,b]= [c,d] -
as sets.
The rational numbers Q as a set = {[a,b]: a,b are integers and b is not 0.}
We then define operations on Q for addition and multiplication- show
these operations are "well defined" and that with them Q is a field.
Finally we characterized what the positive rational numbers are and from this defined an order relation on Q.
The result made Q an ordered field with a 1:1 function from Z to
Q defined by `a ->[a,1]` that preserves the order and algebraic
structures of Z. 12-4
Defining the real numbers:
we looked at ways we describe real numbers- eventually leading to a
discussion of the decimal notation and its connection to infinite
series using powers of 10 which converge by comparison with geometric
series.
This led to a discussion of other sequences that characterize real
numbers- in particular e as the limit of the sums from the Taylor
series `sum 1/(n!)` and the limit of the powers - `(1+1/n)^n`.
To eliminate the assumptions of a limit existing for these sequences of
rational numbers, we can use the theory that characterizes convergence
with the Cauchy condition. This is the start of a more general
definition of a real number using "cauchy sequences" of rational
numbers.
We let R = { S={`a_n`} : where `a_n` is a rational number for each n and S is a Cauchy sequence }
Now we establish an equivalence relation on R , namely S~T [T = {`b_n`}
in R] if for any positive rational number `epsilon` there is a natural
number M where for any k >N, `|a_k - b_k| < epsilon`.
[Thus we indicated why ~ is a reflexive, symmetric, and transitive relation.]
This leads to the partition of R into equivalence classes [S]= {T in R:
S~T}. `R` = {[S]: S is in R} can then be used to define
operations and an order relation [ inherited from Q] so that `R`
is an ordered field which satisfies the least upper bound property.
Thus `R` is a complete ordered field- "the real numbers." 12-8
More on the definition of the real numbers:
Review:Axioms for the Real numbers- Construction of the real numbers- revisited on line- including the proof of the least upper bound property. ["Close to what we outlined last class.]