CMI (BSc II)/BVRao
Analysis, Notes 2
2014 Second week
Theorem (Cantors Characterization of Q): Let (X, ) be a countable linearly ordered set (non-empty) which has no first element; no last
element; between any two distinct elements there is another elemen
CMI (BSc II)/BVRao
Analysis, Notes 1
2014 First week
where are we:
So far we learnt some basic properties of real numbers,
convergence of sequences (helped to approximate unknown numbers
with known fractions; discovered new numbers like Eulers constant, c
CMI (BSc II)/BVRao
Analysis, Notes 3
2014 Third week
I have not given any references. You can consult any book on set theory
and read the topics we are discussing like Halmos, Naive Set Theory. One of
the best sources is the book Real Analysis by Hewitt a
CMI (BSc II)/BVRao
Analysis, Notes 13
2014
Vocabulary:
I withheld some technical names so that you can familiarize and appreciate the concept and meaning before the term is introduced. But it is time to
learn the terms so that you can follow literature (a
CMI (BSc II)/BVRao
Analysis, Notes 6
2014
Holder, Minkowski:
Let us first show that the dp example of last time does indeed satisfy the
triangle inequality. In what follows we have two numbers p, q > 0 such that
1 1
+ = 1.
p q
()
Of course this already im
CMI (BSc II)/BVRao
Analysis, Notes 7
2014
Cantor Construction of Reals:
So far we have:
R0 , the set of Cauchy sequences of rational numbers.
R the set of equivalence classes, where two Cauchy sequences x, y are
equivalent if d(xn , yn ) = |xn yn | 0.
CMI (BSc II)/BVRao
Analysis, Notes 4
2014 Fourth week
Understanding R:
If you go close to the real line you see that you can add and multiply
numbers. But in a superficial look at the line, you only see what comes
after what. We shall now isolate some ord
CMI (BSc II)/BVRao
Analysis, Notes 8
2014
Cantor intersection theorem:
In the real line we have shown that a decreasing sequence of closed nonempty intervals with diameter converging to zero have a point in common.
We shall now generalize this result to m
CMI (BSc II)/BVRao
Analysis, Notes 9
2014
completion continued:
We have a metric space (X, d). We considered the space X 1 of Cauchy
sequences (xn ). Defined an equivalence relation (xn ) (yn ) if they appear to be converging to the same point; more preci
CMI (BSc II)/BVRao
Analysis, Notes 12
2014
Home Assignment:
Given a compact set K contained in an open set U , show an r > 0 such
that
[
B(x, r) U.
xK
Solution : Suppose not. then for every n there are points xn K and
yn U c such that d(xn , yn ) < 1/n. S
CMI (BSc II)/BVRao
Analysis, Notes 15
2014
Plancherel:
We have X = C[0, 1], complex valued continuous functions on [0, 1] with
metric
s
d(f, g) =
Z 1
|f g|2
0
We have l2 , space of (two sided) infinite sequences which are square summable
with metric
r
X
d
CMI (BSc II)/BVRao
Analysis, Notes 10
2014
Banach Contraction mapping principle:
Let (X, d) be a complete metric space. suppose T : X X is a contraction map., that is, there is a number c; 0 c < 1 such that d(T x, T y)
cd(x, y) for all points x, y. Thus
CMI (BSc II)/BVRao
Analysis, Notes 5
2014
R from Q:
We are now in the process of constructing real number system using Q,
the set of rational numbers. A cut x is a non-empty proper subset of Q such
that for any of its elements, everything below is also th
CMI (BSc II)/BVRao
Analysis, Notes 14
2014
what next:
We completed our discussion of metric spaces. We have done some basic
results and applications. There are two possibilities for the remaining period
we can discuss either power series with complex coe
CMI (BSc II)/BVRao
Analysis, Notes 11
2014
Continuous functions:
Let (X, d) and (Y, ) be metric spaces and f : X Y be a function and
a X. We say that f is continuous at a if the following is true: xn a
in X implies f (xn ) f (a) in Y . We say f is continu
CMI (BSc I)/BVRao
Calculus II, Notes3
2014 Third week
A chain rule:
let f : R be a C 1 function. Here R2 is an open set. Suppose
1 and 2 are two real C 1 functions defined on an interval (a, b) such that
for every t, the point (1 (t), 2 (t) . Then it make
CMI (BSc I)/BVRao
Calculus II, Notes14
2014
holomorphic functions:
We have seen last time that if f : C C is complex differentiable, then
Cauchy-Riemann equations are satisfied by the real and imaginary parts of
the function. We shall now show that conver
CMI (BSc I)/BVRao
Calculus II, Notes6
2014 sixth week
unbounded intervals/functions:
We discussed functions of one variable obtained from two variable functions by performing integration etc. We also showed that derivative of integral equals integral of d
CMI (BSc I)/BVRao
Calculus II, Notes 13
2014
an estimation problem:
I have a die, biased. I do not know the chances of the faces appearing in
a throw. I roll the die n times and observe that face i appeared ni times for
P
1 i 6. of course, ni = n. let us
CMI (BSc I)/BVRao
Calculus II, Notes5
2014 Fifth week
We shall prove the implicit function theorem. Recall
Theorem (Implicit function theorem)
Let f : R2 R be a C 1 function. Suppose (a, b) Suppose
f (a, b) = 0 and f2 (a, b) 6= 0. Then
(i) there is a rect
CMI (BSc I)/BVRao
Calculus II, Notes
2014 First week
Prelude:
Last semester we understood some aspects of the set R of real numbers
- rational numbers, irrational numbers, sequences, their convergence, series, absolute convergence, products of series and
CMI (BSc I)/BVRao
Calculus II, Notes7
2014 seventh week
inverse function theorem:
We proved the inverse function theorem assuming that f 0 (a) = I. We
shall now deduce the general case. We start with some auxiliary observations.
Let A be a 2 2 matrix and
CMI (BSc I)/BVRao
Calculus II, Notes8
2014 eighth week
integrability:
we shall now prove the following result:
Q is a rectangle [a, b] [c, d] and f : Q R is a bounded function whose
set of discontinuity points is a small set.
Then f is integrable.
Let > 0
CMI (BSc I)/BVRao
Calculus II, Notes15
2014
We shall complete our discussion on uniform convergence of integrals and
close the chapter. I hope you have got a feeling for the applications I have
outlined last time, namely, to evaluate the characteristic fu
CMI (BSc I)/BVRao
Calculus II, Notes 12
2014
polar coordinates:
Every point (x, y) R2 other than (0, 0) can be uniquely expressed as
x = r cos ; y = r sin for some (r, ) with 0 < r < and 0 < 2. these
(r, ) are called polar coordinates of the cartesian poi
CMI (BSc I)/BVRao
Calculus II, Notes 10
2014
Normal integral:
The following integral appears in several contexts.
I=
Z
2 /2
et
dt.
We can calculate this integral by using some tricks. But the simplest is to
valculate
Z
2
2
2
I =
e(x +y )/2 dxdy.
R2
Note
CMI (BSc I)/BVRao
Calculus II, Notes4
2014 Fourth week
Taylor:
The chain rule will now be applied to derive Taylor formula for function
of several variables. This will be exactly same as the one you learnt last
semester. There is absolutely no change. Fir
CMI (BSc I)/BVRao
Calculus II, Notes 11
2014
Normal integral again:
Here is a tricky way of calculating normal integral. Let us put
an =
Z
2 /2
et
tn dt;
n = 0, 1, 2, .
0
We do not know a0 but can explain all others using it. Integration by
parts gives
a
CMI (BSc I)/BVRao
Calculus II, Notes 9
2014
We have learnt the method of substitution for integrating functions of one
variable. We are now in the process of getting an analogue of that method
for functions of two variables. This goes by the name of chang
CMI (BSc I)/BVRao
Calculus II, Notes2
2014 second week
Using our expertise with functions of one variable, we have defined partial
derivatives; rate of change in the two directions: horizontal and vertical or
equivalently in the x-direction and y-directio