I'll try here to post some of my lectures and solved examples. In near future, I will be posting lectures which would be suitable for undergraduate and graduate students in the following areas:
Reaction Engineering.
Optimization and Optimal Control.
Numerical Method.
I will maintain this section of my personal webpage as an educational center so keep on visiting as you will surely find some materials satisfying your curiosity.
Undergraduate Course Materials:
CHE-201: Introduction to Chemical Engineering
Lecture Notes:
CHAPTER 2: Introduction to Chemical Engineering Calculation
CHAPTER 3: Processes and Process Variables
CHAPTER 5: Single Phase System
Solved Computer Problems:
Application of Bubble Point Calculation Using Newton's Method
Lev Semenovich Pontryagin
(FATHER OF CALCULUS OF VARIATION )
Lev Semenovich Pontryagin's father, Semen Akimovich
Pontryagin was a civil servant. Pontryagin's mother, Tat'yana Andreevna
Pontryagina, was 29 years old when he was born and she was a remarkable
woman who played a crucial role in his path to becoming a mathematician.
Perhaps the description of 'civil servant', although accurate, gives the
wrong impression that the family were reasonably well off. In fact Semen
Akimovich's job left the family without enough money to allow them to give
their son a good education and Tat'yana Andreevna worked using her sewing
skills to help out the family finances.
Pontryagin attended the town school where the standard of education was well
below that of the better schools but the family's poor circumstances put
these well out of reach financially. At the age of 14 years Pontryagin
suffered an accident and an explosion left him blind. This might have meant
an end to his education and career but his mother had other ideas and
devoted herself to help him succeed despite the almost impossible
difficulties of being blind. The help that she gave Pontryagin is described
in [1] and [2]:-
From this moment Tat'yana Andreevna assumed complete responsibility for
ministering to the needs of her son in all aspects of his life. In spite of
the great difficulties with which she had to contend, she was so successful
in her self-appointed task that she truly deserves the gratitude ... of
science throughout the world. For many years she worked, in effect, as
Pontryagin's secretary, reading scientific works aloud to him, writing in
the formulas in his manuscripts, correcting his work and so on. In order to
do this she had, in particular, to learn to read foreign languages. Tat'yana
Andreevna helped Pontryagin in all other respects, seeing to his needs and
taking very great care of him.
It is not unreasonable to pause for a moment and think about how Tat'yana
Andreevna, with no mathematical training or knowledge, made by her
determination and extreme efforts a major contribution to mathematics by
allowing Pontryagin to become a mathematician against all the odds. There
must be many other non-mathematicians, perhaps many of whom are unrecorded
by history, who have also by their unselfish acts allowed mathematics to
flourish. As we try to show in this archive, the development of mathematics
depends on a wide number of influences other than the talents of the
mathematicians themselves: political influences, economic influences, social
influences, and the acts of non-mathematicians like Tat'yana Andreevna.
But how does one read a mathematics paper without knowing any mathematics?
Of course it is full of mysterious symbols and Tat'yana Andreevna, not
knowing their mathematical meaning or name, could only describe them by
their appearance. For example an intersection sign became a 'tails down'
while a union symbol became a 'tails up'. If she read 'A tails right B' then
Pontryagin knew that A was a subset of B!
Pontryagin entered the University of Moscow in 1925 and it quickly became
apparent to his lecturers that he was an exceptional student. Of course that
a blind student who could not make notes yet was able to remember the most
complicated manipulations with symbols was in itself truly remarkable. Even
more remarkable was the fact that Pontryagin could 'see' (if you will excuse
the bad pun) far more clearly than any of his fellow students the depth of
meaning in the topics presented to him. Of the advanced courses he took,
Pontryagin felt less happy with Khinchin's analysis course but he took a
special liking to Aleksandrov's courses. Pontryagin was strongly influenced
by Aleksandrov and the direction of Aleksandrov's research was to determine
the area of Pontryagin's work for many years. However this was as much to do
with Aleksandrov himself as with his mathematics ([1] and [2]):-
Aleksandrov's personal charm, his attention and helpfulness influenced the
formation of Pontryagin's scientific interests to a remarkable extent, as
much in fact as the personal abilities and inclinations of the young scholar
himself.
The year 1927 was the year of the death of Pontryagin's father. By 1927,
although he was still only 19 years old, Pontryagin had begun to produce
important results on the Alexander duality theorem. His main tool was to use
link numbers which had been introduced by Brouwer and, by 1932, he had
produced the most significant of these duality results when he proved the
duality between the homology groups of bounded closed sets in Euclidean
space and the homology groups in the complement of the space.
Pontryagin graduated from the University of Moscow in 1929 and was appointed
to the Mechanics and Mathematics Faculty. In 1934 he became a member of the
Steklov Institute and in 1935 he became head of the Department of Topology
and Functional Analysis at the Institute.
Pontryagin worked on problems in topology and algebra. In fact his own
description of this area that he worked on was:-
... problems where these two domains of mathematics come together.
The significance of this work of Pontryagin on duality ([1] and [2]):-
... lies not merely in its effect on the further development of topology; of
equal significance is the fact that his theorem enabled him to construct a
general theory of characters for commutative topological groups. This
theory, historically the first really exceptional achievement in a new
branch of mathematics, that of topological algebra, was one of the most
fundamental advances in the whole of mathematics during the present
century...
One of the 23 problems posed by Hilbert in 1900 was to prove his conjecture
that any locally Euclidean topological group can be given the structure of
an analytic manifold so as to become a Lie group. This became known as
Hilbert's Fifth Problem. In 1929 von Neumann, using integration on general
compact groups which he had introduced, was able to solve Hilbert's Fifth
Problem for compact groups. In 1934 Pontryagin was able to prove Hilbert's
Fifth Problem for abelian groups using the theory of characters on locally
compact abelian groups which he had introduced.
Among Pontryagin's most important books on the above topics is topological
groups (1938). The authors of [1] and [2] rightly assert:-
This book belongs to that rare category of mathematical works that can truly
be called classical - book which retain their significance for decades and
exert a formative influence on the scientific outlook of whole generations
of mathematicians.
In 1934 Cartan visited Moscow and lectured in the Mechanics and Mathematics
Faculty. Pontryagin attended Cartan's lecture which was in French but
Pontryagin did not understand French so he listened to a whispered
translation by Nina Bari who sat beside him. Cartan's lecture was based
around the problem of calculating the homology groups of the classical
compact Lie groups. Cartan had some ideas how this might be achieved and he
explained these in the lecture but, the following year, Pontryagin was able
to solve the problem completely using a totally different approach to the
one suggested by Cartan. In fact Pontryagin used ideas introduced by Morse
on equipotential surfaces.
Pontryagin's name is attached to many mathematical concepts. The essential
tool of cobordism theory is the Pontryagin-Thom construction. A fundamental
theorem concerning characteristic classes of a manifold deals with special
classes called the Pontryagin characteristic class of the manifold. One of
the main problems of characteristic classes was not solved until Sergei
Novikov proved their topological invariance.
In 1952 Pontryagin changed the direction of his research completely. He
began to study applied mathematics problems, in particular studying
differential equations and control theory. In fact this change of direction
was not quite as sudden as it appeared. From the 1930s Pontryagin had been
friendly with the physicist A A Andronov and had regularly discussed with
him problems in the theory of oscillations and the theory of automatic
control on which Andronov was working. He published a paper with Andronov on
dynamical systems in 1932 but the big shift in Pontryagin's work in 1952
occurred around the time of Andronov's death.
In 1961 he published The Mathematical Theory of Optimal Processes with his
students V G Boltyanskii, R V Gamrelidze and E F Mishchenko. The following
year an English translation appeared and, also in 1962, Pontryagin received
the Lenin prize for his book. He then produced a series of papers on
differential games which extends his work on control theory. Pontryagin's
work in control theory is discussed in the historical survey [3].
Another book by Pontryagin Ordinary differential equations appeared in
English translation, also in 1962.
Pontryagin received many honors for his work. He was elected to the Academy
of Sciences in 1939, becoming a full member in 1959. In 1941 he was of one
the first recipients of the Stalin prizes (later called the State Prizes).
He was honored in 1970 by being elected Vice-President of the International
Mathematical Union.
Article by: J J O'Connor and E F Robertson
References:
P S Aleksandrov, V G Boltyanskii, R V Gamkrelidze and E F Mishchenko, Lev Semenovich Pontryagin (on his sixtieth birthday) (Russian), Uspekhi Mat. Nauk 23 (6) (1968), 187-196.
P S Aleksandrov, V G Boltyanskii, R V Gamkrelidze and E F Mishchenko, Lev Semenovich Pontryagin (on his sixtieth birthday), Russian Math. Surveys 23 (6) (1968), 143-152.
E J McShane, The Calculus of Variations from the beginning through Optimal Control Theory, SIAM Journal on Control and Optimization 27 (5) (1989), 916-939.
Undergraduate Course Materials:
CHE-402: Kinetics and Reactor Design
Lecture Notes:
The remaining materials are under preparation, and will be uploaded once they are done.