This course aims at providing
the student with basic the concepts of calculus (of one and several
variables) which find application in Physics, Chemistry,
and various disciplines in Engineering Sciences. Our emphasis
will be more on the logical developments of the concepts of calculus
in a rigorous fashion. Of course, applications will form an integral
part of the course.
Historically, Integral Calculus developed much before
the Differential Calculus (though, while teaching Differential
Calculus proceeds Integral Calculus). The notation of areas/volumes
of objects attracted the attention of the Greek mathematicians
like Antiphon (430 B.C.), Euclid (300 B.C.) and Archimedes (987
B.C.-212 B.C.). Later, in the 17th century, mathematicians were
faced with various problems: in mechanics, the problem of describing
the motion of a practice mathematically; in optics the need to
analyze the passage of light through a lens, which gave rise to
the problem of defining tangent/normal to a surface; in astronomy,
it was important to know when would and planet be at a maximum/minimum
distance from earth; and so on. These problem along with the problem
of computing areas/volumes gave rise to the subject which we call
today Calculus. The foundation of this was laid by the works of Isaac
Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716),independently.
However, their work made many assumptions which were analyzed
later by many mathematicians. They realized that the process of
giving logical and rigorous proofs to results of calculus, one
has to understand the following concepts:
concept of a real number.
The concept of a function.
The concept of limit.
In this course,
we shall try to clarify these concepts, and then go on to develop
the results of calculus and give their applications. For more
about history, if you have time and feel interested, read the
book by C. H. Edwards, Jr.:"Historical Development of the
Calculus", published by Springer-Verlag.