(Calculus of one and several variables)
  Release: Alpha Version, October 2003

    © Prof. Inder K .Rana
    Department Of Mathematics
    Indian Institute Of Technology Bombay
    Mumbai - 400 076, India



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:

  •  The 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.