The first half of our course, Mathematics-I, concerns itself Calculus or Real Analysis. The second half of the
course is ‘Multi-variable Calculus’. An introduction to the former follows.
To better appreciate this course you need an exposure to the history of Calculus. The video https://www.
youtube.com/watch?v=ObPg3ki9GOI is a start. The book https://www.springer.com/in/book/9780387945514
is more comprehensive. The student is urged to read about history of Calculus on the Web. The roots of this
topic lie with the two problems of calculating tangents to curves and areas under curves. By calulation, we mean
an exact answer, not an approximation. The former is a few hundred years old while the latter predates the birth
of Christ. These problems, at least during those times, were of little practical significance.
As you may be aware, mechanics is the study of motion of objects initiated a few hundred years before Christ.
Even from that era we do have observational data of the motion of planets for instance. However, when it came
to theoretical aspects, it languished as a speculative nonsense for more than a millenia. Following the rationalistic
revival initiated by Galileo, Newton gave his second law of motion using the concept of derivative. The concept
of derivative was inspired by the solution to the problem of calculating tangents. It is of interest to note that
the first rational hypothesis on natural phenomenon is acknowledged to be Newton’s second law of motion. This
marks the beginning of true theoretical science in human history.
Having said that, the present day reliance of other topics from Physics like Electricity, Thermodynamics, etc.
or subjects like Chemistry, Economics etc. on Calculus is near total. And with Engineering having emerged as an
application of basic sciences, this course is compulsory for anyone who wants a degree in Engineering or Science.
Admittedly, the need for Calculus to a Computer Engineer or a Biotech Engineer is questionable. But, Tradition,
which made Calculus a requirement for all Engineering has been slow to mend its ways.
After the initial development of Calculus, mathematicians who came later like Cauchy, Wierstrass, Dedekind,
etc. have taken up the cause of providing a more solid footing. Calculus with an emphasis for rigor is referred to
as Real Analysis. Real Analysis aims to do to Calculus what Euclidean Geometry did to the body of knowledge
developed by sketches on sand.
The main topics in this course are as follows. To name all points on a straight line, or all times or even mass,
rationals are inadequate and we need at least the set of Real Numbers. The arithmetic operations on this set are
noted. Classical treatments of a series, i.e., an infinite sum of real numbers are ill-defined. Making sense of such
a sum leads to the concept of sequences and a definition for their convergence.
Real Analysis also studies functions from the set of reals to itself. Inspired by the definition of sequence con-
vergence, a definition for derivative is given. Physicists were content imagining this as the infinitesimal difference
quotient.
It was known early that the derivative to such functions need not exist – witness examples of functions
whose graphs have a break. This lead to a definition of continuity which although necessary is not sufficient for
differentiability.
One might reverse the process of finding derivative and leave it as anti-differentiation with little practical use.
Recall the second problem from antiquity of calculating areas. This was recognized to be a limit of an infinite
sum. One could calculate such limits for few functions. But a remarkable discovery by Gregory is that such area
calculations are given exactly by evaluating the function’s anti-derivative at end points. This is the fundamental
theorem of calculus connecting two definitions separated by a millenium which is taught as a dumb formula in
most courses.
Most of you will see applications of this course in many of your later courses which are part of your curriculum.
There are many books on this topic written with different viewpoints. Calculus traces some of its roots as
a tool to understand nature. A Newton or a Gregory who gave us such an understanding might consider a text
like the one by Thomas and Finney adequate for Engineers and Scientists. However, the flames of rigor have
quenched such an understanding in every modern day instructor. Consequently, the latter is inclined to choose
books discussing more of Real Analysis than Calculus. You may look at books by Spivak, Victor Bryant, Bartle
and Sherbert and Ghorpade and Limaye.
You should study Chapters 1-8, skipping 8 from BaSh(Bartle and Sherbert). I bet you can pick the sections
if you attend lectures.
MA101 MATHEMATICS I Syllabus
Prerequisite: Nil L-T-P-C: 3-1-0-8
Single variable calculus: Convergence of sequences and series of real numbers; Continuity of functions;
Differentiability, Rolle’s theorem, mean value theorem, Taylor’s theorem; Power series; Riemann inte-
gration, fundamental theorem of calculus, improper integrals; Application to length, area, volume and
surface area of revolution.
