Indian Institute of Technology Kanpur

Department of Mathematics and Statistics

Applications of single variable calculus and several variable calculus (MTH 112)
First Course Handout
Semester 2022-23-I
1. Syllabus
(1) Application of definite integral, Area between two curves, Polar coordinates, Graphs of polar
coordinates.
(2) Area between two curves when their equations are given in polar coordinates, Volumes by
slicing, Volumes by Shells and Washers.
(3) Length of a curve, Area of surface of revolution, Pappus’s theorem, Review of vector algebra,
Equations of lines and planes, Continuity and Differentiability of vector functions.
(4) Arc length for space curves, Unit tangent vector, Unit normal and curvature to plane and
space curves, Binormal, Functions of several variables, Continuity, Partial derivatives, Differ-
entiability, Differentiability implies continuity, Chain rule, Gradient, Directional derivatives.
(5) Tangent plane and Normal line, Mixed derivative theorem, Mean value theorem, Minima and
Saddle point, Necessary and sufficient conditions for Maxima, minima and Saddle point, The
method of Lagrange multipliers.
(6) Double Integral, Fubini’s theorem, Volumes and Areas, Change of variable in double integral.
Special cases: Polar coordinates, Triple integral, Applications, Change of variable in triple
integral. Special cases: Cylindrical and Spherical coordinates.
(7) Surface area, Surface integral, Line integrals, Green’s theorem, Vector fields Divergence and
Curl of a vector field, Stokes theorem, Gauss divergence theorem.

2. Recommended texts
(1) Calculus (Volume I) by Tom. M. Apostol, John Wiley & Sons.
(2) Thomas’s Calculus (12th edition)

3. Attendance policy
(1) A registered student has to maintain at least 75% attendance in both the lectures and tutorials.
(2) Students found not satisfying the attendance criteria mentioned in 1 either in the lectures
or tutorials till the last date of de-registration, given by the Academic Calendar, will be
de-registered by the I/C.
(3) If a student maintains 75% attendance in both the lectures and tutorials till the last date of
de-registration but overall it falls short when counted in the last day of classes, he/she will be
awarded an F grade in the final evaluation.
(4) An exception to the above mentioned criteria can only be made if the student can provide
convincing evidences showing that he/she missed the lecture(s) due to bona fide reasons, and
furthermore, he/she has sanctioned leave for that. The final decision on any such case will
be taken by the instructors.

4. Evaluation policy
First and foremost, the RELATIVE grading system will be adopted for the evaluation of students
in this course. The final grading for this semester will be based on the three parameters as follows:
(1) Performance in the End-Sem. exam.: There will be an End-Semester examination at
the end of this course. Note that this examination is likely to be descriptive in nature, i.e.,
one requires to provide arguments with all relevant details. This examination will be carrying
60% weighatge in the final evaluation.
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(2) Performances in the quiz: A quiz will be conducted during the course. They will carry
40% weightage in the final evaluation.

5. Classroom Decorum
(1) Students have to come to the lectures and tutorials ON TIME. If a student is found to arrive
late in a lecture or tutorial, the concerned instructor or tutor respectively has the authority not
to permit him/her to attend that session. Consequently, he/she will be regarded as ABSENT
for that session.
(2) Once a lecture/tutorial session begins, no student will be allowed to leave it without compelling
reasons. In case of any emergency, the permission from the concerned instructor/tutor must
be taken.
(3) In order to gain the maximum benefit of the discussion, silence must be maintained for the
entire duration of any lecture/tutorial session.
(4) It is expected that a student will behave politely with the instructor/tutor and fellow students
during a lecture/tutorial.
(5) Any student found deliberately breaching the classroom decorum or disturbing others while
the session is in progress, will invite disciplinary actions. Repeat offenders may be punished
with a lower grade.

6. Dropping of the course

A student can apply for dropping the course through Pingala, by the deadline mentioned in the
academic calendar, if he/she thinks that his/her performance has been unsatisfactory in spite of
putting every effort and working hard. This drop request requires the instructor’s consent. In fact,
the instructors may withhold their consent if they feel that the student did not put enough effort in
the course and has, therefore, not ‘earned’ the DROP. Note that, as this is an instructor’s prerogative,
he does not owe any explanation to any student regarding that.

7. Violation of the Student Code of Conduct and Disciplinary Action

As per Ordinance 9 of Section 28, Institutes of Technology Act, 1961, entitled “Student
Conduct and Discipline”, lack of courtesy and decorum; unbecoming conduct, breach of rules and
regulations of the Institute; noisy and unseemly behaviour and similar other undesirable activities
shall constitute violation of the Student Code of Conduct and shall invite disciplinary action and
may merit punishment. Indeed, Ordinance 9.5 says the following:

“The Instructor-in-Charge of a course shall have the power to debar a student from the examina-
tion and/or fail the student in the course in which the student is detected to have indulged in any
act of academic dishonesty. The Instructor/Tutor shall have the power to take appropriate action
against a student who attempts to misbehave in the class.

Use of unfair means in any form, including plagiarism in academic work will constitute violation
of the Student Code of Conduct.”

We shall adopt ‘zero tolerance’ policy against any violation of the Student Code of Conduct through-
out the course. In reaction to any such instance of violation of the above mentioned code of conduct,
the instructors may award the concerned student a lower grade/fail/deregister from the course on
disciplinary grounds. Such a student will not be permitted to drop the course. The record of such
punishment will be preserved by the instructor and as well as by the Academic Section, as per the
recommendation made by the Agenda item 9 of Senate/2010-11/8th minutes, 17/03/2011. Note
that, if the need be, the instructor is also recommended to refer the case to SSAC.
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8. A few words to the registered students

(1) The Department of Mathematics and Statistics, IIT Kanpur, has a great legacy of producing
internationally acclaimed mathematicians. Indeed, many of our country’s leading mathemati-
cians had been students of our department. Even now, we often see some of our students
joining top institutions of India and abroad to pursue their doctoral research. So we are
committed to maintain a good academic standard of our programmes.
(2) You should work hard and strive to learn the subject. Focus must be on understanding the
concepts; put your all efforts to follow every detail.
(3) Solve as many problems as you can. You may try to solve exercises given in the list of
textbooks suggested in 2. Some exercise sets will also be provided to you during the course.
These exercise sets will contain some problems which you may find somewhat challenging.
Undoubtedly solving more and more problems not only enhance your problem solving skill,
it will help you to develop strong intuition in the subject as well. You are also encouraged to
collaborate with your fellow students to solve exercises.
(4) Do attend the tutorials and discussion sessions and take active part in those sessions. Feel free
to tell the instructors and tutors your all doubts, difficulties, questions etc. If the need be and
the students are interested, additional discussion sessions also can be arranged. You can also
request the instructors for an appointment if you wish to discuss further on some topics. It is
the responsibility of the instructors to put every effort to make the students understand and
learn the subject, so you should take the advantage of that. Furthermore, you must engage
yourself in peer discussions.
(5) Study on a regular basis. There is a trend among some students not to study regularly but
only before the examinations. It may not result in a good performance in the examinations,
as a good understanding of the relevant concepts might be necessary to solve the questions
that will be asked in the examinations. So unless you develop that by putting constant efforts,
solving problems etc., it can be difficult to make much progress during the examination.
(6) Your answers must be furnished with all details. Since writing is an important part of Mathe-
matics education, a lot of emphasis will be given in the evaluation on how you write an answer.
The answers have to be necessarily precise and concise. Note that, vagueness, loose bunch
of words have no place in Mathematics; there will be no compromise with the mathematical
rigour.
(7) Do not suggest the instructors how to run the course, evaluation policy, difficulty level of the
question papers etc. It is certainly unprofessional and against the decorum. You can share
your opinion with the instructors only when it has been solicited.
(8) The final grade will be awarded to you based on your performance only. Any attempt that has
been made to influence the instructors to award a certain grade to a student, by any means, will
be considered as a serious breach of the Student Code of Conduct and appropriate disciplinary
actions will be taken.

As the instructor of a course is the sole authority to decide the course policy, it is completely at the
discretion of him/her to modify/revise that whenever deemed necessary. You will be conveyed at once
if there is any change in §1- 8 later on.