Multivariable calculus (Spring 2026)

Course code: MT1213

Course Title: Multivariable Calculus

Credits: 3

Course Coordinators: Anup Biswas and Anindya Goswami

Pre-requisites: Single Variable Calculus 

Objectives (goals, type of students for whom useful, outcome, etc): This course is aimed at providing a concise introduction to the calculus of the vector-valued functions of several variables. It is useful for the students who want to study topics like  Differential Geometry, Topology, Analysis, Lie groups,  Differential Equations, Theory of Relativity, Quantum Mechanics, Mathematical Biology, etc.  The goal is to get the students acquainted with the basic notions of partial and directional derivatives, multiple integrals,  line and surface integrals of functions of several variables. 

Course contents: Euclidean space, Pythagoras' Theorem,  Distance between two points, F unctions on  Rn , Limit and continuity of functions on  Rn , Sufficient conditions for continuity, Partial and directional derivatives, Total derivative,  Sufficient conditions for differentiability, Higher order partial derivatives, Sufficient condition for change of order of differentiation, Equation of tangent plane to a differentiable surface, Chain rule, Jacobian Matrix, Taylor's Theorem, Implicit function theorem, Inverse function theorem, Local optima, Lagrange multiplier, Curves, Line integrals, Regular domain,  Multiple integrals, Fubini's theorem, Differentiation under the integral sign, Green's theorem, Path independent integral,  Change of variables in double integral,  Surface integral of Scalar and Vector field over a smooth surface.

Number of lectures: 29 (excluding tutorials), 10 tutorials

Evaluation:
End-sem examination 35%
Mid-sem examination 35%
Quiz 30%

Text-book:

Calculus Vol. II: Multivariable calculus and linear algebra with applications to differential equations and probability by Tom Apostol (John Wiley and  Sons, 2005). This book has 3 parts. Part 2 is relevant for this course.

Mathematical Analysis by S. C. Malik and Savita Arora (New Age International P.Ltd., 1992). Only Chapters 15, 16, 17, and 18 are  relevant for this course.

Calculus of Several Variables by Serge Lang (Addison-Wesley 1973) E-COPY

Multivariable Calculus by James Stewart Link

Teaching Assistants:

1. Mr. Aakash Coudhary

2. Mr. Abirranga Datta

3. Mr. Aniket Sen 

4. Mr. Niranjan Biswas

5. Ms. Nishtha Sharma

6. Mr. Sandip Mandal 

7. Mr. Mohd. Sharukh

8. Mr. Sooraj M 

9. Ms. Srijonee Chaudhury

Lectures by Anup Biswas

https://drive.google.com/drive/folders/1nk1iKvHOckYetIowlbZFGb3jsG-k2-H2?usp=drive_link 

Lectures by Anindya Goswami

1. 02/03  Curves and their types [01.pdf]

2. 09/03 Notion of Line integral of a scalar field, and notion of arc length. [01.pdf]
   10/03 Tutorial on Assignment 5

3. 12/03 Existence of Line integral wrt arc length and its calculation. Illustration of line integral of a scalar function with two examples. [01.pdf]
   14/03 Tutorial on Assignments 5, 6

4. 16/03 Line integral of a vector field. [02.pdf]
   17/03 Tutorial on Assignment 7, Quiz 4 (on Assignment 5)

5. 19/03 Illustration of the line integral of a vector field with one example. [02.pdf]

6. 23/03 Region, boundary of a region, regular region. Multiple integral. [03.pdf]
   24/03 Quiz 5 (on Assignments 6, 7)

7. 26/03 Integrability of a scalar field, illustration with an example and a non-example, Reduction of double integral into repeated integral. Fubini's Theorem. [03.pdf] 

8. 30/03 Sufficient conditions for integrability. Integration on a regular region.  Example of double integration on a non-regular region. Differentiation under the sign of integral. [.pdf]
   31/03 Holiday

9. 02/04  Green's Theorem, path independent integrals. [.pdf]

10. 06/04  Change of variables in double integral. [.pdf]
    07/04 Tutorial on Assignment 8

11. 09/04 Area formula of a differentiable surface. [.pdf] 

12. 13/04  Surface integral of Scalar and Vector field over a smooth surface. [.pdf]
    14/04 Quiz 6 (on Assignment 8)

13. 16/04 Example of calculation of surface integrals. [.pdf]

Exams

Result