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Multivariate Calculus

By Dr. Bijumon Ramalayathil   |   Mahatma Gandhi College, Iritty, Keezhur PO , Kerala – 670 703.
Learners enrolled: 104
The course “Multivariate Calculus” is proposed for B.Sc. (Hons) Mathematics as the Core course with 5 credits.  The course starts with description of functions of several variables.   The course is also beneficial for all students of Physics, Chemistry, Computer Science and Information Technology.  After the successful completion of the course the students will be familiar with the following concepts:
       Limit and continuity of functions of two variables,  Partial differentiation, total differentiability and differentiability, sufficient condition for differentiability. Chain rule for one and two independent parameters, directional derivatives, the gradient, maximal and normal property of the gradient, tangent planes, Extrema of functions of two variables, method of Lagrange multipliers, constrained optimization problems, Definition of vector field, divergence and curl, Double integration over rectangular region, double integration over non-rectangular region, Double integrals in polar co-ordinates, Triple integrals, Triple integral over a parallelepiped and solid regions, Volume by triple integrals, cylindrical and spherical co-ordinates, Change of variables in double integrals and triple integrals. Line integrals,  Applications of line integrals: Mass and Work. Fundamental theorem for line integrals, conservative vector fields, independence of path, Green’s theorem, surface integrals, integrals over parametrically defined surfaces. Stoke’s theorem, and the Divergence theorem.

Summary
Course Status : Upcoming
Course Type : Core
Language for course content : English
Duration : 16 weeks
Category :
  • Mathematics
Credit Points : 5
Level : Undergraduate
Start Date : 06 Jan 2025
End Date : 26 Apr 2025
Enrollment Ends : 28 Feb 2025
Exam Date : 17 May 2025 IST
NCrF Level   : 4.5
Exam Sift :

Shift 2

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.


Page Visits



Course layout


Week 1
Day 1: Module 1 - Cylinders
Day 2: Module 2 - Quadric Surfaces
Day 3: Module 3 - Functions of Several Variables (Domain, Range, and Graph of Functions of Several Variables)
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 1 

Week 2
Day 1: Module 4 - Graph, Level Curves, and Contour of Functions of More Than One Variable
Day 2: Module 5 - Level Curves and Contour Lines of Functions of Several Variables AND Limit Along Curves of Functions of Two Variables
Day 3: Module 6 - Limit and Continuity of Functions of Several Variables
Day 4: Interaction based on the three modules covered
Day 5: Descriptive Assignment - Assignment 2 

Week 3
Day 1: Module 7 - Partial Differentiation
Day 2: Module 8 - Partial Differentiation of Higher Order
Day 3: Module 9 - Differentiability and Linearization of Functions of Two Variables
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 3 

Week 4
Day 1: Module 10 - Total Differentials of Functions of Two Variables AND Differentiability, Linearization, and Total Differential of Functions of Three Variables
Day 2: Module 11 - Chain Rules for Functions of Several Variables
Day 3: Module 12 - Applications of the Chain Rule - Implicit Differentiation
Day 4: Discussion
Day 5: Descriptive Assignment - Assignment 4

Week 5
Day 1: Module 13 - Directional Derivative of Functions of Two Variables
Day 2: Module 14 - Directional Derivative of Functions of Three Variables
Day 3: Module 15 - Extreme Values of Functions of Several Variables and Saddle Points
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 5 

Week 6
Day 1: Module 16 - Absolute Maxima and Minima on Closed Bounded Regions
Day 2: Module 17 - Lagrange Multipliers with One Constraint
Day 3: Module 18 - Lagrange Multipliers with Two Constraints
Day 4: Interaction based on the three modules covered
Day 5: Descriptive Assignment - Assignment 6 

Week 7
Day 1: Module 19 - Double Integration over Rectangular Regions
Day 2: Module 20 - Double Integration in Non-Rectangular Regions
Day 3: Module 21 - Double Integrals by Reversing the Order of Integration and Area by Double Integrals
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 7 

Week 8
Day 1: Module 22 - Double Integration in Polar Coordinates
Day 2: Module 23 - Triple Integration in Cartesian Coordinates, Triple Integrals over General Surfaces in Cartesian Coordinates AND Triple Integrals in Cartesian Coordinates (Changing the Order of Integration)
Day 3: Module 24 - Cartesian and Cylindrical Coordinate Systems
Day 4: Interaction based on the three modules covered
Day 5: Descriptive Assignment - Assignment 8 

Week 9
Day 1: Module 25 - Spherical Coordinate System in Three Dimensional Space
Day 2: Module 26 - Triple Integrals in Cylindrical Coordinates
Day 3: Module 27 - Triple Integrals in Spherical Coordinates
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 9 

Week 10
Day 1: Module 28 - Change of Variables in Double and Triple Integrals - Jacobians
Day 2: Module 29 - Arc Length - Arcs Given by Vector Valued Functions AND Arc Length Parameter, Speed, and Unit Tangent Vector
Day 3: Module 30 - Line Integrals - Integration Along Curves
Day 4: Interaction based on the three modules covered
Day 5: Descriptive Assignment - Assignment 10 

Week 11
Day 1: Module 31 - Evaluation of Line Integrals
Day 2: Module 32 - Applications of Line Integrals - Finding Centre of Mass
Day 3: Module 33 - Vector Valued Functions, Vector Fields, Divergence, and Curl
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 11 

Week 12
Day 1: Module 34 - Line Integral of Vector Valued Functions
Day 2: Module 35 - The Work Done by a Force Over a Curve in Space
Day 3: Module 36 - Path Independence of Line Integrals - Fundamental Theorem of Line Integrals
Day 4: Interaction based on the three modules covered
Day 5: Descriptive Assignment - Assignment 12 

Week 13
Day 1: Module 37 - The Divergence of a Vector Field AND k-Component of Curl of a Vector Field
Day 2: Module 38 - Green’s Theorem - Normal Form
Day 3: Module 39 - Green’s Theorem - Tangential Form
Day 4: Interaction based on the three modules covered
Day 5: Objective Assignment - Assignment 13 

Week 14
Day 1: Module 40 - Surface Area of Surfaces in Cartesian Form
Day 2: Module 41 - Surface Integrals when Surfaces are given by Cartesian Equations
Day 3: Module 42 - Orientation of Surfaces
Day 4: Interaction based on the three modules covered
Day 5: Descriptive Assignment - Assignment 14 

Week 15
Day 1: Module 43 - Flux of a Vector Field through a Surface
Day 2: Module 44 - Parametrized Surfaces
Day 3: Module 45 - Surface Area of Parametrized Surfaces
Day 4: Module 46 - Surface Integral of Parametrized Surfaces
Day 5: Interaction based on the three modules covered
Day 6: Objective Assignment - Assignment 15 

Week 16
Day 1: Module 47 - Stokes’ Theorem Part A
Day 2: Module 48 - Stokes’ Theorem Part B
Day 3: Module 49 - The Divergence Theorem of Gauss
Day 4: Module 50 - The Divergence Theorem for General Regions
Day 5: Interaction based on the three modules covered
Day 6: Descriptive Assignment - Assignment 16 

Books and references

1. G.B.Thomas and R.L. Finney, Calculus, 9th Ed., Pearson Education, Delhi, 2005.
2. M.J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi, 2007.
3. E. Marsden, A.J. Tromba and A. Weinstein, Basic Multivariable Calculus, Springer (SIE), Indian reprint, 2005.
4. James Stewart, Multivariable Calculus, Concepts and Contexts, 2nd Ed., Brooks /Cole, Thomson Learning, USA, 2001.

Instructor bio

Dr. Bijumon Ramalayathil

Mahatma Gandhi College, Iritty, Keezhur PO , Kerala – 670 703.

Dr Bijumon Ramalayathil (Course Coordinator) has 22 years of experience both in UG and PG level.  He handled the topics Real Analysis, Topology, Complex Analysis and Algebra under PG Level for the last 22years.

He is the PI of the following Two MOOC (SWAYAM) Courses:

·       “Algebra and Trigonometry” for CEC through EMMRC University of Calicut (July-December 2018 Semester, January-June 2021 Semester)

·       Differential Calculus for CEC through EMMRC University of Calicut (July-December 2019 Semester and July-December 2020 Semester). 

·       Subject Expert of more than 80 e-content modules (2011-2012 for CEC) on the Topic Algebra and Trigonometry, Vector Analysis, and Abstract Algebra

·       Subject Matter Expert in the development of course entitled Complex Analysis in Mathematics (30 modules) produced in the month of June 2018 and telecasted the same in the month of August 2018  for Swayamprabha DTH CHANNEL 08 PHYSICAL AND EARTH SCIENCES Aryabhatta by EMMRC, University of Calicut, on behalf of CEC New Delhi, an Inter University centre under UGC.

·       Contributed as Subject Matter Expert in the development of course entitled Vector Calculus (15modules) in Mathematics as a part of Home based recordings produced in the month of june 2020 and telecasted the same in the month of November 2020 for Swayamprabha DTH CHANNEL 08 PHYSICAL AND EARTH SCIENCESAryabhatt Developed study materials for many Universities in Kerala for their School of Distance Education. 

·       Awarded Ph.D. from the University of Calicut in 2010 for thesis titled On the Study of Frame Multiresolution Anaysis in the Superspaces  under the guidance of Prof Dr. M. S. Balasubramani - University of Calicut.

·       Developed study materials for many Universities in Kerala for their School of Distance Education. 

Course certificate

30 Marks will be allocated for internal assessment and 70 Marks will be allocated for end term proctored examination.
Securing 40% in both separately is mandatory to pass the course and get Credit Certificate.


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