College of Engineering Pune - 5
Department of Mathematics,
(MA-20005) : Multivariate Calculus and Differential Equations (Lesson Plan)
S. Y.B.Tech. (for students directly admitted to S.Y. after their Diploma)
Sem IV (All Branches)
2023-24
Teaching Scheme:
There will be 4 theory lectures and 1 tutorial per week. Students are expected to have 100%
attendance both for theory as well as tutorial classes.
Evaluation Scheme:
Internal Evaluation: Maximum 40 marks. T1 and T2: 20 marks each. Tutorials: 10 marks
ESE: 60 marks.
Text Books:
Thomas’ Calculus (11th Edition) by Maurice D. Weir, Joel Hass, Frank R. Giordano,
Pearson Education
Advanced Engineering Mathematics (10th Edition) by Erwin Kreyszig, Wiley eastern Ltd
Unit I: Differential Equations [9 Hrs]
Lesson Topic Section no. of
No. text book II
1&2 Review of first order ODEs 1.1, 1.3, 1.4
3&4 Homogeneous Linear Differential Equations 2.1, 3.1
5&6 Homogeneous differential equations with constant coefficients 2.2, 3.2
7&8 Non Homogeneous ODEs 2.7, 3.3
9 Solution by Variation of Parameters 2.10
Unit II: Laplace Transforms [7 Hrs]
Lesson Topic Section no. of
No. text book II
1&2 Laplace Transforms and Inverse Laplace Transforms 6.1
3 Properties of LT and ILT, ODEs 6.2
4 Unit step function 6.3
5 Dirac Delta function, periodic functions 6.4
6 Convolution theorem 6.5
7 Differentiation and integration of transforms 6.6
Unit III: Functions of Several Variables [6 Hrs]
Lesson Topic Section no. of
No. text book I
1&2 Functions of Several Variables, level curves and level surfaces 14.1
3&4 Partial Derivatives and Chain Rule 14.3, 14.4
5&6 Extreme Values and Saddle Points 14.7
Unit IV: Multiple Integrals [12 Hrs]
Lesson Topic Section no. of
No. text book I
1&2 Double Integrals 15.1
3&4 Areas, Moments and Center of Mass 15.2
5&6 Double Integrals in Polar form 15.3
7&8 Triple Integrals in Rectangular Coordinates 15.4
9&10 Masses and Moments in Three Dimensions 15.5
11&12 Triple Integrals in Cylindrical and Spherical Coordinates 15.6
Unit V: Vector Calculus [11 Hrs]
Lesson Topic Section no. of
No. text book I
1 Directional derivatives and Gradient Vectors 14.5
2 Line Integrals 16.1
3 Path Independence, Potential Functions, and Conservative Fields 16.3
4&5 Green’s Theorem in Plane 16.4
6&7 Surface Area and Surface Integrals 16.5
8&9 Stoke’s Theorem 16.7
10&11 Divergence Theorem 16.8
Unit VI: Partial Differential Equations [7 Hrs]
Lesson Topic Section no. of
No. text book II
1 Basic Concepts of PDE 12.1
2&3 Solution of PDE using Separation of Variables 12.3
4&5 One Dimensional Wave Equation 12.2
6&7 One Dimensional Heat Equation 12.6
Test 1 will be conducted between 12th February to 17th February 2024. Timing will be
announced one week in advance.
Test 2 will be conducted between 11th to 15th March 2024. Timing will be announced one week
in advance.
Tutorials should be submitted within one week after completion of each Unit from the
syllabus.
ESE will be conducted as per the schedule given by the exam cell and will cover the entire
syllabus.
Last day of instruction: 18th April 2024.
Outcomes : Students will be able to
1. know first order ordinary differential equations, list Laplace transform formulae, define
functions of several variables, double / triple integrals, vector differentiation, vector
integration, and partial differential equations.
2. understand basic concepts of higher order ordinary differential equations, level curves and
level surfaces, co-ordinate systems, iterated integrals, gradient, divergence and curl.
3. solve linear differential equations using different methods, find Laplace transforms of
functions using properties and theorems, evaluate directional derivatives and extreme
values, evaluate multiple integrals, find area / mass / volume using multiple integrals,
evaluate line integrals and surface integrals.
4. prove theorems / statements, solve ordinary differential equations using Laplace transforms,
apply Green’s / Stoke’s / Divergence theorem to different type of problems, model one
dimensional heat / wave equations, solve partial differential equations.
5. apply concepts of multivariate calculus and differential equations to various problems
including real life problems.
Note 1 :
To measure CO1, questions may be of the type- define, identify, state, match, list,
name etc.
To measure CO2, questions may be of the type- explain, describe, illustrate,
evaluate, give examples, compute etc.
To measure CO3, questions will be based on applications of core concepts.
To measure CO4, questions may be of the type- true/false with justification,
theoretical fill in the blanks, theoretical problems, prove implications or corollaries
of theorems, etc.
To measure CO5, some questions may be based on self-study topics and also
comprehension of unseen passages.
Note 2 :
All the Course outcomes 1 to 3 will be judged by 75% of the questions and outcomes 4
and 5 will be judged by 25 % of questions.
