Pokhara University

Faculty of Science and Technology

Course Code: MTH 210 Full Marks: 100

Course title: Calculus II (3-2-0) Pass Marks: 45
Nature of the Course: Theory Total Lectures: 45 hours
Level: Bachelor Program: BE

1. Course Description
The Calculus II is designed to develop the competency of the students in the applications of
various mathematical concepts they learned in previous semesters. It is mainly equipped with
Vector Calculus, Laplace transform, Multiple integrals, Differential Equations, Fourier Series
and with introduction of Partial differential equations. The pre-requisite for this course is
Calculus I and Algebra & Geometry. The course will be delivered through lecture method,
assignments on practically based engineering problems and class tests.

2. General Objectives
The course is designed to acquaint the students with applications of mathematics in
engineering.

3. Methods of Instruction
Lecture, tutorials, discussions and assignments

4. Contents in Detail

Specific objectives Contents

Evaluate multiple integrals Unit I: Multiple Integrals (6 Hours)
1.1 Introduction
1.2 Double integrals in Cartesian and polar form, Fubini's
theorem (statement only), change of order of integration,
change of variable from in double integral Jacobian
matrix and reduction into Polar.
1.3 Triple integrals in Cartesian form and Dirichlet’s
Integral, use of cylindrical and spherical coordinates to
evaluate triple integral.
1.4 Application of double and triple integrals to find Area
and volume.
Analyze the concept of solution of Unit II: Series Solution of Differential Equations and
differential equations in terms of Special Functions (6 Hours)
infinite series as power series. 2.1 Power series method of solution of differential equations.
2.2 Legendre's Equation, Legendre's polynomials Pn (x ) of .
Graph of P1 ( x), P2 (x) , P3 (x ) .
2.3 Frobenius method. Bessel's equation, Bessel's function
J ν ( x ) and its properties. Graph of J ν ( x ) for ν=1∧2
Apply the integral transform in Unit III: Laplace Transform and Its Application (8
solving practical problems Hours)
3.1 Laplace Transform (LT), Inverse LT, Linearity of LT,
LT of elementary functions, inverses and first shifting (s-
shifting) theorem. Existence theorem of Laplace
transform (without proof) and uniqueness.
3.2 Transform of Derivative and Integrals of a function.
 3.3 Differentiation and Integration of Laplace transform.
3.4 Unit step function, periodic function and LT, second
shifting (t-shifting) theorem.
3.5 Convolution theorem and its application to find inverse.
3.6 Application of Laplace transform to find the solutions of
ordinary differential equations (IVP).
Solve higher dimensional Unit IV: Advanced Vector Calculus (15 Hours)
(multivariable) calculus problems 4.1 Differentiation of vector function of scalar variable.
4.2 Point functions, Gradient, directional derivative,
divergence and curl with properties (without proof)
4.3 Line integral with physical interpretation and evaluation
of line integrals on various path
4.4 Line integral, potential function and independence of
path
4.5 Green's theorem in plane (without proof) and its various
applications
4.6 Surface integral and evaluation of surface integrals
4.7 Stoke's theorem (without proof) and its applications
4.8 Gauss Divergence theorem (without proof) and its
applications.
Illustrate periodic functions of Unit V: Fourier Series (5 Hours)
practical importance by infinite 5.1 Periodic Functions, odd and even functions
trigonometric series 5.2 Fourier series of 2 π periodic functions in the interval
(α , α +2 π).
5.3 Fourier series of 2 l periodic functions.
5.4 Fourier series of odd and even functions, sine and cosine
series
Interpret physical phenomenon by Unit VII: Partial Differential Equations (5 Hours)
partial differential equations 7.1 Introduction
7.2 Linear constant coefficient equation
7.3 Applications in conservation laws, the breaking time,
shock waves, nonlinear advection equations, and traffic
flow.

Note: The figures in the parentheses indicate the approximate periods for the respective units.

5. List of Tutorials (30 hours)

Tutorial work covers the work to be done in tutorial. This will enable the students to compute
the mathematical problems under the supervision of the course leader. The major tutorial
works are as follows:

Unit Unit name List of Tutorials Tutorial

hours
1 Unit I: Multiple 1.1 Problems on double integral by changing order 2 hrs
Integrals of integration and reduction into polar.
1.2 Triple integral with examples on Dirichlet’s 1 hr
integrals, use Cylindrical and Spherical
coordinates. 1 hr
1.3 Problems on area and volume by double and triple
integral
2 Unit II: Series 2.1 Solve Legendre's polynomials Pn (x ) of different 2 hrs
 Solution of order.
Differential 2.2 Solve Bessel's function J ν ( x ) and their 2 hrs
Equations and properties.
Special Functions
3 Unit III: Laplace 3.1Problems on Laplace and Inverse Laplace 2 hrs
Transform and Its transform of different functions.
Application 3.2 Solution of IVP using Laplace transform. 2 hrs

4 Unit IV: Advanced 4.1 Problems on gradient, Normal vector to a surface, 2 hrs
Vector Calculus Directional derivative, angle between two surfaces.
4.2 Problems on divergence and curl. 2 hrs
4.3Problems on line integrals, Exactness and path 3 hrs
independence.
4.4 Problems based on Green’s theorem, Stoke’s 3 hrs
theorem and Gauss divergence theorem.
5 Unit V: Fourier 5.1 Problems on Fourier series in 2 hrs
Series π 3π
(
(−π , π ) , ( 0,2 π )∧ - ,
2 2 ).
5.2 Problems on Fourier series of odd and even 2 hrs
functions, Fourier series in general interval (−l ,l ) .
6 Unit VII: Partial 6.1 Solve partial differential equations and their 4 hrs
Differential equation applications.

6. Evaluation System and Students’ Responsibilities

Evaluation System
In addition to the formal exam(s), the internal evaluation of a student may consist of quizzes,
assignments, lab reports, projects, class participation, etc. The tabular presentation of the
internal evaluation is as follows.

Internal Evaluation Marks External Evaluation Weigh Marks

t
Attendance & Class Participation 10%
Assignments 20% Semester End Board
Presentations/Quizzes 10% Examination 50% 50
Term exam 60%
Total Internal 50
Full Marks: 50 + 50 = 100

Students’ Responsibilities
Each student must secure at least 45% marks in internal evaluation with 80% attendance in
the class in order to appear in the Semester End Examination. Failing to get such score will
be given NOT QUALIFIED (NQ) and the student will not be eligible to appear the Semester-
End Examinations. Students are advised to attend all the classes, formal exam, test, etc. and
complete all the assignments within the specified time period. Students are required to
complete all the requirements defined for the completion of the course.

7. Prescribed Books and References

Text Books
1. Kreyszig, E. Advance Engineering Mathematics, New Delhi: John Wiley and Sons Inc.
2. Stewart, J. Calculus, Early Transcendental. India; Cengage Learning.

References
1. Dass, H. K. & Verma R. Higher Engineering Mathematics. New Delhi: S Chand
Publishing.
2. Mishra, P., Mishra, R., Mishra, V. P., & Mishra, M. Advance Engineering Mathematics.
New Delhi: V. P. Mishra Publication.
3. Thomas, G. & Finney, R. Calculus and Analytical Geometry. New Delhi: Narosa
Publishing House.