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
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 the 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.
General Objectives
The course is designed to acquaint the students with applications of mathematics in engineering.
Methods of Instruction
Lecture, tutorials, discussions, and assignments
Contents in Detail
Specific Objectives
Contents
Unit I: Multiple Integrals (6 Hours)
Introduction
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.
Triple integrals in Cartesian form and Dirichlet’s Integral, use of cylindrical and spherical coordinates to evaluate triple integral.
Application of double and triple integrals to find Area and volume.
Unit II: Series Solution of Differential Equations and Special Functions (6 Hours)
Power series method of solution of differential equations.
Legendre's Equation, Legendre's polynomials Pn(x) of. Graph of P1(x), P2(x), P3(x).
Frobenius method. Bessel's equation, Bessel's function Jν(x) and its properties. Graph of Jν(x) for ν=1,2.
Unit III: Laplace Transform and Its Application (8 Hours)
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.
Transform of Derivative and Integrals of a function.
Unit IV: Advanced Vector Calculus (15 Hours)
Differentiation of vector function of scalar variable.
Point functions, Gradient, directional derivative, divergence and curl with properties (without proof)
Line integral with physical interpretation and evaluation of line integrals on various paths
Line integral, potential function and independence of path
Green's theorem in plane (without proof) and its various applications
Surface integral and evaluation of surface integrals
Stoke's theorem (without proof) and its applications
Gauss Divergence theorem (without proof) and its applications.
Unit V: Fourier Series (5 Hours)
Periodic Functions, odd and even functions
Fourier series of 2π periodic functions in the interval (α, α + 2π).
Fourier series of 2l periodic functions.
Fourier series of odd and even functions, sine and cosine series
Interpret physical phenomena by partial differential equations.
Unit VII: Partial Differential Equations (5 Hours)
Introduction
Linear constant coefficient equation
Applications in conservation laws, the breaking time, shock waves, nonlinear advection equations, and traffic flow.
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 Integrals | Problems on double integral by changing order of integration and reduction into polar. | 2 hrs |
| 1 | Unit I: Multiple Integrals | Triple integral with examples on Dirichlet’s integrals, use Cylindrical and Spherical coordinates. | 1 hr |
| 1 | Unit I: Multiple Integrals | Problems on area and volume by double and triple integral. | 1 hr |
| 2 | Unit II: Series | Solve Legendre's polynomials Pn(x) of different order. | 2 hrs |
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 | Weight | Marks |
|---|---|---|---|---|
| Attendance & Class Participation | 10% | |||
| Assignments | 20% | |||
| Presentations/Quizzes | 10% | |||
| 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 in the Semester-End Examinations. Students are advised to attend all the classes, formal exams, tests, 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.
Prescribed Books and References
Text Books
- Kreyszig, E. Advance Engineering Mathematics, New Delhi: John Wiley and Sons Inc.
- Stewart, J. Calculus, Early Transcendental. India; Cengage Learning.
References
- Dass, H. K. & Verma R. Higher Engineering Mathematics. New Delhi: S Chand Publishing.
- Mishra, P., Mishra, R., Mishra, V. P., & Mishra, M. Advance Engineering Mathematics. New Delhi: V. P. Mishra Publication.
- Thomas, G. & Finney, R. Calculus and Analytical Geometry. New Delhi: Narosa Publishing House.
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