Mathematics II

Course Title: Mathematics II

Course No: MTHS213 174

Nature of the Course: THEORY

Semester: 2

Full Marks: 50 + 50

Pass Marks: 23 + 23

Credit Hours: 3

Course Description

This course covers essential mathematical concepts starting with complex numbers, including their definition, algebra, properties, and graphical representation. It introduces De Moivre's theorem and applications. The course progresses to infinite sequences and series, focusing on convergence tests and specific tests like P-series and ratio tests. It explores antiderivatives, including definite and improper integrals, quadrature, rectification, and Beta and Gamma functions. Optimization for functions of several variables, partial differentiation, and finding maxima and minima are also covered. Students will study ordinary differential equations, including first-order solutions and second-order equations with constant coefficients, along with initial and boundary value problems. The course addresses number theory, including division, primes, GCD, LCM, and modular arithmetic, with applications in cryptology. Finally, it covers Fourier series and integrals, exploring even and odd functions, periodic functions, and Fourier coefficients, as well as sine and cosine integrals.

Course Objectives

Understand complex numbers, their properties, and algebraic operations. Proficiently analyze infinite sequences and series using convergence tests. Master antiderivatives, including definite and improper integrals, and apply Beta and Gamma functions in problem-solving. Solve ordinary differential equations and optimization problems using techniques like partial differentiation.

Course Contents

1. Complex Numbers

8 hrs

1.1. Definition of a complex number, integral powers of i

1.2. Algebra of complex numbers (sum, difference, multiplication, division)

1.3. Properties of complex numbers (without proof), conjugate of a complex number and its properties

1.4. Modulus of a complex number and its properties (without proof), representation of a complex number by a point in a plane (Argand's diagram)

1.5. Polar representation of a complex number

1.6. Square roots of a complex number (only Cartesian form)

1.7. De Moivre's theorem (statement only) and its application

2. Infinite Sequence and Series

7 hrs

2.1. Introduction

2.2. Convergence test of infinite series (statement only)

2.3. Direct comparison test, Limit comparison test (statement only)

2.4. P-series test, De Alembert's ratio test, and Alternating series test (statement only)

3. Application of Antiderivative

7 hrs

3.1. Definite integral

3.2. Properties of the definite integral

3.3. Improper Integral

3.4. Quadrature

3.5. Rectification

3.6. Beta and Gamma function

4. Optimization: Functions of several variables

6 hrs

4.1. Introduction

4.2. Partial derivative

4.3. Rules of partial differentiation

4.4. Maxima and minima for the function of two variables

5. Ordinary Differential Equation

7 hrs

5.1. Introduction

5.2. Order and degree of differential equation

5.3. Solution of the first order and first-degree differential equation

5.4. Variable separation, homogeneous, linear differential equation

5.5. Second-order linear differential equation with constant coefficients

5.6. Initial and boundary value problems

6. Integers and Division

6 hrs

6.1. Introduction

6.2. Division, primes, the fundamental theorem of arithmetic (statement only)

6.3. The infinitude of primes

6.4. The division algorithm, GCD and LCM

6.5. Modular arithmetic

6.6. Application of congruence's Cryptology

7. Fourier Series and Integrals

7 hrs

7.1. Introduction

7.2. Even and odd function

7.3. Periodic function

7.4. Fourier series and Fourier coefficients (without proof)

7.5. Fourier sine and cosine series

7.6. Fourier integral

7.7. Fourier sine and cosine integral

Text Books

1.Kreyszig, E. (2020). Advanced Engineering Mathematics. New Delhi: John Wiley & Sons Inc.
2.Thomas, G. B. Jr., & Finney, R. L. (2003). Calculus and Analytical Geometry. New Delhi: Narosa Publishing House.
3.Rosen, K. H. (2003). Discrete Mathematics and its Applications (5th ed.). McGraw Hill Companies.

Reference Books

1.Shrestha, K. K., & Thagurathi, R. K. (Year of publication). Applied Mathematics. Kathmandu, Nepal: Buddha Publication

Mathematics II

Mathematics II

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Mathematics II

Course Description

Course Objectives

Course Contents

Text Books

Reference Books