**ENGINEERING MATHEMATICS III**

**SH 501**

**Lecture**

**:**

**3 Year**

**:**

**II**

**Tutorial**

**:**

**2 Part**

**:**

**I**

**Practical**

**:**

**0**

**Course Objective**

**:**

The purpose of this course is to round out
the students’ preparation
for more sophisticated applications with an introduction to linear algebra,
Fourier Series, Laplace Transforms, integral transformation theorems and linear
programming.

**1.**

**Determinants and Matrices**

**(**

**11 hours**

**)**

1.1. Determinant and its properties

1.2. Solution of system of linear equations

1.3. Algebra of matrices

1.4. Complex matrices

1.5. Rank of matrices

1.6. System of linear equations

1.7. Vector spaces

1.8. Linear transformations

1.9. Eigen value and Eigen vectors

1.10. The Cayley-Hamilton theorem and its uses

1.11. Diagonalization of matrices and its
applications

**2.**

**Line, Surface and Volume Integrals**

**(**

**12 hours**

**)**

2.1. Line integrals

2.2. Evaluation of line integrals

2.3. Line integrals independent of path

2.4. Surfaces and surface integrals

2.5. Green’s theorem in the plane and its applications

2.6. Stoke’s theorem (without proof) and its applications

2.7. Volume integrals; Divergence theorem of Gauss
(without
proof) and its
applications

**3.**

**Laplace Transform**

**(**

**8 hours**

**)**

3.1. Definitions and properties of Laplace
Transform

3.2. Derivations of basic formulae of Laplace
Transform

3.3. Inverse Laplace Transform: Definition and standard formulae
of inverse Laplace Transform

3.4. Theorems on Laplace transform and its inverse

3.5. Convolution and related problems

3.6. Applications of Laplace Transform to ordinary
differential equations

**4.**

**Fourier Series `**

**(**

**5 hours**

**)**

4.1. Fourier Series

4.2. Periodic functions

4.3. Odd and even functions

4.4. Fourier series for arbitrary range

4.5. Half range Fourier series

**5.**

**Linear Programming**

**(**

**9 hours**

**)**

5.1. System of Linear Inequalities in two
variables

5.2. Linear Programming in two dimensions: A Geometrical Approach

5.3. A Geometric introduction to the Simplex
method

5.4. The Simplex method: Maximization with Problem
constraints of the form “≤”

5.5. The Dual: Maximization with Problem
Constraints of the form “≥”

5.6. Maximization and Minimization with mixed
Constraints. The two- phase method(An alternative to the Big M
Method)

**References**

__:__

1. E. Kreszig, "Advance Engineering Mathematics", Willey, New York.

2. M.M Gutterman and Z.N.Nitecki, "Differential Equation, a First
Course", 2

^{nd}Edition, saunders, New York.**Evaluation Scheme**

**:**

The questions will cover all the chapters of
the syllabus. The
evaluation scheme will be as indicated in the table below:

Chapters |
Hours |
Marks
distribution* |

1 |
11 |
20 |

2 |
12 |
20 |

3 |
8 |
15 |

4 |
5 |
10 |

5 |
9 |
15 |

Total |
45 |
80 |

*There may be minor deviation in
marks distribution.

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