Engineering Mathematics I
Course Content and Outline
Unit I: Complex Numbers (10 Hours)
Definition – Conjugates - Algebra of complex numbers (geometrical proof not needed) –
Real and Imaginary parts. Simple Problems. Polar form of complex number – Modulus and amplitude form multiplication and division of complex numbers in polar form. Simple Problems Argand plane – Collinear points, four points forming a square, rectangle, rhombus. Simple Problems. Demoivre’s Theorem (statement only) – simple problems. Demoivre’s Theorem related problems.
Unit II: Infinite Series & Fourier series
Infinite Series (7 Hours)
Convergence, divergence, and oscillation of an infinite series, comparison Test, Pseries,
D’Alembert’s ratio test, Raabe’s test, Cauchy’s root test, Cauchy’s integral test (All tests without proof) forseries of positive terms. Alternating series. Absolute and conditional convergence, Leibnitz’s test (without proof)
Fourier series (7 Hours)
Definition, Euler's formula, Conditions (Dirichlet's) for a Fourier expansion, Functions
having points of discontinuity, Change of interval, Odd and even periodic functions,
Expansion of odd and even periodic functions, Half‐range series, Typical wave‐forms,
Parseval's formula, Practical harmonic analysis and Applications to Problems in Engineering.
Uni III: Vector Calculus (10 Hours)
Introduction, Vector addition & subtraction, Resolution of vectors, combination of two
perodic waveforms, plotting periodic functions, determining resultant phasors by calculation. Arithmetic with scalars and vectors, unit vectors.
Dot product of two vectors (scalar& vector products). Reciprocal System of vectors
Application of vectors: Lines and planes. Derivatives – Velocity and acceleration
Divergence and curl operations. Find tangent and normal vectors (to a curve or surface). How to apply Divergence and Green's theorems
Unit IV: Matrices
Introductions: (7 Hours)
Addition, subtraction &multiplication, their properties. The transpose of a matrix. Special
matrices. Minors, the adjoint of matrix, the inverse of a matrix,2 × 2 and 3 × 3 determinants, properties, Cramers’s rule. Consistency of a system of linear equations.
Solutions of Linear Systems (10 Hours)
Simple systems, Homogeneous and Non-homogeneous systems, Gaussian elimination&
Gauss-Jordan elimination, Null Space and Range, Rank and nullity, Consistency conditions in terms of rank, General Solution of a linear system, Elementary Row and Column operations, Row Reduced Form& Row Reduced Echelon Form, Triangular Matrix Factorization and Transition Matrices.
Diagonalizable Matrices (5 Hours)
Diagonalization criterion, the diagonalizing matrix, Cayley-Hamilton theorem, Annihilating
polynomials, Minimal Polynomial, Diagonalizability and Minimal polynomial, Projections & Decomposition of the matrix in terms of projections.
Applications of Matrices (4 Hours)
Optimization and Linear Programming, Network models, Game Theory & Control Theory.
At least a question MUST be set from each unit. Seven questions MUST be set from which five questions are attempted by the students.
MODE OF DELIVERY
- Reading assignments
- Practical assignments
- Whiteboard and Markers
- Flip Charts