The Mathematics – 2 course builds upon the foundational knowledge from Mathematics – 1, introducing advanced mathematical tools like linear algebra, complex analysis, and differential equations. This course equips engineering students with analytical skills to solve complex problems in physics, engineering, and other domains. Below is a detailed, SEO-friendly breakdown of the syllabus, learning outcomes, and applications.
Introduction
Mathematics is essential for modeling, solving, and analyzing engineering problems. The Mathematics – 2 course focuses on matrices, first and higher-order differential equations, and complex variables, combining theoretical insights with practical applications in engineering, physics, and beyond. This course enables students to handle sophisticated calculations and develop a deeper understanding of mathematical theories.
Syllabus Overview
Module 1: Matrices
- Topics Covered:
- Linear independence, Row Echelon Form (REF), and Reduced Row Echelon Form (RREF)
- Rank of a matrix using REF/RREF
- Inverse of a matrix via Gauss-Jordan method
- Solution of linear equations using elementary row operations
- Symmetric, skew-symmetric, and orthogonal matrices
- Eigenvalues and eigenvectors, diagonalization of matrices
- Matrix inversion using the Cayley-Hamilton theorem
- Key Applications:
- System modeling and simulations in engineering, including robotics, mechanics, and control systems.
Module 2: First-Order Ordinary Differential Equations
- Topics Covered:
- Exact differential equations and integrating factors for non-exact equations
- Linear and Bernoulli’s equations
- Non-linear equations solvable for ppp, yyy, or xxx, and Clairaut’s equations
- Key Applications:
- Analyzing fluid flow, thermal systems, and electrical circuits.
Module 3: Higher-Order Ordinary Differential Equations
- Topics Covered:
- Linear differential equations with constant and variable coefficients
- Euler-Cauchy equations
- Methods of undetermined coefficients and variation of parameters
- Classification of ordinary and singular points
- Power series solutions for ordinary points
- Key Applications:
- Modeling mechanical vibrations, electrical networks, and population dynamics.
Module 4: Complex Variables (Differentiation)
- Topics Covered:
- Differentiation and Cauchy-Riemann equations
- Analytic and harmonic functions
- Harmonic conjugates of elementary functions (exponential, trigonometric, logarithmic)
- Key Applications:
- Applications in fluid dynamics, electromagnetic fields, and aerodynamics.
Module 5: Complex Variables (Integration)
- Topics Covered:
- Contour integrals and Cauchy-Goursat theorem
- Cauchy Integral Formula, Taylor and Laurent series
- Zeros of analytic functions, singularities, and residues
- Cauchy Residue Theorem and Rouche’s theorem
- Key Applications:
- Solving engineering problems involving heat conduction, wave propagation, and quantum mechanics.