Engineering Mathematics Course

Lesson 1 • Logic and Proof Techniques

Introduces propositional logic, quantifiers, and standard proof strategies. Enables rigorous justification of mathematical claims used throughout the course.

Lesson 2 • Number Systems and Properties

Covers integers, rationals, reals, and complex numbers with their algebraic properties. Establishes the numeric foundation required for all subsequent engineering calculations.

Lesson 3 • Set Theory and Relations

Defines sets, operations, and binary relations including equivalence and order. Provides the language for describing domains and mappings in engineering models.

Lesson 1 • Eigenvalues and Eigenvectors

Computes eigenvalues, eigenvectors, and diagonalization of square matrices. Applies spectral analysis to vibration modes, stability, and principal component problems.

Lesson 2 • Determinants and Their Applications

Derives determinant properties and Cramer's rule for small systems. Links determinants to area, volume, and invertibility in engineering geometry.

Lesson 3 • Matrix Operations and Systems

Covers matrix arithmetic, row reduction, and solution of linear systems. Provides computational tools for structural, circuit, and network analysis.

Lesson 4 • Vectors and Vector Spaces

Defines vectors, linear independence, span, and basis in n-dimensional space. Connects abstract vector space axioms to concrete engineering coordinate systems.

Lesson 1 • Differentiation Techniques

Derives rules for differentiating algebraic, trigonometric, exponential, and composite functions. Enables computation of rates of change in mechanical and electrical systems.

Lesson 2 • Integration Methods

Covers definite and indefinite integrals using substitution, parts, and partial fractions. Provides tools for computing areas, volumes, and accumulated quantities.

Lesson 3 • Applications of Derivatives

Uses derivatives for optimization, curve sketching, and linear approximation. Connects calculus tools to engineering design and error analysis.

Lesson 4 • Improper Integrals and Series

Evaluates improper integrals and tests convergence of infinite series and power series. Prepares students for Fourier and Laplace transform applications.

Lesson 1 • Partial Derivatives and Gradients

Computes partial derivatives, directional derivatives, and gradient vectors for scalar fields. Links gradient direction to steepest ascent in optimization and heat flow problems.

Lesson 2 • Vector Fields and Line Integrals

Defines vector fields, computes line integrals, and identifies conservative fields. Applies work and circulation concepts to fluid and electromagnetic engineering models.

Lesson 3 • Surface Integrals and Integral Theorems

Evaluates surface integrals of flux and applies Stokes' and Divergence theorems. Unifies vector calculus for use in fluid dynamics and electromagnetic field analysis.

Lesson 4 • Multiple Integrals

Evaluates double and triple integrals in Cartesian, polar, cylindrical, and spherical coordinates. Computes mass, centre of mass, and moments of inertia for engineering bodies.

Lesson 1 • Second-Order Linear ODEs

Derives general solutions for constant-coefficient second-order equations using characteristic roots. Analyses free and forced vibrations in mechanical and electrical systems.

Lesson 2 • Series Solutions and Special Functions

Constructs power series solutions near ordinary and regular singular points. Introduces Bessel and Legendre functions for cylindrical and spherical engineering geometries.

Lesson 3 • Systems of Differential Equations

Converts higher-order ODEs to first-order systems and solves using matrix methods. Analyses coupled oscillators and multi-loop circuit dynamics.

Lesson 4 • First-Order Differential Equations

Solves separable, linear, and exact first-order ODEs with initial conditions. Models exponential growth, decay, and simple circuit transients.

Lesson 1 • Classification and Formulation of PDEs

Classifies second-order PDEs as elliptic, parabolic, or hyperbolic and states boundary conditions. Connects PDE type to physical phenomena such as diffusion, vibration, and steady-state fields.

Lesson 2 • Numerical Methods for PDEs

Introduces finite difference schemes for parabolic and elliptic PDEs and analyses stability. Prepares students to implement computational solutions for complex engineering geometries.

Lesson 3 • Separation of Variables

Applies separation of variables to heat, wave, and Laplace equations on standard domains. Constructs complete solutions as superpositions of eigenfunctions satisfying boundary conditions.

Lesson 4 • Fourier Series and Orthogonal Expansions

Expands periodic functions in Fourier sine, cosine, and full series. Provides the spectral decomposition tool required for PDE solution by separation of variables.

Lesson 1 • Random Variables and Distributions

Characterizes discrete and continuous random variables through PMFs, PDFs, and CDFs. Introduces key distributions used in failure analysis and quality control.

Lesson 2 • Joint Distributions and Correlation

Analyses joint, marginal, and conditional distributions for pairs of random variables. Quantifies dependence through covariance and correlation for multivariate engineering data.

Lesson 3 • Probability Fundamentals

Defines sample spaces, events, and probability axioms including conditional probability. Establishes the probabilistic framework for reliability and quality engineering.

Lesson 4 • Regression and Design of Experiments

Fits linear regression models and interprets coefficients and residuals for engineering data. Introduces factorial experimental designs for systematic process optimization.

Lesson 1 • Discrete Fourier and Z-Transforms

Introduces the DFT, FFT algorithm, and Z-transform for discrete-time engineering systems. Enables digital filter design and analysis of sampled-data control systems.

Lesson 2 • Conformal Mapping and Applications

Uses conformal mappings to transform complex geometries into solvable standard domains. Applies Joukowski and Schwarz-Christoffel mappings to aerodynamic and electrostatic problems.

Lesson 3 • Fourier Transform and Frequency Analysis

Defines the continuous Fourier transform and its properties for signal and system analysis. Applies frequency-domain methods to filtering, convolution, and spectral engineering problems.

Lesson 4 • Complex Integration and Residues

Evaluates contour integrals using Cauchy's integral formula and the residue theorem. Applies residue calculus to evaluate real improper integrals arising in engineering analysis.