Mathematical Engineer Course

Lesson 1 • Logic, Proof Techniques, and Sets

Covers propositional logic, set operations, and formal proof strategies. Develops rigorous reasoning skills essential for deriving and validating engineering theorems.

Lesson 2 • Number Systems and Algebraic Structures

Covers integers, rationals, reals, and complex numbers alongside field and group axioms. Establishes the algebraic language used throughout all subsequent chapters.

Lesson 3 • Vectors and Coordinate Geometry

Introduces vector operations, dot and cross products, and coordinate systems. Connects geometric intuition to algebraic computation for spatial engineering problems.

Lesson 1 • Eigenvalues, Eigenvectors, and Diagonalization

Derives eigenvalue problems and diagonalization procedures for square matrices. Enables modal analysis, stability assessment, and principal component decomposition.

Lesson 2 • Matrix Operations and Systems of Equations

Covers matrix arithmetic, Gaussian elimination, and LU decomposition. Provides computational tools for solving simultaneous engineering equations efficiently.

Lesson 3 • Vector Spaces and Subspaces

Defines vector spaces, bases, dimension, and null and column spaces. Establishes the structural vocabulary for analysing solution spaces of linear systems.

Lesson 4 • Determinants and Invertibility

Defines determinants via cofactor expansion and properties, linking them to matrix invertibility. Connects algebraic conditions to geometric volume scaling in transformations.

Lesson 1 • Limits, Continuity, and Differentiability

Formalizes epsilon-delta limits, continuity criteria, and differentiability conditions. Provides the analytical foundation for all derivative-based engineering computations.

Lesson 2 • Integration Theory and Techniques

Develops Riemann integration, fundamental theorem of calculus, and advanced integration methods. Enables computation of areas, volumes, and accumulated quantities in engineering models.

Lesson 3 • Vector Calculus and Integral Theorems

Covers divergence, curl, line integrals, and surface integrals with Green's, Stokes', and divergence theorems. Directly supports fluid dynamics and electromagnetic field modeling.

Lesson 4 • Multivariable Calculus

Extends differentiation and integration to functions of several variables using partial derivatives and multiple integrals. Supports field analysis and optimisation in higher dimensions.

Lesson 1 • Second-Order Linear ODEs

Solves homogeneous and non-homogeneous second-order equations using undetermined coefficients and variation of parameters. Models mechanical vibrations and electrical circuits.

Lesson 2 • Laplace Transform Methods

Applies Laplace transforms to solve ODEs with discontinuous and impulsive forcing functions. Bridges time-domain ODE analysis with frequency-domain engineering methods.

Lesson 3 • Systems of ODEs and Phase Plane Analysis

Converts higher-order ODEs to first-order systems and analyses equilibria via phase portraits. Enables stability classification of multi-variable dynamic engineering systems.

Lesson 4 • Numerical Methods for ODEs

Implements Euler, Runge-Kutta, and multistep methods for approximating ODE solutions. Prepares students for computational simulation of systems without closed-form solutions.

Lesson 1 • Heat Equation and Diffusion Problems

Solves the heat equation on finite and semi-infinite domains with various boundary conditions. Applies directly to thermal management and mass diffusion in engineering design.

Lesson 2 • Wave Equation and Vibration Analysis

Solves the wave equation for strings, membranes, and acoustic fields using modal superposition. Connects PDE theory to structural vibration and signal propagation engineering.

Lesson 3 • Laplace Equation and Potential Theory

Solves Laplace and Poisson equations in rectangular, cylindrical, and spherical coordinates. Supports electrostatic, gravitational, and steady-state fluid potential field analysis.

Lesson 4 • Separation of Variables and Fourier Series

Applies separation of variables to reduce PDEs to ODEs and expands solutions in Fourier series. Provides the primary analytical method for bounded-domain engineering problems.

Lesson 1 • Complex Integration and Cauchy's Theorem

Covers contour integration, Cauchy's integral theorem, and the residue theorem. Enables exact evaluation of real definite integrals arising in engineering analysis.

Lesson 2 • Fourier Transform and Spectral Analysis

Develops the Fourier transform, its properties, and the convolution theorem for continuous signals. Connects time-domain engineering models to frequency-domain spectral representations.

Lesson 3 • Complex Functions and Analyticity

Defines complex differentiation, Cauchy-Riemann equations, and analytic functions. Establishes the theoretical basis for conformal mapping and complex integration methods.

Lesson 4 • Conformal Mapping and Applications

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

Lesson 1 • Expectation, Variance, and Moment Methods

Defines expectation, variance, covariance, and moment-generating functions for random variables. Enables characterisation of engineering uncertainty and propagation through nonlinear models.

Lesson 2 • Probability Theory and Random Variables

Covers probability axioms, conditional probability, independence, and discrete and continuous distributions. Provides the probabilistic language for modelling uncertain engineering quantities.

Lesson 3 • Regression Analysis and Model Fitting

Develops linear and nonlinear regression, residual analysis, and model selection criteria. Supports data-driven engineering model calibration and predictive performance evaluation.

Lesson 4 • Statistical Inference and Hypothesis Testing

Covers parameter estimation, confidence intervals, and hypothesis testing for engineering data. Connects sample statistics to population parameters for quality and reliability decisions.

Lesson 1 • Numerical Linear Algebra

Covers iterative solvers, condition numbers, and eigenvalue algorithms for large engineering systems. Addresses computational efficiency and numerical stability in matrix-intensive applications.

Lesson 2 • Interpolation and Approximation

Develops Lagrange, Newton, and spline interpolation alongside least-squares polynomial fitting. Supports data reconstruction and function approximation in computational engineering workflows.

Lesson 3 • Finite Difference and Finite Element Methods

Discretises PDEs using finite difference and finite element formulations for engineering field problems. Connects numerical PDE theory to practical simulation of structural and thermal systems.

Lesson 4 • Root Finding and Nonlinear Equations

Covers bisection, Newton-Raphson, secant, and fixed-point iteration methods with convergence analysis. Provides tools for solving implicit engineering equations without closed-form solutions.