Engineering Mathematics
The language of engineering: algebra, calculus, linear algebra, differential equations, probability and numerical methods.
104 topics: 36 beginner, 34 intermediate, 34 advanced.
About this discipline
Engineering mathematics is the shared language of every other discipline. It gives you the tools to describe change (calculus), systems of many quantities (linear algebra), vibration and circuits (differential equations), uncertainty (probability and statistics) and computation (numerical methods).
The course starts with the arithmetic and algebra you need to read a formula with confidence, then builds to calculus, vectors and matrices, differential equations and transforms, and finishes with probability, numerical methods and optimisation used in real design work.
Who it is for
Students starting an engineering degree, technicians who want to strengthen their analysis skills, and professionals who need to check results or read software documentation with understanding.
How the levels build
Beginner topics cover numbers, units, algebra, trigonometry and the first ideas of calculus. Intermediate topics bring in vectors, matrices, ODEs and Laplace transforms. Advanced topics cover PDEs, probability, numerical methods and optimisation.
Use cases
- Checking a hand calculation before you trust a simulation.
- Modelling a vibrating structure or an RC circuit with a differential equation.
- Fitting a calibration line to test data with least squares.
- Estimating the error in a numerical solver before running a long study.
Beginner Engineering Mathematics
Topic 1Number systemsEngineers use natural, integer, rational, real and complex numbers; each extends the previous to solve equations the smaller set cannot.Topic 2Significant figuresDigits that carry measurement meaning; results should not claim more precision than the least precise input.Topic 3Units and dimensional analysisChecking that both sides of an equation have the same dimensions catches errors and helps derive relations.Topic 4Scientific notationWriting numbers as a x 10^n makes very large or small quantities compact and easy to compare.Topic 5Ratios, proportions and percentagesScaling relationships used in mixing, gear ratios, efficiency and error reporting.Topic 6Algebraic manipulationRearranging expressions and equations using field rules to isolate unknowns.Topic 7Linear equationsEquations of the form ax+b=0 whose graphs are straight lines; the base case of all modelling.Topic 8Quadratic equationsax^2+bx+c=0 is solved by x=(-b±√(b²-4ac))/2a; the discriminant tells how many real roots exist.Topic 9Exponents and logarithmsLogs invert exponentials and turn products into sums; essential for decibels, pH and growth laws.Topic 10Polynomials and factoringBreaking polynomials into factors reveals roots and simplifies rational expressions.Topic 11InequalitiesDescribe ranges of acceptable values such as tolerances and safety limits.Topic 12Functions and graphsA function maps each input to one output; graphs reveal trends, intercepts and asymptotes.Topic 13Trigonometry basicsSine, cosine and tangent relate angles to side ratios in right triangles.Topic 14Radians and angular measureOne radian is the angle whose arc equals the radius; 2π rad = 360°.Topic 15Unit circle and identitiessin²θ+cos²θ=1 and related identities simplify oscillation and rotation problems.Topic 16Law of sines and cosinesSolve any triangle; used in surveying, truss geometry and force resolution.Topic 17Coordinate geometryDescribes lines, circles and conics using algebra on a Cartesian plane.Topic 18VectorsQuantities with magnitude and direction, such as force and velocity, added tip-to-tail or by components.Topic 19Dot producta·b=|a||b|cosθ gives projection and is used to compute work and test perpendicularity.Topic 20Cross producta×b gives a vector perpendicular to both, with magnitude equal to the parallelogram area; used for torque.Topic 21Sequences and seriesOrdered lists of numbers and their sums; arithmetic and geometric series appear in finance and signal decay.Topic 22Complex numbersNumbers a+bi with i²=-1 represent amplitude and phase together in AC and wave analysis.Topic 23Polar and exponential formsz=re^{iθ} makes multiplication and powers of complex numbers simple rotations and scalings.Topic 24Euler's formulae^{iθ}=cosθ+i sinθ links exponentials with trigonometry and underpins phasors and Fourier analysis.Topic 25LimitsThe value a function approaches as input approaches a point; the foundation of calculus.Topic 26ContinuityA function is continuous if small input changes cause small output changes with no jumps.Topic 27DerivativesThe instantaneous rate of change of a function; the slope of its tangent line.Topic 28Differentiation rulesPower, product, quotient and chain rules let you differentiate complicated expressions systematically.Topic 29Derivatives of common functionsd/dx sin x = cos x, d/dx e^x = e^x, d/dx ln x = 1/x; the standard table of engineering calculus.Topic 30Applications of derivativesVelocity, acceleration, optimisation, related rates and linear approximation.Topic 31Maxima and minimaSet the derivative to zero and check the second derivative to find optimum designs.Topic 32IntegralsAccumulate quantities such as area, displacement and charge; the reverse of differentiation.Topic 33Fundamental theorem of calculusLinks derivatives and integrals: integrating a rate over an interval gives total change.Topic 34Integration techniquesSubstitution, integration by parts and partial fractions evaluate integrals analytically.Topic 35Definite integrals and areaEvaluate area under curves and work done by variable forces.Topic 36Volumes of revolutionDisk and shell methods compute volumes of rotationally symmetric parts.
Intermediate Engineering Mathematics
Topic 37Centroids and momentsIntegrals locate centres of mass and second moments used in beam design.Topic 38Improper integralsIntegrals with infinite limits or singularities; converge or diverge depending on decay rate.Topic 39Sequences convergence testsRatio, root and comparison tests decide whether infinite series converge.Topic 40Taylor and Maclaurin seriesRepresent smooth functions as polynomials; the basis of small-angle approximations.Topic 41Partial derivativesRates of change of multivariable functions with respect to one variable at a time.Topic 42GradientA vector of partial derivatives pointing in the direction of steepest increase.Topic 43Multiple integralsDouble and triple integrals compute areas, volumes, mass and inertia over regions.Topic 44Coordinate systemsCartesian, cylindrical and spherical coordinates match the symmetry of the problem.Topic 45Line and surface integralsIntegrate fields along curves and over surfaces; used for work, flux and circulation.Topic 46Divergence and curlMeasure how a vector field spreads out and rotates; appear in fluid and electromagnetic equations.Topic 47Green, Stokes and divergence theoremsConvert between boundary and interior integrals; the base of Maxwell equations and conservation laws.Topic 48MatricesRectangular arrays that represent linear transformations and systems of equations compactly.Topic 49Matrix operationsAddition, multiplication, transpose and inverse follow rules that mirror combining transformations.Topic 50DeterminantsA scalar that indicates invertibility and scales volume under a transformation.Topic 51Gaussian eliminationSystematic row operations that solve linear systems and reveal rank.Topic 52Linear independence and rankRank counts independent rows; it decides if a system has unique, many or no solutions.Topic 53Vector spaces and basisA basis is a minimal set of vectors that spans a space; coordinates depend on the chosen basis.Topic 54Eigenvalues and eigenvectorsAv=λv identifies directions a transformation only stretches; they give natural frequencies and stability.Topic 55DiagonalisationWriting A=PDP⁻¹ simplifies powers of matrices and decouples linear systems.Topic 56Orthogonality and projectionsOrthogonal vectors are independent; projections find the closest point in a subspace.Topic 57Least squaresFinds the best-fit solution to overdetermined systems by minimising squared error; used in regression and calibration.Topic 58Singular value decompositionFactorises any matrix into rotations and scaling; used for compression, noise reduction and pseudo-inverses.Topic 59First-order ODEsDifferential equations involving first derivatives; model cooling, charging and decay.Topic 60Separable equationsRearrange so each variable is on one side and integrate both sides.Topic 61Integrating factor methodSolves linear first-order ODEs by multiplying with a factor that makes the left side a derivative.Topic 62Second-order linear ODEsDescribe mass-spring-damper and RLC systems; solutions depend on the characteristic equation roots.Topic 63Homogeneous and particular solutionsGeneral solution is complementary plus particular; the first is natural response and the second is forced response.Topic 64Undetermined coefficientsGuess the form of the particular solution based on the forcing term.Topic 65Variation of parametersA general method to find particular solutions for any continuous forcing function.Topic 66Damping and resonanceUnder-, critically and over-damped responses; resonance occurs when forcing matches natural frequency.Topic 67Laplace transformConverts differential equations in time into algebraic equations in s; handles initial conditions neatly.Topic 68Transfer functionsRatio of output to input in the Laplace domain; characterises a linear system completely.Topic 69Systems of ODEsCoupled equations written as x'=Ax and solved using eigenvalues.Topic 70Fourier seriesRepresents periodic signals as sums of sines and cosines.
Advanced Engineering Mathematics
Topic 71Fourier transformDecomposes non-periodic signals into continuous frequency components.Topic 72Partial differential equationsEquations with several independent variables, such as heat, wave and Laplace equations.Topic 73Heat equationu_t = α u_xx describes diffusion of temperature through a material.Topic 74Wave equationu_tt = c² u_xx models vibrating strings, sound and electromagnetic waves.Topic 75Separation of variables for PDEsAssumes product solutions and reduces a PDE to ODEs with eigenvalue conditions.Topic 76Boundary and initial conditionsConstraints that select the one physical solution from many mathematical ones.Topic 77Probability basicsQuantifies uncertainty; probabilities lie between 0 and 1 and total 1 over all outcomes.Topic 78Conditional probability and BayesP(A|B)=P(B|A)P(A)/P(B); updates belief with evidence, used in diagnostics and filtering.Topic 79Random variablesNumerical outcomes of random processes, either discrete or continuous.Topic 80Expectation and varianceMean measures central value; variance measures spread around it.Topic 81Binomial and Poisson distributionsModel counts of successes in trials and rare events such as failures or arrivals.Topic 82Normal distributionThe bell curve that arises from many small independent effects; central to quality control.Topic 83Central limit theoremSums of many independent variables approach a normal distribution regardless of origin.Topic 84Descriptive statisticsMean, median, standard deviation and percentiles summarise datasets.Topic 85Confidence intervalsA range that would contain the true parameter in a stated fraction of repeated experiments.Topic 86Hypothesis testingUses sample data to decide between competing claims while controlling error rates.Topic 87Regression analysisFits relationships between variables and quantifies prediction error.Topic 88Reliability and failure distributionsExponential and Weibull models predict component life and maintenance intervals.Topic 89Monte Carlo simulationUses random sampling to estimate quantities that are hard to compute analytically.Topic 90Numerical error and conditioningRound-off and truncation errors can grow; ill-conditioned problems amplify them.Topic 91Root-finding methodsBisection, Newton-Raphson and secant methods solve f(x)=0 iteratively.Topic 92InterpolationEstimates values between known data points using polynomials or splines.Topic 93Numerical differentiationApproximates derivatives with finite differences of sampled values.Topic 94Numerical integrationTrapezoidal and Simpson rules approximate integrals from samples.Topic 95Solving ODEs numericallyEuler and Runge-Kutta methods step a differential equation forward in time.Topic 96Finite difference methodReplaces derivatives by differences on a grid to solve PDEs.Topic 97Finite element basicsSplits a domain into small elements with simple shape functions and assembles a global system.Topic 98Optimisation and Lagrange multipliersFind extrema subject to constraints by introducing multipliers that balance gradients.Topic 99Linear programmingOptimises a linear objective under linear constraints; solved by the simplex method.Topic 100Graph theoryNodes and edges model networks, circuits, schedules and routing.Topic 101Z-transform and difference equationsThe discrete-time counterpart of the Laplace transform for sampled systems.Topic 102TensorsMulti-index arrays that generalise vectors and matrices; used for stress, strain and deep learning.Topic 103Calculus of variationsFinds functions that minimise integrals, such as shortest paths and energy functionals.Topic 104Stochastic processesRandom variables evolving in time, such as Markov chains and Brownian motion.
Frequently asked questions
What is Engineering Mathematics?
The language of engineering: algebra, calculus, linear algebra, differential equations, probability and numerical methods.
How many topics are in Engineering Mathematics?
104 topics, ordered from beginner to advanced: 36 beginner, 34 intermediate and 34 advanced.
Where should a beginner start in Engineering Mathematics?
Start with "Number systems" and work through the list in order. Each topic builds on the one before it.
Do I need to be good at maths to study engineering?
You need steady practice rather than a special talent. Work through worked examples, check units and answers, and revisit topics that felt difficult after a few days.
Which maths topics matter most for every discipline?
Algebra, trigonometry, derivatives, integrals, vectors, and basic differential equations are used across almost all branches. Matrices and Laplace transforms become essential in control, circuits and structures.
Should I use a calculator or software?
Use software to check and to handle large problems, but learn the method first so you can judge whether the output is sensible.
How do I know if an answer is wrong?
Check the units, substitute the answer back into the equation, test a limiting case, and compare the magnitude with a rough estimate.
Is numerical methods work still maths?
Yes. It uses the same mathematics, but it adds an understanding of approximation error, stability and conditioning, which decide whether the computed answer can be trusted.
Practise in the interactive course with quizzes, tests and labs.