MathematicsYear 2Quality review 0/4
Mathematics — Year 2
The second-year core of a university mathematics degree: calculus stretched into several dimensions, linear algebra deepened into eigenvalues and inner-product spaces, differential equations, and the numerical methods that make it all computable. Every idea arrives twice — once rigorously, once through an everyday picture — with worked examples grounded in South African life and over fifty practice problems with full solutions.
≈ 13 h 25 min8 sections · 32 lessonsNo ratings yet
What you'll learn
- Differentiate and integrate functions of several variables — partial derivatives, gradients, directional derivatives, and double, triple and line integrals — and interpret each geometrically and physically
- Analyse linear maps through eigenvalues, eigenvectors and diagonalisation, and work confidently in inner-product spaces using norms, orthogonality and projection
- Solve first-order separable and linear ordinary differential equations and second-order linear homogeneous equations with constant coefficients, and use them to model growth, decay and oscillation
- Apply core numerical methods — bisection, Newton's method, the trapezoidal and Simpson's rules, and Euler's method — to find roots, integrals and ODE solutions when no closed form exists
Before you start
- First-year single-variable calculus: limits, differentiation and integration of one-variable functions
- First-year linear algebra: vectors, matrices, determinants and solving systems of linear equations
- Fluency with functions, trigonometry, and the exponential and natural-logarithm functions
Academic references
Use the latest available edition and your institution's prescribed text.
- OpenStax, Calculus, Volumes 1–3
- Gilbert Strang, Introduction to Linear Algebra
- Daniel J. Velleman, How to Prove It
Formal work standard
- State definitions and hypotheses before using a theorem.
- Separate proof, counterexample, and numerical evidence.
- Complete the problem sheet without solution steps before reviewing errors.
Course content
0/32 lessons
1. Multivariable & Vector Calculus4 lessons
- Partial DerivativesLesson25 min
- The Gradient and Directional DerivativesLesson25 min
- Double and Triple IntegralsLesson25 min
- Vector Fields: Divergence, Curl & Line IntegralsLesson30 min
2. Linear Algebra II3 lessons
- Eigenvalues and EigenvectorsLesson25 min
- DiagonalisationLesson25 min
- Inner-Product Spaces & OrthogonalityLesson25 min
3. Ordinary Differential Equations3 lessons
- First-Order Separable ODEsLesson25 min
- First-Order Linear ODEs & the Integrating FactorLesson25 min
- Second-Order Linear Homogeneous ODEsLesson30 min
4. Numerical Methods3 lessons
- Root Finding: Bisection & Newton's MethodLesson25 min
- Numerical Integration: Trapezoidal & Simpson's RulesLesson25 min
- Numerical Solutions of ODEs: Euler's MethodLesson20 min
5. Multivariable and Vector Calculus: Theorems and Geometry5 lessons
- Multivariable Limits and ContinuityLesson25 min
- The Multivariable Chain Rule and JacobiansLesson25 min
- Change of Variables in Multiple IntegralsLesson25 min
- Surface Integrals and FluxLesson25 min
- Green, Stokes and Divergence TheoremsLesson25 min
6. Linear Algebra II: Decompositions and Canonical Structure5 lessons
- Characteristic PolynomialsLesson25 min
- The Spectral TheoremLesson25 min
- Gram–Schmidt and QR FactorisationLesson25 min
- Jordan Form and Generalised EigenvectorsLesson25 min
- Quadratic Forms and DefinitenessLesson25 min
7. Ordinary Differential Equations: Methods and Systems5 lessons
- Existence, Uniqueness and Phase LinesLesson25 min
- Exact and Bernoulli EquationsLesson25 min
- Laplace Transforms for Initial-Value ProblemsLesson25 min
- Linear Systems of ODEsLesson25 min
- Power-Series SolutionsLesson25 min
8. Numerical Analysis: Error, Linear Systems and Approximation4 lessons
- Error, Conditioning and StabilityLesson25 min
- LU Factorisation and PivotingLesson25 min
- Polynomial and Spline InterpolationLesson25 min
- Numerical OptimisationLesson25 min