A precise definition
Applied mathematics uses mathematical tools — calculus, linear algebra, differential equations, optimisation, statistics, numerical methods — to model and solve problems in science, engineering, economics, and technology. Its sub-areas include: mathematical physics, fluid mechanics, mathematical biology, operations research, numerical analysis, and mathematical finance. The 'applied' label means the mathematics is motivated by external problems — but the mathematics itself is often profound.
The problem it was invented to solve
Newton's mechanics, Maxwell's electromagnetism, and Einstein's relativity all developed as applied mathematics — the mathematics was invented to describe physical phenomena. The 20th century saw applied mathematics expand to economics (game theory, Nash, 1950), biology (evolutionary game theory, Maynard Smith), and computation (complexity theory, algorithms). The distinction from pure mathematics is increasingly blurry: many of the deepest pure results arose from applied investigations.
Where you find it in the world — including South Africa
These are not contrived textbook examples. Each application below is currently in use, driven by real institutions, and producing real outcomes.
Climate modelling: South Africa's SAWS and academic centres
The South African Weather Service and climate research groups at UCT, Wits, and UKZN use coupled atmosphere-ocean models to simulate and forecast SA climate under different emissions scenarios. These models are systems of PDEs solved on supercomputing infrastructure at CHPC in Cape Town.
Operations research: Eskom, Transnet, SAA
Operations research (OR) — the application of mathematical optimisation to operational decisions — is used by every major SA utility and transport company. Eskom uses OR to schedule generation and maintenance. Transnet uses it to schedule train movements. SAA used it for crew scheduling.
Mathematical finance: JSE and SA banking
The quantitative finance groups at Standard Bank, FirstRand, Investec, and Rand Merchant Bank use applied mathematics daily: stochastic calculus for derivative pricing, optimisation for portfolio management, PDEs for risk calculation.
Epidemiology: UKZN Centre for Epidemiological Modelling
Mathematical epidemiology — using differential equations and stochastic models to understand disease spread — is a major research area at SA universities. UKZN's SACEMA (South African Centre for Epidemiological Modelling and Analysis) has produced pandemic response models used by the SA government.
You've already encountered this
The Wolfram-Physics Project attempts to derive all of physics from a simple computational rule — a deeply applied mathematics project that is also deeply foundational. The dividing line between pure and applied mathematics is a sociological distinction, not a mathematical one.
Where it connects in the map of mathematics
Related topics and institutions
Applied mathematics starts at Grade 12 and runs through every STEM career.
The Continuum develops the mathematical modelling mindset — translating real problems into equations — that makes applied mathematics accessible from the moment you start university.
No card required. South African curriculum. Grade 8–12.