Plain-language explanation.
Calculus is the mathematics of change. It has two main branches: differential calculus (how things change — rates of change, slopes, derivatives) and integral calculus (how to find total quantities — areas, volumes, accumulated change). Together they form the foundation of modern science and engineering.
Core concepts and standard treatment.
Differential calculus starts with limits (the formal definition of the derivative), then rules for differentiation (power rule, product rule, quotient rule, chain rule), and applications (optimisation — finding maxima/minima, related rates, curve sketching). Integral calculus covers the definite and indefinite integral, the Fundamental Theorem of Calculus (connecting differentiation and integration), integration techniques (substitution, integration by parts, partial fractions), and applications (areas, volumes of revolution, work).
Deeper theory, debates and edge cases.
Advanced calculus extends to multivariable calculus (partial derivatives, gradient, divergence, curl — vector calculus; line integrals, surface integrals, Green's, Stokes', and Gauss's theorems), differential equations (ODEs — separable, linear, systems; PDEs — heat, wave, Laplace equations), and real analysis (epsilon-delta proofs, uniform convergence, Lebesgue integration — measure theory). Calculus of variations and Hamiltonian/Lagrangian mechanics extend it to physics.
How it is applied in practice.
At the applied mathematician, engineer, and quantitative analyst level, calculus underpins every field: structural analysis (stress equations), signal processing (Fourier transforms — FFT), control theory (Laplace transforms — PID controllers), financial mathematics (Black-Scholes PDE for options pricing — Itô calculus), machine learning (gradient descent — backpropagation), and physics simulation (finite element methods — FEA, CFD).