| Calculus | |
|---|---|
| Room | Systems |
| Field | Mathematics |
| Known for | Derivative, integral, fundamental theorem, differential equations |
| Key figures | Newton, Leibniz, Euler, Cauchy, Riemann, Weierstrass |
Calculus is the mathematics of continuous change. It provides the language for physics, the tools for engineering, the framework for machine learning, and the analytical foundation for economics and finance.
Eudoxus (c. 370 BC) developed the method of exhaustion — approximating curved shapes with an infinite sequence of polygons — a direct precursor to the concept of a limit. Archimedes (c. 225 BC) computed the area of a parabolic segment by summing an infinite geometric series — the first known summation of an infinite series.
Zeno of Elea's paradoxes (c. 450 BC) articulated the philosophical problem of infinity. The Kerala school in India (14th–16th centuries) discovered infinite series expansions for trigonometric functions 200 years before Europe. Nicole Oresme (c. 1350) drew the first velocity-time graphs, anticipating integration.
Isaac Newton (1643–1727) developed his method of fluxions during the plague years (1665–66), applying it to derive Kepler's laws from his inverse-square law of gravitation. He published in the Principia (1687) in geometric form. Gottfried Leibniz (1646–1716) independently developed calculus in the 1670s. His notation — dx for differentials, ∫ for integration — proved far more supple and made calculus teachable.
Leonhard Euler (1707–1783) transformed calculus into a systematic algebraic discipline. His Introductio (1748) defined functions as the central object. He unified differential and integral calculus and developed the calculus of variations. The Bernoulli family extended calculus to fluid dynamics and the brachistochrone problem.
Bishop Berkeley mocked infinitesimals as "the ghosts of departed quantities." Augustin-Louis Cauchy (1789–1857) gave calculus its modern foundations, defining the derivative and integral through the limit. Bernhard Riemann refined the integral (1854). Karl Weierstrass completed the rigorization with the ε-δ definition.
Supporting figures: Bernard Bolzano (anticipated limits), Richard Dedekind (real numbers via cuts), Joseph Fourier (heat equation forced the rigor crisis).
Henri Lebesgue (1875–1941) generalized the integral to measure theory. Abraham Robinson (1918–1974) made infinitesimals rigorous through non-standard analysis. Andrey Kolmogorov axiomatized probability using the Lebesgue integral.
The Fundamental Theorem bridges differentiation and integration. Part 1: For continuous f, F(x) = ∫ₐˣ f(t)dt is differentiable with F'(x) = f(x). Part 2: If F is an antiderivative of f, then ∫ₐᵇ f(x)dx = F(b) − F(a). This makes calculus computable — finding areas reduces to finding antiderivatives.
Physics: Newton's second law F = ma is the ODE m·d²x/dt² = F. Maxwell's equations are four PDEs. The Schrödinger equation governs quantum behavior.
Engineering: The Euler-Bernoulli beam equation EI·d⁴y/dx⁴ = w(x) designs every bridge. RLC circuits follow L·d²I/dt² + R·dI/dt + I/C = dV/dt. The Fourier Transform powers signal processing (JPEG, MP3). PID controllers use derivative and integral terms.
Machine Learning: Neural networks train via gradient descent — θₜ₊₁ = θₜ − η·∇L(θₜ). Backpropagation uses the chain rule dL/dw = dL/dy·dy/dz·dz/dw across millions of parameters.
Finance: The Black-Scholes PDE ∂V/∂t + ½σ²S²·∂²V/∂S² + rS·∂V/∂S − rV = 0 launched modern derivatives markets.