Physics does not just use calculus as a tool — it is written in calculus. Newton's laws, Maxwell's equations, Schrödinger's equation, Einstein's field equations — all of them are differential equations. Understanding what they say, not just how to use them, requires understanding what a derivative and integral actually mean.
Newton's Laws in Calculus Form
F = ma = m·d²x/dt². Force is mass times second derivative of position. Velocity: v = dx/dt. Acceleration: a = dv/dt = d²x/dt². From F(x), solving the DE gives the full trajectory — this is classical mechanics.
Work, Energy, and Power
Work: W = ∫F·dx (integral of force over displacement). Kinetic energy: KE = ½mv² (integral of force for constant-mass system). Power: P = dW/dt (derivative of work). The work-energy theorem: W_net = ΔKE.
Maxwell's Equations
Gauss's Law: ∇·E = ρ/ε₀. Faraday's Law: ∇×E = −∂B/∂t. Ampère-Maxwell: ∇×B = μ₀J + μ₀ε₀·∂E/∂t. These four PDEs — all in the language of vector calculus — describe all electromagnetic phenomena, including light.
Thermodynamics and Heat
The heat equation ∂T/∂t = α∇²T describes how temperature diffuses. It is a PDE whose solutions involve Fourier series — themselves computed using integrals. Every heat transfer problem in engineering requires this calculus.
Physics is the original motivation for calculus. Newton invented calculus specifically to describe motion — every law of classical mechanics is a differential equation. Modern physics from quantum mechanics to general relativity is written entirely in the language of advanced calculus.
Classical Mechanics — The Original Application
Newton's second law F = ma is, in full notation, F(x, t) = m·d²x/dt². Given a force law, solving this second-order ODE gives the trajectory x(t). For gravity near Earth's surface, F = −mg (constant), and the ODE gives x(t) = x₀ + v₀t − ½gt² — the familiar projectile motion formula. For a spring, F = −kx (Hooke's Law), and the ODE gives x(t) = A cos(ωt + φ) — simple harmonic motion. Every classical mechanics problem reduces to setting up and solving a differential equation.
Work, Energy, and Conservation Laws
Work is defined as W = ∫F·dx — the integral of force over displacement. This definition handles variable forces correctly; the constant-force formula W = F·d is just the special case of a constant integrand. The work-energy theorem — W_net = ΔKE — is proved by integrating Newton's second law. Conservation of energy (when total work by non-conservative forces is zero) follows directly from path-independence of conservative force integrals.
Electromagnetism — Maxwell's Equations
The four Maxwell's equations, in differential (local) form, are: ∇·E = ρ/ε₀ (Gauss's Law for electricity — charge creates diverging electric field), ∇·B = 0 (no magnetic monopoles — magnetic field lines are closed loops), ∇×E = −∂B/∂t (Faraday's Law — changing magnetic field creates electric field), ∇×B = μ₀J + μ₀ε₀·∂E/∂t (Ampère-Maxwell Law — current and changing electric field create magnetic field).
From these four PDEs, Maxwell derived in 1865 that electromagnetic waves exist and travel at speed c = 1/√(μ₀ε₀) ≈ 3×10⁸ m/s — the speed of light. The identification of light as an electromagnetic wave was a purely mathematical deduction. This is the most spectacular application of vector calculus in history.
Quantum Mechanics — The Schrödinger Equation
The time-dependent Schrödinger equation: iℏ·∂ψ/∂t = Ĥψ, where Ĥ = −ℏ²/(2m)·∇² + V(x). This is a partial differential equation for the wave function ψ(x,t). The probability of finding a particle in region R at time t is P = ∫_R |ψ|² dV — a triple integral of the squared magnitude. All measurable quantities (energy, momentum, position) are computed as integrals: ⟨A⟩ = ∫ψ*Âψ dV. Quantum mechanics is, mathematically, a theory of Hilbert spaces, differential operators, and integration.
Thermodynamics and Statistical Mechanics
The first law of thermodynamics: dU = δQ − δW, where dU is the change in internal energy, δQ is heat added, and δW = ∫P dV is work done by the system (an integral over volume change). The entropy S satisfies dS = δQ/T, with total entropy change ΔS = ∫dQ/T. The Maxwell-Boltzmann distribution for molecular speeds is a continuous probability density function whose moments (mean speed, rms speed) are integrals. Statistical mechanics — connecting microscopic particles to macroscopic thermodynamic quantities — is built entirely on integration over phase space.
Fluid Dynamics
The Navier-Stokes equations, describing viscous fluid flow: ρ(∂v/∂t + v·∇v) = −∇P + μ∇²v + f. These nonlinear PDEs involve divergence, gradient, Laplacian, and material derivatives. Whether smooth solutions always exist for the 3D Navier-Stokes equations is one of the seven Millennium Prize Problems — a $1 million unsolved question in mathematics. Every simulation of weather, airplane aerodynamics, or blood flow through arteries numerically solves a version of these equations.
- Kleppner, D. & Kolenkow, R. (2014). An Introduction to Mechanics, 2nd ed. Cambridge UP.
- Griffiths, D.J. (2017). Introduction to Electrodynamics, 4th ed. Cambridge UP.
- Feynman, R.P. (2011). The Feynman Lectures on Physics, Vol. 2. Basic Books.