1.3 Taylor series and linearisation

The previous two lessons developed derivatives and integrals as the two inverse operations of calculus. This third lesson sits on top of both: a Taylor series uses derivatives at a single point to approximate a function in a neighbourhood. The first-order truncation — keeping only the constant and linear terms — is linearisation, and it is the single most-used technique in all of physics. Acoustics is the linearised theory of fluid mechanics. Optics is the linearised theory of Maxwell’s equations. The harmonic oscillator is the linearised pendulum. Newton’s law of cooling is the linearised Stefan–Boltzmann law.

This lesson develops Taylor series and the linearisation it makes precise.

The Taylor expansion

For a smooth function $f$ near a base point $t_0$ , the Taylor expansion is

\boxed{\;f(t_0 + \varepsilon) \;=\; f(t_0) \;+\; \varepsilon\, f'(t_0) \;+\; \tfrac{1}{2!} \varepsilon^2\, f''(t_0) \;+\; \tfrac{1}{3!} \varepsilon^3\, f'''(t_0) \;+\; \cdots\;}

In more compact form,

f(t_0 + \varepsilon) \;=\; \sum_{n=0}^{\infty} \frac{\varepsilon^n}{n!}\, f^{(n)}(t_0).

The series gives the value of $f$ at the displaced point $t_0 + \varepsilon$ in terms of the value of $f$ and all its derivatives at the base point $t_0$ . The remarkable claim is that all the information about a smooth $f$ near $t_0$ is contained in those derivatives.

Truncating the series at degree $N$ produces the Taylor polynomial $T_N(t; t_0)$ , a polynomial of degree $N$ that matches $f$ and its first $N$ derivatives at $t_0$ . The error in this approximation is $\mathcal{O}(\varepsilon^{N+1})$ — vanishingly small if $\varepsilon$ is small and the function is well-behaved, but growing as you move away from $t_0$ .

function:

expansion point x₀ = 0.00 degree N = 3

The interactive builds the Taylor polynomial $T_N(x; x_0)$ approximation of a chosen function. Slide the expansion point $x_0$ and the degree $N$ and watch the red approximation track the black true curve near $x_0$ — and diverge from it as you move far away. Two things to feel for:

Near the expansion point, even low-order Taylor polynomials are nearly perfect approximations. Linear ( $N=1$ ) suffices for very small $\varepsilon$ ; quadratic ( $N=2$ ) extends the agreement; higher orders are needed only for larger neighbourhoods or for the very tails of the function.
Far from the expansion point, the approximation diverges. The radius over which the series converges is the famous radius of convergence; some functions (like $1/(1-x)$ around $x_0 = 0$ ) have a finite radius and diverge outside.

The Maclaurin series: Taylor at zero

When $t_0 = 0$ , the Taylor series is sometimes called the Maclaurin series. The most-used cases in physics:

e^x \;=\; 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots

\sin x \;=\; x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots

\cos x \;=\; 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots

(1 + x)^p \;=\; 1 + p x + \frac{p(p-1)}{2!} x^2 + \frac{p(p-1)(p-2)}{3!} x^3 + \cdots \qquad (\text{binomial series})

\ln(1 + x) \;=\; x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots

The radii of convergence: $e^x$ , $\sin x$ , $\cos x$ converge for all $x$ . The binomial and logarithmic series converge only for $|x| < 1$ .

The Maclaurin series for $e^x$ with $x = i\theta$ is what gives Euler’s formula — see Foundations 3.1. The cancellation of every fourth term in $(i\theta)^n / n!$ produces the sine and cosine series exactly.

Linearisation: first-order Taylor

The first-order truncation of the Taylor expansion is

f(t_0 + \varepsilon) \;\approx\; f(t_0) + \varepsilon\, f'(t_0) + \mathcal{O}(\varepsilon^2).

This is the linearisation of $f$ around $t_0$ . The approximation is excellent for small $\varepsilon$ and useless for large.

Linearisation is the move that takes a nonlinear physics problem and converts it to a linear one. The full pendulum equation,

\ddot \theta \;+\; \frac{g}{L} \sin\theta \;=\; 0,

is nonlinear because $\sin\theta$ depends nonlinearly on $\theta$ . But for small angles,

\sin\theta \;=\; \theta - \frac{\theta^3}{6} + \cdots \;\approx\; \theta + \mathcal{O}(\theta^3),

so the linearised pendulum is

\ddot \theta \;+\; \frac{g}{L}\, \theta \;=\; 0,

which is the simple-harmonic-motion equation of Foundations 5.3 and has the closed-form solution $\theta(t) = A \cos(\omega_0 t + \varphi)$ with $\omega_0 = \sqrt{g/L}$ . For amplitudes below about $30°$ this is essentially exact; above that, the cubic correction starts to matter.

Linearisation is the master move of acoustics

Every time the Sound book writes “small perturbation” or " $|p'| \ll p_0$ " or “linearised theory,” it is quietly invoking a first-order Taylor expansion. The wave equation itself is the result of linearising the full nonlinear fluid-mechanics equations around a still-air equilibrium:

The continuity equation $\partial_t \rho + \nabla \cdot (\rho \mathbf{v}) = 0$ is exact but nonlinear (because of the $\rho \mathbf{v}$ product). Linearise around equilibrium $\rho = \rho_0 + \rho'$ with $|\rho'| \ll \rho_0$ and the equation becomes $\partial_t \rho' + \rho_0 \nabla \cdot \mathbf{v} = 0$ , linear in the small perturbations.
Euler’s equation $\rho (\partial_t \mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v}) = -\nabla p$ has a nonlinear convective term $\mathbf{v} \cdot \nabla \mathbf{v}$ . Linearising drops it (since it is quadratic in the small velocity), leaving $\rho_0 \partial_t \mathbf{v} = -\nabla p'$ .
The adiabatic equation of state $p \propto \rho^\gamma$ becomes $p' = c^2 \rho'$ with $c^2 = \gamma p_0 / \rho_0$ — a one-line Taylor expansion of pressure-as-a-function-of-density around equilibrium.

Combine the three linearised equations and you get the wave equation $\partial_t^2 p = c^2 \nabla^2 p$ . Everything in the Sound book is downstream of this single linearisation. See Sound 4.5 — Linearisation and the wave equation for the full development.

The same move runs through:

Optics. Maxwell’s equations are linear in the EM fields but nonlinear in the medium-dependent permittivity; nonlinear optics treats the higher-order terms.
Elasticity. Hooke’s law $\sigma = E \epsilon$ is the linearisation of stress-strain around the unstressed state.
Cosmology. Linearised general relativity around a Friedmann–Robertson–Walker background gives the equations for cosmological perturbations.
Chemistry. Linearisation of chemical kinetics around a reaction equilibrium gives the linear-response theory of fluctuations.

In every case, the underlying physics is nonlinear; linearisation is what makes it tractable.

Higher-order corrections

When linearisation is insufficient — when the perturbation is large enough that the $\varepsilon^2$ term cannot be ignored — you keep more terms. The second-order Taylor expansion is

f(t_0 + \varepsilon) \;\approx\; f(t_0) + \varepsilon\, f'(t_0) + \tfrac{1}{2} \varepsilon^2\, f''(t_0).

This is the quadratic approximation, used whenever the linear term vanishes (e.g. at an extremum) or when the first-order correction is too coarse. In acoustics, quadratic terms appear in energy and intensity calculations — kinetic energy is $\tfrac12 \rho v^2$ , quadratic in velocity — and they are exactly what links the linear acoustic field to the nonlinear quantities (energy, momentum, intensity) one ultimately wants to measure.

The third-order corrections appear in nonlinear acoustics — when amplitudes are large enough that even the quadratic terms aren’t sufficient. See Sound 10.4 — Wave steepening for what happens when those corrections compound.

What we use this for

Calculus operations across the bookshelf:

The wave equation comes from linearising fluid mechanics around a still-air equilibrium (Sound 4.5).
The speed of sound $c = \sqrt{(\partial p / \partial \rho)_s}$ is a derivative evaluated at equilibrium.
Energy, intensity, and power are integrals of pressure-times-velocity over time.
The Fourier coefficients are integrals; the Fourier transform itself is an improper integral (Foundations 7).
Euler’s formula $e^{i\theta} = \cos\theta + i\sin\theta$ is derived from the Taylor series of all three functions (Foundations 3.1).
Every “small-angle approximation,” “weak-field limit,” or “leading-order analysis” in physics is a first-order Taylor expansion.

If any of the above looks unfamiliar, work through a textbook chapter on single-variable calculus before continuing. Spivak’s Calculus is the unsentimental favourite; Strang’s Calculus is the most physics-friendly.