Glossary

Short definitions of the terms this book uses, from A to Z, each with a link to the chapter that teaches it.

A

Absolute convergence A series \(\sum a_n\) converges absolutely if \(\sum |a_n|\) converges. Absolute convergence implies convergence, and an absolutely convergent series can be rearranged without changing its sum. See Chapter 12.

Absolute value The function \(|x|\), equal to \(x\) for \(x \ge 0\) and \(-x\) for \(x < 0\). It is continuous everywhere but has a corner at \(0\), where it is not differentiable. See Chapter 3 and Chapter 4.

Absolutely continuous A function \(F\) on \([a, b]\) for which small total change in \(x\) forces small total change in \(F\): for every \(\varepsilon > 0\) there is \(\delta > 0\) such that disjoint intervals of total length less than \(\delta\) have \(\sum |F(b_k) - F(a_k)| < \varepsilon\). These are exactly the functions \(F(x) = F(a) + \int_a^x f\) with \(f\) Lebesgue integrable. See Chapter 21.

Acceleration The rate of change of velocity, the second derivative of position: \(a(t) = v'(t) = s''(t)\). Near the Earth's surface a falling stone has acceleration about \(9.8\) m/s², and on Mars about \(3.71\) m/s². See Chapter 4 and Chapter 7.

Accumulation function A function \(F(x) = \int_a^x f(t)\,dt\) that records the signed area under \(f\) from a fixed point \(a\) up to \(x\). It is always continuous, and the fundamental theorem says its derivative is \(f\) where \(f\) is continuous. See Chapter 9.

Almost everywhere A property holds almost everywhere (a.e.) if it fails only on a set of measure zero. Functions equal almost everywhere have the same Lebesgue integral. See Chapter 21.

Alternating harmonic series The series \(1 - \frac12 + \frac13 - \frac14 + \cdots\), which converges to \(\ln 2\) but not absolutely. Rearranging its terms can change its sum. See Chapter 12 and Chapter 12.

Alternating series test If \(b_n\) decreases to \(0\), then \(\sum (-1)^{n+1} b_n\) converges, and the error after \(N\) terms is at most \(b_{N+1}\). It is due to Leibniz. See Chapter 12.

Antiderivative A function \(F\) with \(F' = f\) on an interval. Any two antiderivatives of \(f\) on an interval differ by a constant, and by the fundamental theorem any one of them computes \(\int_a^b f = F(b) - F(a)\). See Chapter 9 and Chapter 9.

Approximate identity A family of kernels, such as the Fejér kernels, that are nonnegative, have total integral \(1\) (after normalising) and concentrate their mass near \(0\). Convolving with them reproduces a continuous function in the limit. See Chapter 21.

Arc length The length of a curve, \(\int_a^b \sqrt{1 + f'(x)^2}\,dx\) for the graph of \(f\), found by adding the lengths of small straight pieces. See Chapter 11.

Archimedean property For every real number \(x\) there is a whole number \(n > x\); equivalently \(\frac1n \to 0\). It follows from completeness, and it is why the rationals are dense in the reals. See Chapter 12 and Chapter 18.

Archimedes' quadrature of the parabola Archimedes' theorem that a parabolic segment has \(\frac43\) the area of its inscribed triangle, proved by the method of exhaustion and a double contradiction. See Chapter 1.

Attracting fixed point A fixed point \(x^*\) of a map \(f\) with \(|f'(x^*)| < 1\): points that start near it are pulled in under iteration. See Chapter 22.

Average speed Distance covered divided by time taken over an interval: \(\frac{s(b) - s(a)}{b - a}\). It is the slope of a secant line on the distance-time graph. See Chapter 1.

Average value The average of \(f\) on \([a, b]\), \(\frac{1}{b - a}\int_a^b f\). By the mean value theorem for integrals a continuous function takes its average value somewhere. For a region in the plane, divide the double integral by the area. See Chapter 8, Chapter 9 and Chapter 16.

B

Baire category theorem A complete metric space is not a countable union of nowhere dense sets. It underlies the uniform boundedness principle, the open mapping theorem and the closed graph theorem. See Chapter 22.

Banach space A normed vector space that is complete in the distance \(\|v - w\|\), such as \(\mathbb{R}^n\), \(C[a, b]\) with the sup norm, or \(L^1\). See Chapter 22.

Basel problem The problem of summing \(\sum \frac{1}{n^2}\). Euler found the answer \(\frac{\pi^2}{6}\) in 1734. See Chapter 12.

Bernoulli's inequality For \(r \ge 1\) and \(x > -1\), \((1 + x)^r \ge 1 + rx\), proved with the mean value theorem. See Chapter 6.

Bernstein polynomials The polynomials \(B_n f(x) = \sum_{k=0}^{n} f(\frac kn)\binom nk x^k(1 - x)^{n-k}\), which converge uniformly to a continuous \(f\) on \([0, 1]\) and so prove the Weierstrass approximation theorem. See Chapter 19.

Bessel's inequality For a square-integrable \(f\), the sum of the squares of its Fourier coefficients is at most \(\frac{1}{2\pi}\int_{-\pi}^{\pi} |f|^2\). See Chapter 21.

Bijection A map between two sets that is one-to-one and onto, pairing their elements exactly. Two sets have the same cardinality when there is a bijection between them. See Chapter 18.

Binomial series The series \((1 + x)^\alpha = \sum_{n \ge 0} \binom{\alpha}{n} x^n\) for any real \(\alpha\), valid for \(|x| < 1\). See Chapter 13.

Bisection method A way to find a root of a continuous function: halve an interval where the sign changes and keep the half where it still changes. After \(n\) steps the root is pinned to an interval of length \(\frac{b - a}{2^n}\). See Chapter 3.

Bolzano-Weierstrass theorem Every bounded sequence of real numbers has a convergent subsequence. See Chapter 18.

Bounded A function is bounded on a set if some number \(K\) has \(|f(x)| \le K\) there; a sequence or set is bounded if all its terms or points lie within some fixed distance of \(0\). A continuous function on \([a, b]\) is bounded. See Chapter 3 and Chapter 12.

Bounded above A set \(S\) is bounded above if some number \(M\) has \(s \le M\) for every \(s\) in \(S\); such an \(M\) is an upper bound. See Chapter 18.

Bump function A smooth function that is positive on an interval and zero outside it, built from \(e^{-1/x}\). Such functions show that smooth functions need not be analytic. See Chapter 13.

C

Cantor function The "devil's staircase": a continuous increasing function from \(0\) to \(1\) that is constant on every interval removed in building the Cantor set, so its derivative is \(0\) almost everywhere. See Chapter 20.

Cantor set What is left of \([0, 1]\) after removing the open middle third, then the middle thirds of the two pieces left, and so on forever. It has length zero, yet it is uncountable, compact, nowhere dense and perfect. See Chapter 20.

Cantor's theorem The real numbers are uncountable: no list contains every number in \((0, 1)\), as Cantor's diagonal argument shows. See Chapter 18.

Cardinality The size of a set. Two sets have the same cardinality if there is a bijection between them; this is how infinite sets are compared. See Chapter 18.

Catenary The shape of a chain hanging under its own weight, the graph of \(y = a\cosh(x/a)\). See Chapter 11.

Cauchy integral formula For \(f\) holomorphic inside and on a closed curve \(\gamma\), \(f(a) = \frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{z - a}\,dz\) at every point \(a\) inside. The values on the boundary determine the values inside. See Chapter 22.

Cauchy principal value The symmetric limit \(\lim_{b \to \infty} \int_{-b}^{b} f\), which can exist even when the improper integral \(\int_{-\infty}^{\infty} f\) does not. See Chapter 11.

Cauchy sequence A sequence whose terms get close to each other: for every \(\varepsilon > 0\), \(|a_m - a_n| < \varepsilon\) for all large \(m\) and \(n\). In \(\mathbb{R}\) a sequence converges if and only if it is Cauchy; a metric space where this holds is complete. See Chapter 12, Chapter 18 and Chapter 20.

Cauchy-Hadamard formula The radius of convergence of \(\sum c_n (x - a)^n\) is \(R = 1/\limsup |c_n|^{1/n}\). See Chapter 13.

Cauchy-Riemann equations The equations \(u_x = v_y\) and \(u_y = -v_x\) that the real and imaginary parts of a holomorphic function \(f = u + iv\) satisfy. See Chapter 22.

Cauchy's mean value theorem If \(f\) and \(g\) are continuous on \([a, b]\) and differentiable inside, there is a \(c\) with \((f(b) - f(a))g'(c) = (g(b) - g(a))f'(c)\). It is the key to L'Hôpital's rule. See Chapter 6.

Cauchy's theorem If \(f\) is holomorphic on an open set containing a closed curve \(\gamma\) and the region it encloses, then \(\oint_\gamma f(z)\,dz = 0\). See Chapter 22.

Cavalieri's principle Two solids with cross-sections of equal area at every height have equal volumes. See Chapter 11.

Centre of mass The balance point of a body, the average position weighted by mass: \(\bar x = \frac{\int x\,dm}{\int dm}\). For a uniform plane region it is the centroid. See Chapter 11 and Chapter 16.

Centroid The centre of mass of a uniform plane region: the average of position over the region. See Chapter 11.

Cesàro means The averages \(\sigma_N f = \frac{1}{N + 1}(S_0 f + \cdots + S_N f)\) of the partial sums of a Fourier series. Fejér proved they converge uniformly to any continuous periodic \(f\). See Chapter 21.

Chain rule The derivative of a composition: \((f \circ g)'(x) = f'(g(x))\,g'(x)\), or \(\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}\). Rates multiply. In several variables the Jacobian matrices multiply. See Chapter 5 and Chapter 15.

Chaos Behaviour of a deterministic dynamical system that shows sensitive dependence on initial conditions, as the logistic map does at \(r = 4\). See Chapter 22.

Characteristic equation For \(ay'' + by' + cy = 0\), the quadratic \(ar^2 + br + c = 0\). Its roots \(r\) give the solutions \(e^{rx}\), and real, repeated or complex roots give the three kinds of solution. See Chapter 14.

Circulation The line integral \(\oint_C \mathbf F \cdot d\mathbf r\) of a vector field around a closed curve: how much the field pushes along the curve. See Chapter 17.

Clairaut's theorem If the mixed partial derivatives \(f_{xy}\) and \(f_{yx}\) are continuous, they are equal. Also credited to Schwarz. See Chapter 15.

Closed graph theorem A linear map between Banach spaces whose graph is closed is bounded. One of the four big theorems of functional analysis. See Chapter 22.

Closed set A set that contains all its limit points; equivalently, its complement is open. A set is closed exactly when every convergent sequence in it has its limit in it. See Chapter 20.

Closure A set together with all its limit points: the smallest closed set containing it. See Chapter 20.

Common refinement A partition containing all the points of two given partitions, used to compare their upper and lower sums. See Chapter 8.

Compact A set is compact if every open cover has a finite subcover; in a metric space this is the same as sequentially compact. In \(\mathbb{R}^n\) the compact sets are the closed bounded ones (Heine-Borel). See Chapter 3 and Chapter 20.

Comparison test If \(0 \le a_n \le b_n\) and \(\sum b_n\) converges, so does \(\sum a_n\); if \(\sum a_n\) diverges, so does \(\sum b_n\). The same holds for improper integrals. See Chapter 11 and Chapter 12.

Complete metric space A metric space in which every Cauchy sequence converges. \(\mathbb{R}\) is complete; \(\mathbb{Q}\) is not. See Chapter 20.

Completeness axiom Every nonempty set of real numbers that is bounded above has a least upper bound. It is the property that separates \(\mathbb{R}\) from \(\mathbb{Q}\), and every big theorem of calculus rests on it. See Chapter 3 and Chapter 18.

Composition The function \(f \circ g\), given by \((f \circ g)(x) = f(g(x))\): apply \(g\), then \(f\). A composition of continuous functions is continuous. See Chapter 3 and Chapter 5.

Concave up / concave down A graph is concave up where it bends upwards, like a cup (\(f'' > 0\) there), and concave down where it bends downwards (\(f'' < 0\)). See Chapter 7.

Conditional convergence A series converges conditionally if it converges but \(\sum |a_n|\) diverges, like the alternating harmonic series. By Riemann's rearrangement theorem, its terms can be reordered to give any sum. See Chapter 12.

Connected A metric space or set is connected if it cannot be split into two disjoint nonempty open pieces. The connected subsets of \(\mathbb{R}\) are exactly the intervals. See Chapter 20.

Conservative vector field A field \(\mathbf F\) that is the gradient of a potential, \(\mathbf F = \nabla f\). Its line integrals depend only on the endpoints and vanish around closed curves. See Chapter 17.

Constraint An equation such as \(g(x, y) = c\) that the points must satisfy in an optimisation problem. See Chapter 15.

Continuous at a point \(f\) is continuous at \(a\) if \(\lim_{x \to a} f(x) = f(a)\): the value at \(a\) is what the nearby values are heading towards. In epsilon-delta form, for every \(\varepsilon > 0\) there is \(\delta > 0\) with \(|f(x) - f(a)| < \varepsilon\) whenever \(|x - a| < \delta\). See Chapter 3 and Chapter 18.

Continuous extension Filling in a removable discontinuity by giving the function its limit as its value there, such as setting \(\frac{\sin x}{x}\) equal to \(1\) at \(0\). See Chapter 3.

Continuous function A function continuous at every point of its domain (or of an interval). In a metric space, \(f\) is continuous exactly when the preimage of every open set is open. See Chapter 3 and Chapter 20.

Continuously differentiable A function whose partial derivatives all exist and are continuous, written \(C^1\). Such a function is differentiable. See Chapter 15.

Continuum hypothesis The statement that there is no set whose size is strictly between that of \(\mathbb{N}\) and that of \(\mathbb{R}\). It can be neither proved nor disproved from the usual axioms. See Chapter 18.

Contour map A picture of a function of two variables drawn with its level curves, as on a map of hills. See Chapter 15.

Contraction A map \(T\) of a metric space with \(d(Tx, Ty) \le k\,d(x, y)\) for some constant \(k < 1\). See Chapter 20.

Contraction mapping theorem A contraction of a nonempty complete metric space has exactly one fixed point, and iterating from any starting point converges to it. It was proved by Banach in 1922. See Chapter 14 and Chapter 20.

Convergent improper integral An improper integral converges if the limit that defines it exists and is finite, and diverges otherwise. See Chapter 11.

Convergent sequence A sequence \((a_n)\) converges to \(L\) if for every \(\varepsilon > 0\) there is \(N\) with \(|a_n - L| < \varepsilon\) for all \(n \ge N\). Otherwise it diverges. See Chapter 12.

Convergent series A series converges if its sequence of partial sums converges; its sum is that limit. Otherwise it diverges. See Chapter 12.

Corner A point where a continuous graph changes direction abruptly, so that the left and right derivatives differ, as for \(|x|\) at \(0\). See Chapter 4.

Countable A set is countable if it is finite or can be listed as \(x_1, x_2, x_3, \ldots\), that is, put in bijection with the whole numbers. The integers, the rationals and the algebraic numbers are countable. See Chapter 18.

Countably additive A measure is countably additive if the measure of a union of countably many disjoint measurable sets is the sum of their measures. See Chapter 21.

Cover-up method A quick way to find the constants in partial fractions: put a root of the denominator into the equation, which covers up all but one term. See Chapter 10.

Critical point A point where the derivative is zero or does not exist; in several variables, where the gradient is zero. Interior extrema can only occur at critical points. See Chapter 6, Chapter 7 and Chapter 15.

Critically damped A damped spring whose characteristic equation has a repeated root: it returns to rest as fast as possible without oscillating. See Chapter 14.

Curl For a vector field \(\mathbf F\) in space, \(\operatorname{curl}\mathbf F = \nabla \times \mathbf F\), which measures how much the field rotates about each point. A gradient field has zero curl. See Chapter 17 and Chapter 17.

Curse of dimensionality The way volume behaves in high dimensions: almost all of a ball's volume lies near its surface, and the unit ball takes up a vanishing fraction of the unit cube. It makes data in many dimensions sparse. See Chapter 16.

Cusp A point where the graph comes to a sharp point with tangents from both sides vertical, as for \(y = x^{2/3}\) at \(0\); the function is not differentiable there. See Chapter 4.

Cylindrical coordinates Coordinates \((r, \theta, z)\) in space: polar coordinates in the plane plus the height \(z\). The volume element is \(dV = r\,dr\,d\theta\,dz\). See Chapter 16.

D

Darboux's theorem A derivative has the intermediate value property, even when it is not continuous: derivatives cannot jump. See Chapter 6.

De Rham cohomology A way of measuring the holes in a space by asking which curl-free (closed) forms fail to be gradients (exact). See Chapter 17.

Decreasing A function is decreasing (strictly decreasing) if \(x < y\) implies \(f(x) \ge f(y)\) (\(f(x) > f(y)\)); a sequence is decreasing if each term is at most the one before. A function with \(f' < 0\) on an interval is strictly decreasing. See Chapter 3, Chapter 6 and Chapter 12.

Dedekind cut A set \(\alpha\) of rationals that is nonempty, not all of \(\mathbb{Q}\), closed downwards and without a largest element. Each cut is a real number, and the cuts give a construction of \(\mathbb{R}\) from \(\mathbb{Q}\). See Chapter 18.

Definite integral The number \(\int_a^b f(x)\,dx\): the common value of the supremum of the lower sums and the infimum of the upper sums, when they agree. For \(f \ge 0\) it is the area under the graph. See Chapter 8.

Density of the rationals Between any two real numbers there is a rational number. Between any two there is also an irrational one. See Chapter 18.

Derivative The instantaneous rate of change \(f'(a) = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}\), the slope of the tangent line at \(a\). In several variables the derivative is the linear map that best approximates the function near the point. See Chapter 1, Chapter 4 and Chapter 15.

Derivative function The function \(f'\) whose value at each \(x\) is the derivative of \(f\) at \(x\). It is also written \(\frac{dy}{dx}\) or \(\frac{d}{dx}f\). See Chapter 4.

Difference quotient The slope \(\frac{f(a + h) - f(a)}{h}\) of a secant line; its limit as \(h \to 0\) is the derivative. See Chapter 4.

Differentiable \(f\) is differentiable at \(a\) if \(f'(a)\) exists. In several variables, \(f\) is differentiable at \(\mathbf a\) if a linear map \(L\) makes \(f(\mathbf a + \mathbf h) = f(\mathbf a) + L\mathbf h + o(|\mathbf h|)\). Differentiable functions are continuous. See Chapter 4, Chapter 4 and Chapter 15.

Differential equation An equation relating an unknown function to its derivatives, such as \(y' = ky\). An ordinary differential equation has one independent variable; a partial differential equation has several. See Chapter 14.

Differential form An object that can be integrated over curves, surfaces or solids: 0-forms are functions, 1-forms are like \(P\,dx + Q\,dy + R\,dz\), and so on. With the exterior derivative they unify gradient, curl and divergence. See Chapter 17.

Differentials The quantities \(dx\) and \(dy = f'(x)\,dx\), which estimate the change in \(y\) caused by a small change \(dx\) in \(x\). See Chapter 7.

Dini's theorem If continuous functions on \([a, b]\) decrease to a continuous limit at every point, the convergence is uniform. See Chapter 19.

Directional derivative The rate of change of \(f\) at \(\mathbf a\) in the direction of a unit vector \(\mathbf u\), equal to \(\nabla f(\mathbf a) \cdot \mathbf u\). See Chapter 15.

Dirichlet kernel The function \(D_N(t) = \sum_{n=-N}^{N} e^{int}\), whose convolution with \(f\) gives the \(N\)th partial sum of its Fourier series. See Chapter 21.

Dirichlet's function The function equal to \(1\) at rational numbers and \(0\) at irrational ones. It is discontinuous everywhere, not Riemann integrable, and has Lebesgue integral \(0\) on \([0, 1]\). See Chapter 3, Chapter 8 and Chapter 21.

Dirichlet's theorem If \(f\) is \(2\pi\)-periodic and piecewise continuously differentiable, its Fourier series converges at every point to the average of the left and right limits of \(f\) there. Dirichlet proved it in 1829. See Chapter 21.

Discontinuity A point where a function is not continuous. It may be removable (a hole), a jump, infinite, or oscillating (essential). See Chapter 3.

Discrete metric The metric in which every two distinct points are at distance \(1\). See Chapter 20.

Divergence For a vector field, \(\operatorname{div}\mathbf F = \nabla \cdot \mathbf F = P_x + Q_y + R_z\), which measures how much the field spreads out from each point (its source density). See Chapter 17 and Chapter 17.

Divergence test If \(a_n\) does not tend to \(0\), the series \(\sum a_n\) diverges. See Chapter 12.

Divergence theorem The total flux of a field out of a closed surface equals the integral of its divergence over the solid inside: \(\iint_{\partial E} \mathbf F \cdot d\mathbf S = \iiint_E \operatorname{div}\mathbf F\,dV\). It is due to Gauss and Ostrogradsky. See Chapter 17.

Dominated convergence theorem If \(f_n \to f\) almost everywhere and \(|f_n| \le g\) for one integrable \(g\), then \(\int f_n \to \int f\). See Chapter 21.

Double integral The integral \(\iint_R f\,dA\) of a function of two variables over a region, defined by Riemann sums over small rectangles. For \(f \ge 0\) it is the volume under the graph. See Chapter 16.

Doubling map The map \(\theta \mapsto 2\theta \pmod 1\) of the circle; the logistic map at \(r = 4\) is this map in disguise. See Chapter 22.

Dummy variable The variable of integration, as \(t\) in \(\int_a^x f(t)\,dt\). Its name does not matter. See Chapter 8 and Chapter 9.

Dynamical system A rule for how a state changes in time, either by iterating a map \(x_{n+1} = f(x_n)\) or by a differential equation. See Chapter 22.

E

e (the number) The base \(e = 2.71828\ldots\) for which \(\lim_{h \to 0}\frac{e^h - 1}{h} = 1\), so that \(e^x\) is its own derivative. It is also \(\lim_{n \to \infty}(1 + \frac1n)^n\). See Chapter 5 and Chapter 5.

Eccentricity The number \(e\) in the polar equation of a conic, measuring how far it is from a circle: \(0\) for a circle, between \(0\) and \(1\) for an ellipse. See Chapter 14.

Elementary functions Functions built from \(x\) and constants by finitely many arithmetic operations, roots, exponentials, logarithms, trigonometric functions and their inverses, and compositions. Some, like \(e^{-x^2}\), have no elementary antiderivative. See Chapter 10.

Elliptic integral An integral such as \(\int \sqrt{1 - k^2\sin^2\theta}\,d\theta\), which gives the arc length of an ellipse and has no elementary antiderivative. See Chapter 10.

Entire function A function holomorphic on the whole complex plane. By Liouville's theorem a bounded entire function is constant. See Chapter 22.

Epsilon-delta definition \(\lim_{x \to a} f(x) = L\) means: for every \(\varepsilon > 0\) there is a \(\delta > 0\) such that \(0 < |x - a| < \delta\) implies \(|f(x) - L| < \varepsilon\). See Chapter 2.

Escape velocity The launch speed needed to leave a planet for good, \(\sqrt{2GM/R}\), found by integrating the work against gravity out to infinity. For Mars it is about \(5.0\) km/s. See Chapter 11 and Chapter 17.

Euler-Mascheroni constant The limit \(\gamma = \lim (H_N - \ln N) \approx 0.5772\), where \(H_N\) is the \(N\)th harmonic number. Nobody knows whether it is rational. See Chapter 12.

Euler's formula For every real \(\theta\), \(e^{i\theta} = \cos\theta + i\sin\theta\). It follows from the power series of the three functions. See Chapter 13.

Euler's method A numerical way to solve \(y' = F(x, y)\): from \((x_n, y_n)\) step to \(y_{n+1} = y_n + hF(x_n, y_n)\), following the slope field in steps of size \(h\). Its error is of first order in \(h\). See Chapter 14.

Existence theorem A theorem that says something exists without saying how to find it, like the intermediate value theorem. See Chapter 3.

Expected value The mean of a random variable with density \(f\): \(\int x f(x)\,dx\). See Chapter 11.

Exterior derivative The operation \(d\) on differential forms that takes a \(k\)-form to a \((k+1)\)-form. On 0-, 1- and 2-forms in space it is the gradient, curl and divergence. See Chapter 17.

Extreme value theorem A continuous function on a closed bounded interval \([a, b]\) attains a maximum and a minimum value. More generally, a continuous real function on a compact space does. See Chapter 3, Chapter 18 and Chapter 20.

F

Fast Fourier transform An algorithm that computes the discrete Fourier transform of \(N\) values in about \(N\log N\) steps instead of \(N^2\). See Chapter 22.

Fatou's lemma For measurable \(f_n \ge 0\), \(\int \liminf f_n \le \liminf \int f_n\). See Chapter 21.

Feigenbaum's constant The number \(\delta \approx 4.669\), the limiting ratio of the gaps between successive period doublings, the same for a wide family of maps. See Chapter 22.

Fejér kernel The kernel \(F_N\) whose convolution with \(f\) gives the Cesàro means of its Fourier series. It is nonnegative, which is why the Cesàro means behave better than the partial sums. See Chapter 21.

Fermat's lemma If \(f\) has a local maximum or minimum at an interior point \(c\) and \(f'(c)\) exists, then \(f'(c) = 0\). See Chapter 6.

Finite subcover A finite selection from an open cover that still covers the set. See Chapter 20.

First-order method A numerical method whose global error is at most a constant times the step size \(h\), like Euler's method. See Chapter 14.

Fixed point A point \(x\) with \(f(x) = x\). Continuous maps of \([0, 1]\) to itself have one, and contractions of a complete space have exactly one. See Chapter 3, Chapter 20 and Chapter 22.

Flow line A curve whose velocity at each point is the value of a vector field there; the path a particle carried by the field follows. See Chapter 17.

Flux The surface integral \(\iint_S \mathbf F \cdot \mathbf n\,dS\): how much of a field flows through an oriented surface. See Chapter 17.

Fourier coefficients The numbers \(c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)e^{-inx}\,dx\) (or the matching \(a_n\) and \(b_n\) for cosines and sines), found using orthogonality. See Chapter 21.

Fourier series The trigonometric series \(\sum c_n e^{inx}\), or \(\frac{a_0}{2} + \sum (a_n\cos nx + b_n\sin nx)\), built from the Fourier coefficients of a periodic function \(f\). See Chapter 21.

Fourier transform The continuous version of Fourier series for functions on the whole line, \(\hat f(\xi) = \int f(x)e^{-2\pi i x\xi}\,dx\), which describes a function by its frequencies. See Chapter 22.

Fractal A set with detail at every scale and, often, a fractional dimension. The Cantor set, with dimension \(\log 2/\log 3\), is the simplest. See Chapter 20.

Fresnel integral The function \(\int_0^x \sin(t^2)\,dt\), which has no elementary formula but, by the fundamental theorem, a simple derivative. It is used in optics. See Chapter 9.

Fubini's theorem A double integral of a continuous function over a rectangle can be computed as an iterated integral in either order. See Chapter 16.

Function A rule that gives one output for each input, such as the distance fallen after \(t\) seconds. See Chapter 1.

Function of two variables A rule \(f(x, y)\) giving a number for each point of a region of the plane. See Chapter 15.

Fundamental theorem for line integrals If \(\mathbf F = \nabla f\), then \(\int_C \mathbf F \cdot d\mathbf r = f(\text{end}) - f(\text{start})\). See Chapter 17.

Fundamental theorem of calculus Finding areas is undoing rates. Part 1: if \(f\) is continuous, \(\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\). Part 2: if \(F' = f\) and \(f\) is integrable, \(\int_a^b f = F(b) - F(a)\). See Chapter 1, Chapter 9 and Chapter 9.

G

Gabriel's horn The solid formed by rotating \(y = 1/x\), \(x \ge 1\), about the \(x\)-axis. It has finite volume \(\pi\) but infinite surface area. See Chapter 11.

Gamma function \(\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt\) for \(s > 0\). It satisfies \(\Gamma(n + 1) = n!\), extending the factorial, and \(\Gamma(\frac12) = \sqrt\pi\). See Chapter 11 and Chapter 16.

Gaussian integral The integral \(\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt\pi\), computed by squaring it and changing to polar coordinates. See Chapter 16.

Gaussian quadrature A numerical integration rule that places its sample points cleverly so as to integrate polynomials of the highest possible degree exactly. See Chapter 10.

Generalised Stokes theorem The single statement \(\int_{\partial M} \omega = \int_M d\omega\) for differential forms, which contains the fundamental theorem of calculus, Green's theorem, Stokes's theorem and the divergence theorem. See Chapter 17.

Geometric series The series \(\sum_{n \ge 0} ar^n\), which converges to \(\frac{a}{1 - r}\) when \(|r| < 1\) and diverges otherwise (for \(a \ne 0\)). See Chapter 12.

Gibbs phenomenon Near a jump of \(f\), the partial sums of its Fourier series overshoot by about 9 per cent of the jump, and the overshoot never shrinks as more terms are added. See Chapter 21.

Gradient The vector of partial derivatives \(\nabla f = (f_x, f_y, \ldots)\). It points in the direction of fastest increase of \(f\), its length is that fastest rate, and it is perpendicular to the level sets. See Chapter 15 and Chapter 15.

Gradient descent Minimising a function by repeatedly taking a small step in the direction \(-\nabla f\), downhill. See Chapter 15.

Gradient field A vector field of the form \(\nabla f\); \(f\) is then called a potential. See Chapter 17.

Green's theorem For a region \(D\) in the plane with positively oriented boundary \(C\), \(\oint_C P\,dx + Q\,dy = \iint_D (Q_x - P_y)\,dA\). See Chapter 17.

Grönwall's inequality An inequality bounding a function by an exponential when its growth is controlled by its own size; it gives uniqueness for differential equations with Lipschitz right-hand side. See Chapter 14.

H

Hahn-Banach theorem A bounded linear functional on a subspace of a normed space extends to the whole space without increasing its norm. See Chapter 22.

Half-life The time after which half of a quantity decaying exponentially is left, \(\frac{\ln 2}{\lambda}\); for an exponential distribution it is the median. See Chapter 11.

Hamming distance The number of positions in which two strings of bits differ: a metric on bit strings, used in error-correcting codes. See Chapter 20.

Harmonic numbers The partial sums \(H_N = 1 + \frac12 + \cdots + \frac1N\) of the harmonic series, which grow like \(\ln N\). See Chapter 12.

Harmonic series The series \(\sum \frac1n\), which diverges even though its terms tend to \(0\). See Chapter 12.

Heat equation The partial differential equation \(u_t = \kappa u_{xx}\) for the temperature \(u(x, t)\) in a rod, where \(\kappa\) is the diffusivity. Fourier invented his series to solve it. See Chapter 21 and Chapter 22.

Heine-Borel theorem A subset of \(\mathbb{R}\) or \(\mathbb{R}^n\) is compact if and only if it is closed and bounded. See Chapter 20.

Heine-Cantor theorem A function continuous on a closed bounded interval is uniformly continuous there. See Chapter 18.

Hessian matrix The matrix of second partial derivatives of \(f\). At a critical point its signs decide between a local minimum, a local maximum and a saddle. See Chapter 15.

Higher derivatives The derivatives of the derivative: \(f''\), \(f'''\), ..., \(f^{(n)}\). The second derivative of position is acceleration. See Chapter 4.

Hilbert space A Banach space whose norm comes from an inner product, so that angles and orthogonal projections make sense, such as \(\ell^2\) or \(L^2[-\pi, \pi]\). See Chapter 22.

Hilbert's sixteenth problem An open problem, in its second part: how many isolated periodic orbits (limit cycles) can a polynomial vector field in the plane of a given degree have? See Chapter 22.

Hohmann transfer The cheapest simple route between two circular orbits: half of an ellipse touching both. It is the basic route from Earth to Mars. See Chapter 14.

Holomorphic A complex function is holomorphic (complex differentiable) at \(z_0\) if \(\lim_{h \to 0}\frac{f(z_0 + h) - f(z_0)}{h}\) exists with \(h\) complex. A holomorphic function has derivatives of all orders. See Chapter 22.

Homogeneous linear equation A linear differential equation with zero right-hand side, such as \(ay'' + by' + cy = 0\). Sums and multiples of its solutions are solutions (superposition). See Chapter 14.

Horizontal asymptote A line \(y = L\) that the graph of \(f\) approaches as \(x \to \infty\) or \(x \to -\infty\), when \(\lim f(x) = L\) there. See Chapter 2.

I

Implicit differentiation Differentiating both sides of an equation like \(x^2 + y^2 = 1\) with \(y\) treated as a function of \(x\), then solving for \(\frac{dy}{dx}\). See Chapter 5 and Chapter 15.

Implicit function theorem If \(F(x, y) = 0\) and \(F_y \ne 0\) at a point, then near that point the equation defines \(y\) as a differentiable function of \(x\). It justifies implicit differentiation. See Chapter 5 and Chapter 15.

Improper integral An integral over an infinite interval (type 1) or of a function that is unbounded near a point (type 2), defined as a limit of ordinary integrals. See Chapter 11.

Increasing A function is increasing (strictly increasing) if \(x < y\) implies \(f(x) \le f(y)\) (\(f(x) < f(y)\)); a sequence is increasing if each term is at least the one before. A function with \(f' > 0\) on an interval is strictly increasing. See Chapter 3, Chapter 6 and Chapter 12.

Indefinite integral The notation \(\int f(x)\,dx = F(x) + C\) for the family of all antiderivatives of \(f\) on an interval. See Chapter 9.

Indeterminate form A limit of a form such as \(\frac00\), \(\frac\infty\infty\), \(0 \cdot \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\) or \(\infty^0\), whose value cannot be read off from the limits of the parts. See Chapter 2 and Chapter 6.

Indicator function The function \(\mathbf 1_E\) equal to \(1\) on the set \(E\) and \(0\) off it. See Chapter 21.

Infimum The greatest lower bound of a set of real numbers, written \(\inf S\). Every nonempty set that is bounded below has one. See Chapter 8 and Chapter 18.

Inflection point A point where a graph changes concavity, from concave up to concave down or back. See Chapter 7.

Initial value problem A differential equation together with a starting value, such as \(y' = f(x, y)\) with \(y(x_0) = y_0\). See Chapter 9 and Chapter 14.

Injective A function is injective (one-to-one) if different inputs give different outputs. A continuous injective function on an interval is strictly monotone. See Chapter 3.

Instantaneous rate of change The derivative \(f'(a)\): the limit of average rates of change over shorter and shorter intervals. See Chapter 4.

Instantaneous speed The speed at a single moment: the value the average speeds approach as the time interval shrinks to nothing. See Chapter 1.

Integrable A bounded function on \([a, b]\) is Riemann integrable if its lower and upper integrals are equal. A measurable function is Lebesgue integrable if \(\int |f| < \infty\). See Chapter 8 and Chapter 21.

Integral equation An equation in which the unknown function appears under an integral sign, such as \(y(t) = y_0 + \int_{t_0}^{t} F(s, y(s))\,ds\), which is equivalent to an initial value problem. See Chapter 14.

Integral test If \(f\) is positive, continuous and decreasing on \([1, \infty)\), then \(\sum f(n)\) and \(\int_1^\infty f\) converge or diverge together. See Chapter 12.

Integrating factor A function \(\mu(x)\), for \(y' + p(x)y = q(x)\) the function \(e^{\int p}\), that turns the left side into the derivative of \(\mu y\) so the equation can be integrated. See Chapter 14.

Integration by parts The rule \(\int u\,dv = uv - \int v\,du\), the product rule read backwards. See Chapter 10.

Interior The set of points of a set that have a whole open ball around them inside the set: the largest open set inside it. See Chapter 20.

Intermediate value theorem If \(f\) is continuous on \([a, b]\) and \(y\) lies between \(f(a)\) and \(f(b)\), then \(f(c) = y\) for some \(c\) in \([a, b]\). To show an equation has a solution, find a sign change. See Chapter 3 and Chapter 18.

Invariant subspace problem The open problem whether every bounded operator on a separable infinite-dimensional Hilbert space has a closed invariant subspace other than \(\{0\}\) and the whole space. See Chapter 22.

Inverse function The function \(f^{-1}\) that undoes a one-to-one function \(f\): \(f^{-1}(y) = x\) exactly when \(f(x) = y\). The inverse of a continuous one-to-one function on an interval is continuous, and \((f^{-1})'(y) = 1/f'(x)\). See Chapter 3 and Chapter 5.

Iterated integral An integral computed one variable at a time, such as \(\int_a^b \left(\int_c^d f(x, y)\,dy\right) dx\). See Chapter 16.

J

Jacobian For a change of variables \((u, v) \mapsto (x, y)\), the determinant \(\frac{\partial(x, y)}{\partial(u, v)}\) of the Jacobian matrix. Its absolute value is the factor by which the map scales small areas. See Chapter 16.

Jacobian matrix The matrix of all first partial derivatives of a map \(\mathbf f : \mathbb{R}^n \to \mathbb{R}^m\): the matrix of its derivative. See Chapter 15.

Jump discontinuity A discontinuity where the left and right limits both exist but differ. See Chapter 3.

K

Kakeya conjecture The conjecture that a set in \(\mathbb{R}^n\) containing a unit segment in every direction, which can have measure zero, must still have Hausdorff dimension \(n\). It is known in the plane and in space and open in four or more dimensions. See Chapter 22.

Kepler's laws Planets move in ellipses with the Sun at a focus; the line from the Sun sweeps out equal areas in equal times; the square of the period is proportional to the cube of the mean distance. All three follow from Newton's law of gravity. See Chapter 14.

L

Lagrange multiplier At a maximum or minimum of \(f\) subject to \(g = c\), \(\nabla f = \lambda \nabla g\) for some number \(\lambda\), the Lagrange multiplier. It measures how fast the best value changes as the constraint is relaxed. See Chapter 15.

Laplace equation The partial differential equation \(u_{xx} + u_{yy} + u_{zz} = 0\), whose solutions are the harmonic functions: steady temperatures and electric potentials in empty space. See Chapter 22.

Laplacian The operator \(\Delta f = \nabla \cdot \nabla f = f_{xx} + f_{yy} + f_{zz}\), the divergence of the gradient. See Chapter 17.

Laurent series A series \(\sum_{n=-\infty}^{\infty} c_n (z - a)^n\) with negative powers allowed, which represents a holomorphic function near an isolated singularity. See Chapter 22.

Least upper bound See supremum. See Chapter 18.

Lebesgue integral The integral built from measure: approximate \(f \ge 0\) from below by simple functions and take the supremum of their integrals. It slices the range instead of the domain, integrates far more functions than Riemann's, and behaves well under limits. See Chapter 21.

Lebesgue measurable A set \(E\) is Lebesgue measurable if \(m^*(A) = m^*(A \cap E) + m^*(A \setminus E)\) for every set \(A\). The measurable sets form a \(\sigma\)-algebra on which measure is countably additive. See Chapter 21.

Lebesgue's criterion A bounded function on \([a, b]\) is Riemann integrable if and only if its set of discontinuities has measure zero. See Chapter 8 and Chapter 21.

Left-hand and right-hand limits The limits \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\), in which \(x\) approaches \(a\) only from below or only from above. The two-sided limit exists exactly when both exist and agree. See Chapter 2.

Leibniz notation Writing the derivative as \(\frac{dy}{dx}\) and the integral as \(\int f(x)\,dx\). It is not a fraction, but it makes the chain rule and substitution look like algebra. See Chapter 4 and Chapter 10.

Leibniz series The series \(1 - \frac13 + \frac15 - \frac17 + \cdots = \frac\pi4\), the arctangent series at \(x = 1\). See Chapter 13.

Level curve The set of points where a function of two variables has a given value, \(f(x, y) = c\). For three variables the level sets are level surfaces. See Chapter 15.

L'Hôpital's rule If \(f(x)\) and \(g(x)\) both tend to \(0\) (or both to \(\pm\infty\)) and \(\frac{f'(x)}{g'(x)} \to L\), then \(\frac{f(x)}{g(x)} \to L\). See Chapter 6.

Limit \(\lim_{x \to a} f(x) = L\) means that \(f(x)\) can be made as close to \(L\) as you want by taking \(x\) close enough to \(a\), but not equal to \(a\). The value at \(a\) itself is ignored. See Chapter 2 and Chapter 2.

Limit at infinity \(\lim_{x \to \infty} f(x) = L\) means \(f(x)\) can be made as close to \(L\) as you want by taking \(x\) large enough. See Chapter 2.

Limit comparison test If \(a_n, b_n > 0\) and \(a_n/b_n \to L\) with \(0 < L < \infty\), then \(\sum a_n\) and \(\sum b_n\) converge or diverge together. The same holds for improper integrals. See Chapter 11 and Chapter 12.

Limit laws The limit of a sum, difference, product or quotient is the sum, difference, product or quotient of the limits, provided those limits exist (and the limit of the denominator is not \(0\)). See Chapter 2 and Chapter 2.

Limit point A point \(p\) is a limit point of a set \(S\) if every open ball around \(p\) contains a point of \(S\) other than \(p\). See Chapter 20.

Limit superior \(\limsup a_n\): the largest number that some subsequence of a bounded sequence converges to. See Chapter 13.

Limits of integration The endpoints \(a\) and \(b\) in \(\int_a^b f(x)\,dx\). See Chapter 8.

Line integral The integral of a function along a curve, \(\int_C f\,ds\), or of a vector field, \(\int_C \mathbf F \cdot d\mathbf r\), the work done by the field along the curve. See Chapter 17.

Linear approximation Approximating \(f\) near \(a\) by its tangent line: \(f(x) \approx f(a) + f'(a)(x - a)\). In several variables, the tangent plane does the same. See Chapter 4 and Chapter 7.

Linear first-order equation A differential equation \(y' + p(x)y = q(x)\), solved with an integrating factor. See Chapter 14.

Liouville's theorem (complex analysis) A bounded entire function is constant. It gives a short proof of the fundamental theorem of algebra. See Chapter 22.

Liouville's theorem (integration) A criterion for when an integral such as \(\int fe^g\) with rational \(f\) and \(g\) has an elementary answer; it proves that \(\int e^{-x^2}\,dx\) has none. See Chapter 10.

Lipschitz A function is Lipschitz with constant \(K\) if \(|f(x) - f(y)| \le K|x - y|\) for all \(x\) and \(y\). A function with \(|f'| \le K\) on an interval is Lipschitz, and Lipschitz functions are uniformly continuous. See Chapter 6, Chapter 9 and Chapter 18.

Lipschitz in y A function \(F(t, y)\) is Lipschitz in \(y\) if \(|F(t, y_1) - F(t, y_2)| \le K|y_1 - y_2|\). This condition guarantees existence and uniqueness for \(y' = F(t, y)\). See Chapter 14 and Chapter 20.

Local error The error a numerical method makes in a single step; for Euler's method it is of order \(h^2\). See Chapter 14.

Local maximum / local minimum A value \(f(c)\) that is the largest (smallest) among the values at nearby points. Both are local extrema. See Chapter 6 and Chapter 15.

Logarithmic derivative The quantity \(\frac{y'}{y} = (\ln y)'\), the relative rate of change of \(y\). See Chapter 5.

Logarithmic differentiation Finding \(y'\) by differentiating \(\ln y\), which turns products, quotients and powers into sums. See Chapter 5.

Logistic map The map \(x \mapsto rx(1 - x)\) on \([0, 1]\). As \(r\) grows its behaviour passes from a fixed point through period doubling to chaos. See Chapter 22.

Lower integral / upper integral The supremum of all lower sums and the infimum of all upper sums of \(f\) on \([a, b]\). \(f\) is integrable when they are equal. See Chapter 8.

Lower sum The sum of the areas of rectangles whose heights are the smallest (infimum) values of \(f\) on each piece of a partition. It is never more than the area. See Chapter 1 and Chapter 8.

M

Maclaurin polynomial A Taylor polynomial centred at \(0\). See Chapter 13.

Mandelbrot set The set of complex numbers \(c\) for which iterating \(z \mapsto z^2 + c\) from \(0\) stays bounded. Whether it is locally connected (MLC) is an open problem. See Chapter 22.

Marginal cost The derivative of cost with respect to quantity produced: the approximate cost of one more unit. See Chapter 4.

Maxwell's equations The four equations of electromagnetism: Gauss's law, no magnetic charges, Faraday's law and Ampère's law with Maxwell's correction. Together they predict waves that travel at the speed of light. See Chapter 17.

Mean value theorem If \(f\) is continuous on \([a, b]\) and differentiable inside, there is a \(c\) between \(a\) and \(b\) with \(f'(c) = \frac{f(b) - f(a)}{b - a}\): some tangent is parallel to the secant. See Chapter 6.

Mean value theorem for integrals If \(f\) is continuous on \([a, b]\), there is a \(c\) with \(f(c) = \frac{1}{b - a}\int_a^b f\). See Chapter 8 and Chapter 9.

Measurable function A function \(f\) such that every set \(\{x : f(x) > y\}\) is measurable. See Chapter 21.

Measure zero A set has measure zero if it can be covered by countably many intervals of total length as small as you like. Every countable set has measure zero; so does the Cantor set. See Chapter 8 and Chapter 21.

Memoryless A waiting time is memoryless if having waited does not change the chance of waiting longer; the exponential distribution is the only continuous one with this property. See Chapter 11.

Mesh The largest width of the subintervals of a partition \(P\), written \(\|P\|\). See Chapter 8.

Method of exhaustion The Greek method of finding an area by filling it with ever more pieces of known area and proving by contradiction that the answer can be neither more nor less than a given value. See Chapter 1.

Metric A distance function \(d\) on a set: \(d(x, y) \ge 0\) with equality only when \(x = y\), \(d(x, y) = d(y, x)\), and the triangle inequality \(d(x, z) \le d(x, y) + d(y, z)\). See Chapter 20.

Metric space A set together with a metric. The real line, \(\mathbb{R}^n\), spaces of functions with the sup norm and bit strings with the Hamming distance are examples. See Chapter 18 and Chapter 20.

Moment The product of a mass and its signed distance from an axis, summed or integrated over a body; dividing by the total mass gives the centre of mass. See Chapter 11.

Moment of inertia The integral \(\iiint r^2\,dm\) of mass times squared distance from an axis: how hard a body is to spin. The moment of inertia factor \(I/(MR^2)\) of a planet shows how its density increases with depth. See Chapter 16.

Monotone Increasing or decreasing. A strictly monotone function is one-to-one. See Chapter 3 and Chapter 12.

Monotone convergence theorem (integrals) If \(0 \le f_1 \le f_2 \le \cdots\) are measurable and \(f_n \to f\), then \(\int f_n \to \int f\). See Chapter 21.

Monotone convergence theorem (sequences) An increasing sequence that is bounded above converges, to the supremum of its terms. See Chapter 12 and Chapter 18.

N

Natural frequency The frequency \(\omega_0 = \sqrt{k/m}\) at which a spring oscillates with no friction and no forcing. See Chapter 14.

Natural logarithm The function \(\ln x\), the inverse of \(e^x\). Its derivative is \(\frac1x\). See Chapter 5.

Navier-Stokes equations The partial differential equations for the motion of a viscous fluid. Whether smooth solutions in three dimensions stay smooth for all time is a Millennium Prize problem. See Chapter 22 and Chapter 22.

Nested interval theorem If \([a_1, b_1] \supseteq [a_2, b_2] \supseteq \cdots\) are closed bounded intervals, there is a point in all of them. See Chapter 18.

Net change theorem The integral of a rate of change is the net change: \(\int_a^b F'(t)\,dt = F(b) - F(a)\). See Chapter 9.

Newton's law of cooling An object cools at a rate proportional to the difference between its temperature and that of its surroundings: \(T' = -k(T - T_s)\). See Chapter 14.

Newton's method A way to solve \(f(x) = 0\): follow the tangent line to where it crosses the axis, \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), and repeat. Near a simple root it converges quadratically. See Chapter 7.

Norm A length \(\|v\|\) on a vector space that is positive for \(v \ne 0\), scales as \(\|cv\| = |c|\,\|v\|\) and satisfies the triangle inequality. See Chapter 22.

Normal equations The two linear equations that give the least-squares line through a set of data points, found by setting partial derivatives to zero. See Chapter 15.

Normal vector A vector perpendicular to a curve or surface; for a parametrised surface \(\mathbf r(u, v)\) it is \(\mathbf r_u \times \mathbf r_v\). See Chapter 7 and Chapter 17.

Nowhere dense A set is nowhere dense if its closure contains no interval (no open ball), like the Cantor set. See Chapter 20.

Nowhere differentiable function A continuous function with no derivative at any point, such as the Takagi function or Weierstrass's function. See Chapter 19.

Numerical integration Approximating a definite integral by a weighted sum of values of the function, as in the midpoint, trapezoid and Simpson's rules. See Chapter 10.

O

One-to-one See injective. See Chapter 3.

Open ball The set \(B(a, r)\) of points at distance less than \(r\) from \(a\). See Chapter 20.

Open cover A collection of open sets whose union contains a given set. See Chapter 20.

Open interval of convergence The interval \((a - R, a + R)\) on which a power series centred at \(a\) with radius \(R\) converges absolutely. See Chapter 13.

Open mapping theorem A bounded linear map from one Banach space onto another takes open sets to open sets. See Chapter 22.

Open set A set that contains an open ball around each of its points. See Chapter 20.

Operator A linear map between vector spaces, especially infinite-dimensional ones such as spaces of functions. See Chapter 22.

Optimisation Finding the largest or smallest value of a quantity, by locating critical points and comparing them with the endpoints. See Chapter 7.

Order of a differential equation The order of the highest derivative in it. See Chapter 14.

Ordinary differential equation A differential equation for a function of one variable. See Chapter 14.

Orthogonal Two functions are orthogonal on \([-\pi, \pi]\) if the integral of their product is \(0\); the functions \(\sin nx\) and \(\cos nx\) are orthogonal to each other. See Chapter 21.

Orthonormal basis A set of mutually orthogonal unit vectors whose combinations approximate every vector; the functions \(e^{inx}\) form one for \(L^2[-\pi, \pi]\). See Chapter 21.

Outer measure For any set \(A \subseteq \mathbb{R}\), the infimum \(m^*(A)\) of the total lengths of countable families of open intervals covering \(A\). See Chapter 21.

Overdamped A damped spring whose characteristic equation has two distinct real roots: it creeps back to rest without oscillating. See Chapter 14.

P

Pappus's theorem The volume swept out by rotating a plane region about an axis that does not cross it equals its area times the distance travelled by its centroid. See Chapter 11.

Parametrised surface A surface given as the image of a map \(\mathbf r(u, v)\) from a region of the plane into space. See Chapter 17.

Parseval's identity For square-integrable \(f\), the sum of the squares of the moduli of its Fourier coefficients equals \(\frac{1}{2\pi}\int_{-\pi}^{\pi} |f|^2\). See Chapter 21.

Partial derivative The derivative of a function of several variables with respect to one variable, the others held fixed: \(f_x = \frac{\partial f}{\partial x}\). See Chapter 15.

Partial differential equation A differential equation for a function of several variables, involving its partial derivatives, such as the heat, wave and Laplace equations. See Chapter 22.

Partial fractions Writing a proper rational function as a sum of simple fractions \(\frac{A}{(x - a)^k}\) and \(\frac{Bx + C}{(x^2 + px + q)^k}\), each easy to integrate. See Chapter 10.

Partial sums The sums \(s_N = a_1 + \cdots + a_N\) of the first \(N\) terms of a series. See Chapter 12.

Particular solution One solution of a nonhomogeneous linear equation; adding the general solution of the homogeneous equation gives all solutions. See Chapter 14.

Partition A finite set of points \(a = x_0 < x_1 < \cdots < x_n = b\) cutting \([a, b]\) into subintervals. See Chapter 8.

Path-connected A set is path-connected if any two of its points can be joined by a continuous path inside it. Path-connected sets are connected, but not conversely. See Chapter 20.

Perfect set A closed set with no isolated points: every point is a limit of other points of the set, as in the Cantor set. See Chapter 20.

Period doubling The way a fixed point of the logistic map gives way to a 2-cycle, then a 4-cycle, and so on, as \(r\) increases. See Chapter 22.

Picard iteration Solving \(y' = F(t, y)\), \(y(t_0) = y_0\) by repeatedly substituting into \(y_{n+1}(t) = y_0 + \int_{t_0}^{t} F(s, y_n(s))\,ds\); the iterates converge to the solution. See Chapter 14 and Chapter 20.

Picard-Lindelöf theorem If \(F\) is continuous and Lipschitz in \(y\) near \((t_0, y_0)\), the problem \(y' = F(t, y)\), \(y(t_0) = y_0\) has exactly one solution on some interval around \(t_0\). See Chapter 14 and Chapter 20.

Planimeter A mechanical instrument that measures the area of a region by tracing its boundary, an application of Green's theorem. See Chapter 17.

Pointwise convergence A sequence of functions \(f_n\) converges pointwise to \(f\) on \(D\) if \(f_n(x) \to f(x)\) for each \(x\) in \(D\) separately. See Chapter 19.

Polar coordinates Coordinates \((r, \theta)\) in the plane, with \(x = r\cos\theta\) and \(y = r\sin\theta\). The area element is \(dA = r\,dr\,d\theta\). See Chapter 16.

Pole An isolated singularity \(a\) of \(f\) where \(|f(z)| \to \infty\); near it the Laurent series has only finitely many negative powers. A simple pole has just the power \((z - a)^{-1}\). See Chapter 22.

Positive definite A quadratic form \(Q\) is positive definite if \(Q(\mathbf h) > 0\) for every \(\mathbf h \ne \mathbf 0\). A critical point with positive definite Hessian is a local minimum. See Chapter 15.

Potential A function \(f\) whose gradient is a given vector field \(\mathbf F\). See Chapter 17 and Chapter 17.

Power rule \(\frac{d}{dx}x^n = nx^{n-1}\), proved first for whole numbers, then rational exponents, then every real exponent. See Chapter 4, Chapter 4 and Chapter 5.

Power series A series \(\sum_{n \ge 0} c_n (x - a)^n\), a polynomial of infinite degree centred at \(a\). Inside its radius of convergence it can be differentiated and integrated term by term. See Chapter 13 and Chapter 13.

Probability density function A function \(f \ge 0\) with total integral \(1\) such that the probability of landing in \([a, b]\) is \(\int_a^b f\). See Chapter 11.

Product rule \((fg)' = f'g + fg'\). See Chapter 4.

p-series The series \(\sum \frac{1}{n^p}\), which converges if and only if \(p > 1\). Likewise \(\int_1^\infty \frac{dx}{x^p}\) converges if and only if \(p > 1\). See Chapter 11 and Chapter 12.

Q

Quadratic convergence Convergence in which the error is roughly squared each step, so the number of correct digits roughly doubles, as in Newton's method. See Chapter 7.

Quadratic form A homogeneous polynomial of degree 2, such as \(ah^2 + 2bhk + ck^2\); the second-order term of a function at a critical point. See Chapter 15.

Quotient rule \(\left(\frac fg\right)' = \frac{f'g - fg'}{g^2}\) where \(g \ne 0\). See Chapter 4.

R

Radians The measure of an angle by the length of arc it cuts on a unit circle, so that \(2\pi\) radians make a full turn. Only in radians is \(\frac{d}{dx}\sin x = \cos x\). See Chapter 2 and Chapter 5.

Radius of convergence The number \(R\) such that a power series converges absolutely for \(|x - a| < R\) and diverges for \(|x - a| > R\). See Chapter 13.

Rate of change How fast one quantity changes compared with another, such as metres per second. See Chapter 1.

Ratio test If \(a_n > 0\) and \(\frac{a_{n+1}}{a_n} \to \rho\), then \(\sum a_n\) converges if \(\rho < 1\) and diverges if \(\rho > 1\); \(\rho = 1\) decides nothing. See Chapter 12.

Rational function A quotient of two polynomials. It is continuous wherever the denominator is not zero. See Chapter 2 and Chapter 10.

Real analytic A function that equals a convergent power series near every point of its domain, written \(C^\omega\). See Chapter 13.

Real number An element of the complete ordered field \(\mathbb{R}\), which can be constructed from the rationals by Dedekind cuts or by classes of Cauchy sequences. See Chapter 18 and Chapter 18.

Rearrangement theorem Riemann's theorem: the terms of a conditionally convergent series can be reordered to converge to any chosen real number, or to diverge. Absolutely convergent series have the same sum in every order. See Chapter 12.

Reduction formula A formula, usually from integration by parts, that expresses an integral with a power \(n\) in terms of one with a smaller power, such as \(\int_0^{\pi/2}\sin^n x\,dx = \frac{n-1}{n}\int_0^{\pi/2}\sin^{n-2} x\,dx\). See Chapter 10.

Refinement A partition \(Q\) is a refinement of \(P\) if it contains all the points of \(P\). Refining raises lower sums and lowers upper sums. See Chapter 8.

Related rates Problems in which several quantities change with time and an equation linking them gives, by differentiation, an equation linking their rates. See Chapter 7.

Relative error The error divided by the size of the quantity, often given as a percentage. See Chapter 7.

Remainder The difference \(R_n(x) = f(x) - T_n(x)\) between a function and its Taylor polynomial. Taylor's theorem bounds it. See Chapter 13.

Removable discontinuity A point where the limit exists but is not equal to the value of the function (or the function is undefined); redefining one value removes it. See Chapter 3.

Repelling fixed point A fixed point \(x^*\) of a map \(f\) with \(|f'(x^*)| > 1\): points near it are pushed away under iteration. See Chapter 22.

Residue The coefficient of \((z - a)^{-1}\) in the Laurent series of \(f\) at \(a\). See Chapter 22.

Residue theorem If \(f\) is holomorphic inside and on a closed curve \(\gamma\) except at finitely many poles inside, then \(\oint_\gamma f\,dz\) is \(2\pi i\) times the sum of the residues there. It evaluates many real integrals. See Chapter 22.

Resonance The large response of an oscillator forced at its natural frequency; without friction the amplitude grows without bound. See Chapter 14.

Riemann hypothesis The conjecture that all nontrivial zeros of the Riemann zeta function have real part \(\frac12\). It is the most famous open problem in mathematics. See Chapter 22.

Riemann sum A sum \(\sum f(t_k)\,\Delta x_k\) over a tagged partition: the total area of rectangles with heights taken at the tags. In several variables the same is done over small rectangles or boxes. See Chapter 8 and Chapter 16.

Riemann zeta function \(\zeta(s) = \sum_{n \ge 1} \frac{1}{n^s}\) for \(s > 1\), so that \(\zeta(2) = \frac{\pi^2}{6}\). See Chapter 12.

Riemann-Lebesgue lemma The Fourier coefficients of an integrable function tend to \(0\). See Chapter 21.

Riesz-Fischer theorem The spaces \(L^1\) and \(L^2\) of Lebesgue integrable and square-integrable functions are complete. See Chapter 21 and Chapter 21.

Rocket equation Tsiolkovsky's equation \(\Delta v = v_e\ln\frac{m_0}{m_1}\): the change in speed of a rocket from its exhaust speed and its initial and final masses. See Chapter 14.

Rolle's theorem If \(f\) is continuous on \([a, b]\), differentiable inside, and \(f(a) = f(b)\), then \(f'(c) = 0\) for some \(c\) between \(a\) and \(b\). See Chapter 6.

Root test If \(a_n \ge 0\) and \(\sqrt[n]{a_n} \to \rho\), then \(\sum a_n\) converges if \(\rho < 1\) and diverges if \(\rho > 1\). See Chapter 12.

S

Saddle point A critical point that is neither a local maximum nor a local minimum, where the graph rises in some directions and falls in others, like \(x^2 - y^2\) at the origin. See Chapter 15.

Scale height The height over which the pressure or density of an atmosphere falls by a factor of \(e\), in the law \(p = p_0e^{-h/H}\). For Mars it is about 11 km. See Chapter 5 and Chapter 14.

Secant line The straight line through two points of a graph. Its slope is an average rate of change. See Chapter 1 and Chapter 4.

Second derivative The derivative of the derivative, \(f''\). It measures how the slope changes, and gives concavity and acceleration. See Chapter 4 and Chapter 7.

Second derivative test At a critical point, \(f'' > 0\) gives a local minimum and \(f'' < 0\) a local maximum. In several variables the Hessian matrix plays the role of \(f''\). See Chapter 7 and Chapter 15.

Second-order linear equation A differential equation \(ay'' + by' + cy = g(x)\); with constant coefficients it is solved through its characteristic equation. See Chapter 14.

Sendov's conjecture If all zeros of a polynomial lie in the closed unit disc, each zero is within distance \(1\) of some zero of the derivative. See Chapter 22.

Sensitive dependence on initial conditions Nearby starting points move apart exponentially fast under a map, so that small errors ruin long-term prediction. It is the hallmark of chaos. See Chapter 22.

Separable equation A differential equation \(y' = g(x)h(y)\), solved by separating the variables and integrating \(\int \frac{dy}{h(y)} = \int g(x)\,dx\). See Chapter 14.

Separation of variables (PDE) Looking for solutions of a partial differential equation as products such as \(X(x)T(t)\), which turns it into ordinary differential equations. See Chapter 21.

Sequence An infinite list \(a_1, a_2, a_3, \ldots\) of numbers, that is, a function on the whole numbers. See Chapter 12.

Sequentially compact A set in which every sequence has a subsequence converging to a point of the set. See Chapter 20.

Series An infinite sum \(\sum a_n = a_1 + a_2 + \cdots\), defined as the limit of its partial sums. See Chapter 12.

Shadow price The economist's name for a Lagrange multiplier: the gain in the best value per unit relaxation of a constraint. See Chapter 15.

Shell method Finding a volume of revolution by slicing it into thin cylindrical shells, giving \(\int 2\pi x\,h(x)\,dx\). See Chapter 11.

\(\sigma\)-algebra A family of sets closed under complements and countable unions (hence countable intersections), such as the Lebesgue measurable sets. See Chapter 21.

Simple function A function taking finitely many values, each on a measurable set: \(\sum c_i \mathbf 1_{E_i}\). See Chapter 21.

Simply connected A region in which every closed curve can be shrunk to a point without leaving it; on such a region a field with zero curl is conservative. See Chapter 17.

Simpson's rule Numerical integration that replaces \(f\) on each pair of subintervals by the parabola through three points. Its error falls like \(\frac{1}{n^4}\). See Chapter 10.

Slope field A picture of a differential equation \(y' = F(x, y)\) drawn as short line segments of slope \(F(x, y)\) at many points; solutions follow the segments. See Chapter 14.

Smith-Volterra-Cantor set A "fat" Cantor set: removing middle intervals of lengths \(\frac14\), \(\frac{1}{16}\), ... leaves a set of measure \(\frac12\) that contains no interval. See Chapter 21.

Smooth A function with derivatives of every order, written \(C^\infty\). See Chapter 13.

Snell's law The law of refraction \(\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}\), derived from Fermat's principle that light takes the path of least time. See Chapter 7.

Spectral theorem A self-adjoint operator on a Hilbert space can be diagonalised, the infinite-dimensional version of diagonalising a symmetric matrix. See Chapter 22.

Spherical coordinates Coordinates \((\rho, \phi, \theta)\) in space: distance from the origin, angle from the \(z\)-axis and polar angle. The volume element is \(\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\). See Chapter 16.

Squeeze theorem If \(g(x) \le f(x) \le h(x)\) near \(a\) and \(g\) and \(h\) both tend to \(L\), then \(f\) tends to \(L\) too. See Chapter 2 and Chapter 2.

Stokes's theorem The circulation of a field around the boundary of a surface equals the flux of its curl through the surface: \(\oint_{\partial S} \mathbf F \cdot d\mathbf r = \iint_S \operatorname{curl}\mathbf F \cdot d\mathbf S\). See Chapter 17.

Subsequence A sequence \(a_{n_1}, a_{n_2}, \ldots\) with \(n_1 < n_2 < \cdots\), taken from a given sequence by skipping terms. See Chapter 12 and Chapter 18.

Substitution The rule \(\int f(g(x))g'(x)\,dx = \int f(u)\,du\) with \(u = g(x)\), the chain rule read backwards. See Chapter 10.

Sup norm The norm \(\|g\|_\infty = \sup_{x \in D} |g(x)|\) of a bounded function. \(f_n \to f\) uniformly exactly when \(\|f_n - f\|_\infty \to 0\). See Chapter 19.

Superposition For a homogeneous linear equation, any sum of multiples of solutions is again a solution. See Chapter 14 and Chapter 21.

Supremum The least upper bound of a set of real numbers, written \(\sup S\): an upper bound that is at most every other upper bound. By completeness, every nonempty set bounded above has one. See Chapter 8, Chapter 12 and Chapter 18.

Surface area of revolution The area of the surface swept out by rotating a curve about an axis, \(\int 2\pi f(x)\sqrt{1 + f'(x)^2}\,dx\). See Chapter 11.

Surface integral The integral of a function over a surface, \(\iint_S f\,dS\), or of a vector field across it, which is the flux. See Chapter 17.

T

Tag A choice of one point \(t_k\) in each subinterval of a partition, at which a Riemann sum takes its heights. See Chapter 8.

Takagi function The continuous, nowhere differentiable function \(\sum_{k \ge 0} \frac{s(2^k x)}{2^k}\), where \(s(x)\) is the distance from \(x\) to the nearest integer. See Chapter 19.

Tangent line The line through \((a, f(a))\) with slope \(f'(a)\): the limit of secant lines, and the best straight-line approximation to the graph near \(a\). See Chapter 1 and Chapter 4.

Tangent plane The plane \(z = f(\mathbf a) + \nabla f(\mathbf a) \cdot (\mathbf x - \mathbf a)\) that best approximates the graph of a differentiable function of two variables near \(\mathbf a\). See Chapter 15.

Taylor polynomial The polynomial \(T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^k\), which has the same value and first \(n\) derivatives as \(f\) at \(a\). See Chapter 13.

Taylor series The power series \(\sum_{k \ge 0} \frac{f^{(k)}(a)}{k!}(x - a)^k\) of a smooth function. It need not converge to \(f\), as \(e^{-1/x^2}\) shows. See Chapter 13 and Chapter 13.

Taylor's theorem The remainder of the Taylor polynomial is \(R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}\) for some \(c\) between \(a\) and \(x\) (Lagrange form), or an explicit integral (integral form). See Chapter 13.

Telescoping series A series of the form \(\sum (b_n - b_{n+1})\), whose partial sums collapse to \(b_1 - b_{N+1}\). See Chapter 12.

Terminal velocity The constant speed a falling body approaches when air resistance balances gravity. See Chapter 14.

Thomae's function The function equal to \(\frac1q\) at a rational \(\frac pq\) in lowest terms and \(0\) at irrationals. It is continuous at every irrational and discontinuous at every rational, yet integrable with integral \(0\). See Chapter 8.

Topologist's sine curve The graph of \(\sin(1/x)\) for \(0 < x \le 1\) together with the segment of the \(y\)-axis from \(-1\) to \(1\): a set that is connected but not path-connected. See Chapter 20.

Topology The collection of open sets of a space, which is all one needs to talk about continuity, compactness and connectedness. See Chapter 20.

Totally disconnected A set whose only connected pieces are single points, like the Cantor set. See Chapter 20.

Trapezoid rule Numerical integration that joins neighbouring points of the graph by straight lines and adds the areas of the trapezoids. Its error falls like \(\frac{1}{n^2}\). See Chapter 10.

Triangle inequality \(|a + b| \le |a| + |b|\); in a metric space, \(d(x, z) \le d(x, y) + d(y, z)\). See Chapter 2 and Chapter 20.

Trigonometric series A series \(\frac{a_0}{2} + \sum (a_n\cos nx + b_n\sin nx)\) in sines and cosines. See Chapter 21.

Trigonometric substitution Replacing \(x\) by \(a\sin\theta\), \(a\tan\theta\) or \(a\sec\theta\) to simplify integrals containing \(\sqrt{a^2 - x^2}\), \(\sqrt{a^2 + x^2}\) or \(\sqrt{x^2 - a^2}\). See Chapter 10.

Triple integral The integral \(\iiint_E f\,dV\) of a function of three variables over a solid region. See Chapter 16.

U

Uncertainty principle A function and its Fourier transform cannot both be sharply concentrated: the product of their spreads is bounded below. See Chapter 22.

Uncountable An infinite set that cannot be listed as \(x_1, x_2, \ldots\), such as \(\mathbb{R}\) or the Cantor set. See Chapter 18.

Underdamped A damped spring whose characteristic equation has complex roots: it oscillates with decaying amplitude. See Chapter 14.

Uniform boundedness principle A family of bounded linear maps on a Banach space that is bounded at each point is bounded in norm. It shows that some continuous function has a divergent Fourier series. See Chapter 22.

Uniform Cauchy criterion A sequence of functions converges uniformly if and only if for every \(\varepsilon > 0\) there is \(N\) with \(|f_m(x) - f_n(x)| < \varepsilon\) for all \(m, n \ge N\) and all \(x\). See Chapter 19.

Uniform convergence \(f_n \to f\) uniformly on \(D\) if for every \(\varepsilon > 0\) one \(N\) works for all \(x\) at once: \(|f_n(x) - f(x)| < \varepsilon\) for all \(n \ge N\) and all \(x\) in \(D\). Uniform limits of continuous functions are continuous, and integrals of uniform limits are limits of integrals. See Chapter 13, Chapter 19, Chapter 19 and Chapter 19.

Uniformly continuous \(f\) is uniformly continuous on \(D\) if for every \(\varepsilon > 0\) one \(\delta\) works everywhere: \(|x - y| < \delta\) implies \(|f(x) - f(y)| < \varepsilon\) for all \(x, y\) in \(D\). See Chapter 8 and Chapter 18.

Unit ball The set of points of \(\mathbb{R}^n\) at distance at most \(1\) from the origin. Its volume is \(\frac{\pi^{n/2}}{\Gamma(n/2 + 1)}\), which tends to \(0\) as \(n\) grows. See Chapter 16.

Upper bound A number \(M\) with \(s \le M\) for every \(s\) in a set \(S\). See Chapter 18.

Upper sum The sum of the areas of rectangles whose heights are the largest (supremum) values of \(f\) on each piece of a partition. It is never less than the area. See Chapter 1 and Chapter 8.

V

Variance The spread of a random variable with density \(f\) and mean \(\mu\): \(\int (x - \mu)^2 f(x)\,dx\). See Chapter 11.

Vector field A rule assigning a vector \(\mathbf F(\mathbf x)\) to each point, such as a wind or a gravitational field. See Chapter 17.

Velocity The rate of change of position, \(v(t) = s'(t)\). Speed is its absolute value. See Chapter 4 and Chapter 7.

Vertical asymptote A line \(x = a\) near which \(f(x)\) tends to \(\infty\) or \(-\infty\) as \(x\) approaches \(a\) from one side or both. See Chapter 2.

Volterra's function A function, found by Volterra in 1881, that is differentiable everywhere with a bounded derivative that is not Riemann integrable. See Chapter 9 and Chapter 21.

Volume by slicing The volume of a solid as \(\int A(x)\,dx\), where \(A(x)\) is the area of its cross-section at \(x\); discs and washers are special cases. See Chapter 11.

W

Wallis's product The infinite product \(\frac\pi2 = \frac21 \cdot \frac23 \cdot \frac43 \cdot \frac45 \cdot \frac65 \cdot \frac67 \cdots\), found by John Wallis in 1656 and proved with reduction formulas. See Chapter 10.

Washer The ring-shaped cross-section of a solid of revolution with a hole; its area is \(\pi(R^2 - r^2)\). See Chapter 11.

Wave equation The partial differential equation \(u_{tt} = c^2 \Delta u\) for waves moving at speed \(c\). Maxwell's equations give it for light. See Chapter 17 and Chapter 22.

Wavelets Families of functions made from one small wave by shifting and scaling, used to analyse signals that change over time and to compress images. See Chapter 22.

Weak solution A solution of a partial differential equation in an averaged sense, which may not be differentiable. Leray proved that weak solutions of the Navier-Stokes equations exist for all time. See Chapter 22.

Wedge product The product \(\wedge\) of differential forms, with \(dx \wedge dy = -dy \wedge dx\). See Chapter 17.

Weierstrass approximation theorem Every continuous function on a closed bounded interval is a uniform limit of polynomials. See Chapter 19.

Weierstrass M-test If \(|g_k(x)| \le M_k\) for all \(x\) and \(\sum M_k\) converges, then \(\sum g_k\) converges uniformly (and absolutely). See Chapter 19.

Weierstrass substitution The half-angle substitution \(t = \tan\frac x2\), which turns any rational function of \(\sin x\) and \(\cos x\) into a rational function of \(t\). See Chapter 10.

Work The integral \(\int_a^b F(x)\,dx\) of force over distance; for a vector field along a curve, the line integral \(\int_C \mathbf F \cdot d\mathbf r\). See Chapter 11 and Chapter 17.