Glossary

This glossary collects the key terms of the two-to-three-semester calculus sequence covered in this book. Each entry gives a precise, plain-language definition, the relevant notation, and the chapter where the term is introduced (or most fully developed). Where a term has both a formal and an informal name, both are given. Terms in italics within a definition appear as their own entries.

A

Absolute (global) extremum — The largest (absolute maximum) or smallest (absolute minimum) value a function attains over its entire domain or a specified set. The Extreme Value Theorem guarantees both exist for a continuous function on a closed, bounded interval. (Ch. 10)

Absolute convergence — A series $\sum a_n$ converges absolutely if the series of absolute values $\sum |a_n|$ converges. Absolute convergence implies convergence, and absolutely convergent series may be rearranged without changing their sum. (Ch. 22)

Alternating series — A series whose terms alternate in sign, written $\sum (-1)^n b_n$ with $b_n \ge 0$. (Ch. 22)

Alternating series test — If $b_n$ is positive, decreasing, and $b_n \to 0$, then the alternating series $\sum (-1)^n b_n$ converges; the truncation error is bounded by the first omitted term. (Ch. 22)

Antiderivative — A function $F$ whose derivative is $f$, that is $F' = f$. Any two antiderivatives of $f$ differ by a constant, so the general antiderivative is written $F(x) + C$. (Ch. 12)

Arc length — The length of a curve. For $y = f(x)$ on $[a,b]$ it is $\int_a^b \sqrt{1 + (f'(x))^2}\,dx$; for a parametric or vector curve it is $\int \|\mathbf{r}'(t)\|\,dt$. (Ch. 18, 25, 28)

Asymptote — A line a curve approaches arbitrarily closely. Vertical asymptotes occur where the function blows up; horizontal asymptotes describe end behavior; slant (oblique) asymptotes are non-horizontal lines approached as $x \to \pm\infty$. (Ch. 3, 9)

Average value — The mean height of $f$ over $[a,b]$, defined as $\frac{1}{b-a}\int_a^b f(x)\,dx$. The Mean Value Theorem for Integrals guarantees $f$ attains this value at some point in the interval. (Ch. 14)

B

Basel problem — The problem of summing $\sum_{n=1}^\infty 1/n^2$, famously solved by Euler, who showed the value is $\pi^2/6$. (Ch. 24)

Bisection method — A root-finding algorithm that repeatedly halves an interval on which a continuous function changes sign, relying on the Intermediate Value Theorem to trap a root. (Ch. 4)

C

Cardioid — A heart-shaped polar curve of the form $r = a(1 - \cos\theta)$ (or with $\sin$), a special case of the limaçon. (Ch. 26)

Centroid — The geometric center (center of mass for uniform density) of a region. For a plane region its coordinates are weighted averages computed with definite integrals. (Ch. 18)

Chain rule — The rule for differentiating a composition: $(f \circ g)'(x) = f'(g(x))\,g'(x)$, or in Leibniz form $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$. (Ch. 7)

Comparison test — For positive-term series, if $0 \le a_n \le b_n$ and $\sum b_n$ converges then $\sum a_n$ converges; if $a_n \ge b_n \ge 0$ and $\sum b_n$ diverges then $\sum a_n$ diverges. The limit comparison test uses $\lim a_n/b_n$ instead. (Ch. 22)

Concavity — The bending direction of a graph, governed by the sign of $f''$. The curve is concave up (holds water) where $f'' > 0$ and concave down where $f'' < 0$. (Ch. 9)

Conditional convergence — A series that converges but does not converge absolutely, such as the alternating harmonic series $\sum (-1)^{n+1}/n$. Its sum can be changed by rearrangement (Riemann rearrangement theorem). (Ch. 22)

Conic section — A curve obtained by slicing a cone: the parabola, ellipse, and hyperbola (the circle being a special ellipse). Each can be described by a focus, directrix, and eccentricity. (Ch. 27)

Conservative vector field — A vector field that is the gradient of some scalar potential function, $\mathbf{F} = \nabla \varphi$. Line integrals of conservative fields are path-independent, and (on simply connected domains) a field is conservative exactly when its curl is zero. (Ch. 35)

Continuity — A function $f$ is continuous at $a$ if $\lim_{x \to a} f(x) = f(a)$; this requires the limit to exist, $f(a)$ to be defined, and the two to agree. A function continuous at every point of an interval is continuous on that interval. (Ch. 4)

Convergence — The property of a limit, sequence, series, or improper integral settling on a single finite value. A sequence $\{a_n\}$ converges to $L$ if its terms get arbitrarily close to $L$; a series converges if its partial sums converge. (Ch. 3, 20, 21)

Critical point (critical number) — A point in the domain of $f$ where $f'(x) = 0$ or $f'(x)$ does not exist; in several variables, where $\nabla f = \mathbf{0}$ or is undefined. Interior extrema can occur only at critical points. (Ch. 9, 31)

Curl — A measure of the infinitesimal rotation of a vector field, written $\nabla \times \mathbf{F}$. A field with zero curl everywhere on a simply connected region is conservative. (Ch. 34)

Curvature — How sharply a curve bends, $\kappa = \|\mathbf{r}'(t) \times \mathbf{r}''(t)\| / \|\mathbf{r}'(t)\|^3$; the reciprocal $1/\kappa$ is the radius of the best-fitting circle. (Ch. 28)

Cycloid — The curve traced by a point on the rim of a wheel rolling along a straight line, a classic parametric example. (Ch. 25)

Cylindrical coordinates — A 3D coordinate system $(r, \theta, z)$ combining polar coordinates in the plane with a height $z$; useful for integrals with circular symmetry about an axis. (Ch. 32)

D

Definite integral — The signed area accumulated by $f$ over $[a,b]$, defined as the limit of Riemann sums, written $\int_a^b f(x)\,dx$. It is a number, not a function. (Ch. 13)

Derivative — The instantaneous rate of change of $f$, defined as the limit of the difference quotient: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Geometrically it is the slope of the tangent line. (Ch. 5, 6)

Difference quotient — The average rate of change $\frac{f(x+h) - f(x)}{h}$, whose limit as $h \to 0$ defines the derivative. (Ch. 5)

Differentiable — A function is differentiable at a point if its derivative exists there; this requires the graph to have a well-defined non-vertical tangent (no corner, cusp, or break). Differentiability implies continuity, but not conversely. (Ch. 6)

Differential — The linear quantity $dy = f'(x)\,dx$ expressing how the output changes for an infinitesimal input change $dx$; the foundation of linear approximation and the notation for integrals. (Ch. 11)

Differential equation — An equation relating a function to its derivatives, such as $y' = ky$. Solving one means finding the function(s) that satisfy it. (Ch. 19)

Differential form — An object such as $P\,dx + Q\,dy$ that can be integrated over a curve or surface; differential forms unify Green's, Stokes', and the Divergence theorems into a single generalized statement. (Ch. 38)

Directional derivative — The rate of change of $f$ at a point in the direction of a unit vector $\mathbf{u}$, computed as $D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}$. (Ch. 30)

Discontinuity — A point where a function fails to be continuous. A removable discontinuity is a hole that could be patched; a jump discontinuity has differing finite one-sided limits; an infinite discontinuity occurs where the function diverges to $\pm\infty$. (Ch. 4)

Divergence — A measure of the net outward flux per unit volume of a vector field, written $\nabla \cdot \mathbf{F}$; positive divergence indicates a source, negative a sink. (Ch. 34)

Divergence (of a series/sequence) — Failure to converge: the values do not approach a single finite limit. (Ch. 20, 21)

Divergence Theorem (Gauss's theorem) — Relates the flux of a vector field out of a closed surface to the divergence inside: $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F})\,dV$. (Ch. 37)

Double integral — An integral of a function of two variables over a planar region, $\iint_R f(x,y)\,dA$, giving signed volume under a surface; evaluated as an iterated integral by Fubini's theorem. (Ch. 32)

E

$e$ — Euler's number, $\approx 2.71828$, the base of the natural exponential and logarithm; the unique base for which $\frac{d}{dx}e^x = e^x$. (Ch. 7)

Eccentricity — A number $e$ describing the shape of a conic section: $e = 0$ for a circle, $0 < e < 1$ for an ellipse, $e = 1$ for a parabola, and $e > 1$ for a hyperbola. (Ch. 26, 27)

Ellipse — A conic whose points have constant total distance to two foci; in standard form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. (Ch. 27)

$\varepsilon$–$\delta$ definition (of a limit) — The formal 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 $|f(x) - L| < \varepsilon$. It makes "arbitrarily close" precise. (Ch. 3)

Euler's formula — The identity $e^{i\theta} = \cos\theta + i\sin\theta$, linking exponential and trigonometric functions; derived from Taylor series. (Ch. 24)

Euler's identity — The special case $e^{i\pi} + 1 = 0$, uniting $e$, $i$, $\pi$, $1$, and $0$; a recurring anchor example of the book. (Ch. 24)

Euler's method — The simplest numerical scheme for solving an initial value problem, stepping forward with $y_{n+1} = y_n + h\,f(t_n, y_n)$. (Ch. 19)

Extreme Value Theorem (EVT) — A function continuous on a closed, bounded interval $[a,b]$ attains both an absolute maximum and an absolute minimum on that interval. (Ch. 4)

F

Finite difference — A discrete approximation to a derivative using nearby function values, such as the forward difference $\frac{f(x+h) - f(x)}{h}$; the basis of numerical differentiation. (Ch. 6)

First derivative test — A method to classify a critical point by examining the sign change of $f'$: from $+$ to $-$ gives a local max, from $-$ to $+$ a local min. (Ch. 9)

Flux — The net amount of a vector field passing through a surface, computed as the surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$. (Ch. 36)

Focus (plural foci) — A special point defining a conic section; reflection and distance properties of the ellipse, parabola, and hyperbola are stated relative to their foci. (Ch. 27)

Fubini's theorem — Under mild conditions a double or triple integral equals an iterated integral, and the order of integration may be exchanged. (Ch. 32)

Function — A rule assigning to each input exactly one output, with a specified domain and range. (Ch. 2)

Fundamental Theorem of Calculus (FTC) — The central result tying differentiation and integration together. Part 1 states $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$; Part 2 states $\int_a^b f(x)\,dx = F(b) - F(a)$ for any antiderivative $F$. (Ch. 14)

G

Gamma function — A continuous extension of the factorial, $\Gamma(n) = \int_0^\infty t^{n-1} e^{-t}\,dt$, satisfying $\Gamma(n) = (n-1)!$ for positive integers; a key improper integral. (Ch. 17)

Gaussian integral — The integral $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$, evaluated by a polar change of variables; central to the normal distribution. (Ch. 32)

Geometric series — A series $\sum_{n=0}^\infty a r^n$ with constant ratio $r$. It converges to $\frac{a}{1-r}$ when $|r| < 1$ and diverges otherwise. (Ch. 21)

Gradient — The vector of partial derivatives $\nabla f = \langle f_x, f_y, \ldots \rangle$; it points in the direction of steepest ascent and is perpendicular to level curves/surfaces. (Ch. 30)

Gradient descent — An iterative optimization algorithm that steps opposite the gradient, $\mathbf{x}_{n+1} = \mathbf{x}_n - \alpha\,\nabla f(\mathbf{x}_n)$, to minimize a function; a recurring anchor example tied to machine learning. (Ch. 6, 30)

Green's theorem — Relates a line integral around a simple closed curve to a double integral over the enclosed region: $\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dA$. (Ch. 35)

H

Harmonic series — The series $\sum_{n=1}^\infty 1/n$, which diverges despite its terms tending to zero — a fundamental cautionary example. (Ch. 21)

Hessian — The matrix of second-order partial derivatives of a multivariable function; its definiteness drives the second derivative test for classifying critical points. (Ch. 31)

Higher-order derivative — A derivative taken more than once, such as $f''$, $f'''$, or $f^{(n)}$; the second derivative measures concavity and acceleration. (Ch. 9)

Hyperbola — A conic whose points have constant difference of distances to two foci; standard form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. (Ch. 27)

I

Implicit differentiation — Differentiating an equation that defines $y$ implicitly in terms of $x$, treating $y$ as a function of $x$ and applying the chain rule, then solving for $dy/dx$. (Ch. 8)

Improper integral — An integral with an infinite limit of integration or an unbounded integrand, defined as a limit of proper integrals; it converges if that limit is finite. (Ch. 17)

Indefinite integral — The general antiderivative of $f$, written $\int f(x)\,dx = F(x) + C$; unlike a definite integral, it is a family of functions, not a number. (Ch. 12)

Inflection point — A point where a curve's concavity changes sign, typically where $f'' = 0$ or is undefined and the sign of $f''$ flips. (Ch. 9)

Initial value problem (IVP) — A differential equation together with initial condition(s) that pin down a unique solution. (Ch. 12, 19)

Integral test — For a positive, decreasing function $f$, the series $\sum f(n)$ and the integral $\int_1^\infty f(x)\,dx$ both converge or both diverge. (Ch. 22)

Integration by parts — A technique reversing the product rule: $\int u\,dv = uv - \int v\,du$. (Ch. 15)

Intermediate Value Theorem (IVT) — If $f$ is continuous on $[a,b]$ and $N$ lies between $f(a)$ and $f(b)$, then $f(c) = N$ for some $c$ in $(a,b)$. (Ch. 4)

J

Jacobian — The determinant of the matrix of partial derivatives of a change-of-variables map; it measures local area/volume distortion and is the factor that appears when transforming a multiple integral. (Ch. 33)

K

Kepler's laws — Three empirical laws of planetary motion (elliptical orbits, equal areas in equal times, harmonic period–distance relation) derivable from Newtonian gravity using calculus. (Ch. 27)

L

Lagrange multipliers — A method for constrained optimization in which extrema of $f$ subject to $g = c$ occur where $\nabla f = \lambda \nabla g$ for some scalar multiplier $\lambda$. (Ch. 31)

Level curve (level set) — The set of points where a multivariable function is constant, $f(x,y) = k$; the gradient is everywhere perpendicular to level curves. (Ch. 29)

L'Hôpital's rule — For an indeterminate form $0/0$ or $\infty/\infty$, $\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$ provided the latter limit exists. (Ch. 9)

Limit — The value a function approaches as the input approaches a target: $\lim_{x \to a} f(x) = L$ means $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ (but not equal). Made precise by the $\varepsilon$–$\delta$ definition. (Ch. 3)

Line integral — An integral of a scalar function or vector field along a curve, $\int_C f\,ds$ or $\int_C \mathbf{F} \cdot d\mathbf{r}$; for vector fields it measures work done along the path. (Ch. 35)

Linear approximation (linearization) — Approximating $f$ near $a$ by its tangent line, $f(x) \approx f(a) + f'(a)(x - a)$; the simplest case of Taylor approximation and an instance of "approximation is the soul of calculus." (Ch. 11)

Logistic equation — The differential equation $\frac{dy}{dt} = ry\left(1 - \frac{y}{K}\right)$ modeling bounded population growth toward a carrying capacity $K$. (Ch. 19)

M

Maclaurin series — A Taylor series expanded about $a = 0$: $\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$. (Ch. 23)

Mean Value Theorem (MVT) — If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there is a $c$ in $(a,b)$ with $f'(c) = \frac{f(b) - f(a)}{b - a}$ — the instantaneous rate equals the average rate somewhere. (Ch. 9)

N

Net change — The accumulated change of a quantity, recovered by integrating its rate of change: $\int_a^b F'(x)\,dx = F(b) - F(a)$. A direct reading of FTC Part 2. (Ch. 14)

Newton's method — A fast root-finding iteration using tangent lines: $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$. (Ch. 11)

Normal distribution — The bell-shaped probability density $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ (standard form); its tail areas are computed by integration and approximated by Taylor series, a recurring anchor example. (Ch. 13, 32)

Numerical integration — Approximating a definite integral when no elementary antiderivative exists, using rules such as the midpoint, trapezoidal, and Simpson's rules. (Ch. 13)

O

One-sided limit — The value a function approaches from one side only: $\lim_{x \to a^+} f(x)$ (from the right) or $\lim_{x \to a^-} f(x)$ (from the left). A two-sided limit exists exactly when both one-sided limits exist and agree. (Ch. 3)

Optimization — Finding the maximum or minimum value of a quantity, in one variable via critical points and the EVT, in several variables via the Hessian or Lagrange multipliers. (Ch. 10, 31)

P

Parabola — A conic consisting of points equidistant from a focus and a directrix; standard form $y^2 = 4px$ (or rotated). (Ch. 27)

Parameter — An auxiliary variable (often $t$) used to describe a curve's coordinates separately, as in $x = x(t),\ y = y(t)$. (Ch. 25)

Parametric equations — A description of a curve by coordinate functions of a parameter, $x = x(t),\ y = y(t)$, allowing motion, direction, and self-intersection to be expressed. (Ch. 25)

Partial derivative — The derivative of a multivariable function with respect to one variable while holding the others fixed, written $\frac{\partial f}{\partial x}$ or $f_x$. (Ch. 29)

Partial fractions — A technique that rewrites a rational function as a sum of simpler fractions so each piece can be integrated. (Ch. 16)

Partial sum — The sum of the first $n$ terms of a series, $s_n = \sum_{i=1}^n a_i$; a series converges precisely when its sequence of partial sums converges. (Ch. 21)

Polar coordinates — A plane coordinate system $(r, \theta)$ locating points by distance from the origin and angle from the positive $x$-axis. (Ch. 26)

Power rule — The basic differentiation rule $\frac{d}{dx} x^n = n x^{n-1}$, valid for any real exponent $n$. (Ch. 7)

Power series — A series of the form $\sum_{n=0}^\infty c_n (x - a)^n$ that defines a function on its interval of convergence. (Ch. 23)

Product rule — The rule $(fg)' = f'g + fg'$ for differentiating a product. (Ch. 7)

Q

Quotient rule — The rule $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ for differentiating a quotient. (Ch. 7)

R

Radius of convergence — The number $R$ such that a power series $\sum c_n (x-a)^n$ converges for $|x - a| < R$ and diverges for $|x - a| > R$; endpoints must be checked separately to find the full interval of convergence. (Ch. 23)

Ratio test — Using $L = \lim |a_{n+1}/a_n|$: the series converges absolutely if $L < 1$, diverges if $L > 1$, and the test is inconclusive if $L = 1$. Especially effective for series with factorials or exponentials. (Ch. 22)

Related rates — Problems in which several quantities change in time and are linked by an equation, so differentiating that equation relates their rates. (Ch. 8)

Removable discontinuity — A discontinuity where the limit exists but differs from (or replaces a missing) function value — a "hole" that can be patched by redefining one point. (Ch. 4)

Riemann sum — An approximation of a definite integral by summing rectangle areas $\sum f(x_i^*)\,\Delta x$; the definite integral is the limit as the partition is refined. (Ch. 13)

Rolle's theorem — If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then $f'(c) = 0$ for some $c$ in $(a,b)$ — the special case of the MVT with equal endpoints. (Ch. 9)

Root test — Using $L = \lim \sqrt[n]{|a_n|}$: the series converges absolutely if $L < 1$, diverges if $L > 1$, inconclusive if $L = 1$. Useful when terms involve $n$th powers. (Ch. 22)

S

Saddle point — A critical point of a multivariable function that is a maximum along one direction and a minimum along another; detected when the Hessian is indefinite. (Ch. 31)

Second derivative test — A classification of a critical point using $f''$ (one variable: $f'' > 0$ ⇒ local min, $f'' < 0$ ⇒ local max) or the Hessian in several variables. (Ch. 9, 31)

Sequence — An ordered list of numbers indexed by the naturals, a function $\{a_n\}$ on $\mathbb{N}$. (Ch. 20)

Series — The sum of the terms of a sequence, $\sum_{n=1}^\infty a_n$, given meaning through its sequence of partial sums. (Ch. 21)

SIR model — A system of differential equations modeling an epidemic by tracking Susceptible, Infected, and Recovered populations; a recurring anchor example. (Ch. 19)

Solid of revolution — A 3D solid formed by rotating a plane region about an axis; its volume is found by the disk, washer, or shell methods. (Ch. 18)

Spherical coordinates — A 3D coordinate system $(\rho, \theta, \phi)$ using distance from the origin and two angles; ideal for integrals with spherical symmetry. (Ch. 32)

Squeeze theorem (sandwich theorem) — If $g(x) \le f(x) \le h(x)$ near $a$ and $\lim g = \lim h = L$, then $\lim f = L$; used to evaluate limits like $\lim_{x\to 0} x^2 \sin(1/x)$. (Ch. 3)

Stokes' theorem — Relates the circulation of a vector field around a closed curve to the flux of its curl through any surface it bounds: $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$; a generalization of Green's theorem to surfaces. (Ch. 37)

Substitution rule ($u$-substitution) — Reversing the chain rule for integration: $\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ with $u = g(x)$. (Ch. 15)

Surface integral — An integral of a scalar function or vector field over a surface in space, $\iint_S f\,dS$ or $\iint_S \mathbf{F} \cdot d\mathbf{S}$ (the latter giving flux). (Ch. 36)

T

Tangent line — The line touching a curve at a point with the same slope, $y = f(a) + f'(a)(x - a)$; the best linear approximation to the curve there and the geometric meaning of the derivative. (Ch. 1, 5)

Tangent plane — The plane that best approximates the graph of $z = f(x,y)$ at a point, $z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$; the two-variable analog of the tangent line. (Ch. 30)

Taylor series — The power series representation of a smooth function about $a$: $\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$, expressing the function as an "infinite polynomial." (Ch. 23)

Taylor's theorem with remainder — Gives the error in approximating $f$ by its degree-$n$ Taylor polynomial, e.g. the Lagrange remainder $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$, making "close enough" quantitative. (Ch. 23)

Telescoping series — A series whose partial sums collapse because consecutive terms cancel, leaving only a few terms to evaluate the limit. (Ch. 21)

Triple integral — An integral of a function of three variables over a solid region, $\iiint_E f\,dV$; used for volume, mass, and center of mass. (Ch. 32)

Trigonometric substitution — An integration technique substituting a trig function (e.g. $x = a\sin\theta$) to simplify integrands containing $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$. (Ch. 16)

U

$u$-substitution — See Substitution rule. (Ch. 15)

V

Vector field — An assignment of a vector to each point of space, $\mathbf{F}(x,y,z)$, used to model velocity, force, and flow. (Ch. 34)

Vector-valued function — A function $\mathbf{r}(t) = \langle x(t), y(t), z(t)\rangle$ tracing a curve in space; its derivative gives velocity and its second derivative acceleration. (Ch. 28)

Volume of revolution — See Solid of revolution; the volume of the solid generated by rotating a region about an axis. (Ch. 18)

W

Work — The energy expended by a force over a displacement, computed as a definite integral $W = \int_a^b F(x)\,dx$ (variable force) or as a line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ of a force field. (Ch. 18, 35)