Glossary

absolute convergence | if the series $\displaystyle \sum^∞_{n=1}|a_n|$ converges, the series $\displaystyle \sum^∞_{n=1}a_n$ is said to converge absolutely

absolute error | if $B$ is an estimate of some quantity having an actual value of $A$, then the absolute error is given by $|A−B|$

absolute extremum | if $f$ has an absolute maximum or absolute minimum at $c$, we say $f$ has an absolute extremum at $c$

absolute maximum | if $f(c)≥f(x)$ for all $x$ in the domain of $f$, we say $f$ has an absolute maximum at $c$

absolute minimum | if $f(c)≤f(x)$ for all $x$ in the domain of $f$, we say $f$ has an absolute minimum at $c$

absolute value function | $f(x)=\begin{cases}−x, & \text{if } x<0\x, & \text{if } x≥0\end{cases}$

acceleration | is the rate of change of the velocity, that is, the derivative of velocity

acceleration vector | the second derivative of the position vector

algebraic function | a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable $x$

alternating series | a series of the form $\displaystyle \sum^∞_{n=1}(−1)^{n+1}b_n$ or $\displaystyle \sum^∞_{n=1}(−1)^nb_n$, where $b_n≥0$, is called an alternating series

alternating series test | for an alternating series of either form, if $b_{n+1}≤b_n$ for all integers $n≥1$ and $b_n→0$, then an alternating series converges

amount of change | the amount of a function $f(x)$ over an interval $[x,x+h] is f(x+h)−f(x)$

angular coordinate | $θ$ the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise

antiderivative | a function $F$ such that $F′(x)=f(x)$ for all $x$ in the domain of $f$ is an antiderivative of $f$

arc length | the arc length of a curve can be thought of as the distance a person would travel along the path of the curve

arc-length function | a function $s(t)$ that describes the arc length of curve $C$ as a function of $t$

arc-length parameterization | a reparameterization of a vector-valued function in which the parameter is equal to the arc length

arithmetic sequence | a sequence in which the difference between every pair of consecutive terms is the same is called an arithmetic sequence

asymptotically semi-stable solution | $y=k$ if it is neither asymptotically stable nor asymptotically unstable

asymptotically stable solution | $y=k$ if there exists $ε>0$ such that for any value $c∈(k−ε,k+ε)$ the solution to the initial-value problem $y′=f(x,y),y(x_0)=c$ approaches $k$ as $x$ approaches infinity

asymptotically unstable solution | $y=k$ if there exists $ε>0$ such that for any value $c∈(k−ε,k+ε)$ the solution to the initial-value problem $y′=f(x,y),y(x_0)=c$ never approaches $k$ as $x$approaches infinity

autonomous differential equation | an equation in which the right-hand side is a function of $y$ alone

average rate of change | is a function $f(x)$ over an interval $[x,x+h]$ is $\frac{f(x+h)−f(a)}{b−a}$

average value of a function | (or $f_{ave})$ the average value of a function on an interval can be found by calculating the definite integral of the function and dividing that value by the length of the interval

average velocity | the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval [$t,a$] (if $t<a$ or [$a,t$] if $t>a$), with a position given by $s(t)$, that is $v_{ave}=\dfrac{s(t)−s(a)}{t−a}$

base | the number $b$ in the exponential function $f(x)=b^x$ and the logarithmic function $f(x)=\log_bx$

binomial series | the Maclaurin series for $f(x)=(1+x)^r$; it is given by $(1+x)^r=\sum_{n=0}^∞(^r_n)x^n=1+rx+\dfrac{r(r−1)}{2!}x^2+⋯+\dfrac{r(r−1)⋯(r−n+1)}{n!}x^n+⋯$ for $|x|<1$

binormal vector | a unit vector orthogonal to the unit tangent vector and the unit normal vector

boundary conditions | the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times

boundary point | a point $P_0$ of $R$ is a boundary point if every $δ$ disk centered around $P_0$ contains points both inside and outside $R$

boundary-value problem | a differential equation with associated boundary conditions

bounded above | a sequence $\displaystyle {a_n}$ is bounded above if there exists a constant $\displaystyle M$ such that $\displaystyle a_n≤M$ for all positive integers $\displaystyle n$

bounded below | a sequence $\displaystyle {a_n}$ is bounded below if there exists a constant $\displaystyle M$ such that $\displaystyle M≤a_n$ for all positive integers $\displaystyle n$

bounded sequence | a sequence $\displaystyle {a_n}$ is bounded if there exists a constant $\displaystyle M$ such that $\displaystyle |a_n|≤M$ for all positive integers $\displaystyle n$

cardioid | a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is $r=a(1+\sin θ)$ or $r=a(1+\cos θ)$

carrying capacity | the maximum population of an organism that the environment can sustain indefinitely

catenary | a curve in the shape of the function $y=a\cdot\cosh(x/a)$ is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary

center of mass | the point at which the total mass of the system could be concentrated without changing the moment

centroid | the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of the lamina depends only on the shape of the corresponding planar region; in this case, the center of mass of the lamina corresponds to the centroid of the representative region

chain rule | the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function

change of variables | the substitution of a variable, such as $u$, for an expression in the integrand

characteristic equation | the equation $aλ^2+bλ+c=0$ for the differential equation $ay″+by′+cy=0$

circulation | the tendency of a fluid to move in the direction of curve $C$. If $C$ is a closed curve, then the circulation of $\vecs F$ along $C$ is line integral $∫_C \vecs F·\vecs T \,ds$, which we also denote $∮_C\vecs F·\vecs T \,ds$.

closed curve | a curve for which there exists a parameterization $\vecs r(t), a≤t≤b$, such that $\vecs r(a)=\vecs r(b)$, and the curve is traversed exactly once

closed curve | a curve that begins and ends at the same point

closed set | a set $S$ that contains all its boundary points

comparison test | If $0≤a_n≤b_n$ for all $n≥N$ and $\displaystyle \sum^∞_{n=1}b_n$ converges, then $\displaystyle \sum^∞_{n=1}a_n$ converges; if $a_n≥b_n≥0$ for all $n≥N$ and $\displaystyle \sum^∞_{n=1}b_n$ diverges, then $\displaystyle \sum^∞_{n=1}a_n$ diverges.

complementary equation | for the nonhomogeneous linear differential equation $$a+2(x)y″+a_1(x)y′+a_0(x)y=r(x), \nonumber$$ the associated homogeneous equation, called the complementary equation, is $$a_2(x)y''+a_1(x)y′+a_0(x)y=0 \nonumber$$

component | a scalar that describes either the vertical or horizontal direction of a vector

component functions | the component functions of the vector-valued function $\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}$ are $f(t)$ and $g(t)$, and the component functions of the vector-valued function $\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}+h(t)\hat{\mathbf{k}}$ are $f(t)$, $g(t)$ and $h(t)$

composite function | given two functions $f$ and $g$, a new function, denoted $g∘f$, such that $(g∘f)(x)=g(f(x))$

computer algebra system (CAS) | technology used to perform many mathematical tasks, including integration

concave down | if $f$ is differentiable over an interval $I$ and $f'$ is decreasing over $I$, then $f$ is concave down over $I$

concave up | if $f$ is differentiable over an interval $I$ and $f'$ is increasing over $I$, then $f$ is concave up over $I$

concavity | the upward or downward curve of the graph of a function

concavity test | suppose $f$ is twice differentiable over an interval $I$; if $f''>0$ over $I$, then $f$ is concave up over $I$; if $f''<$ over $I$, then $f$ is concave down over $I$

conditional convergence | if the series $\displaystyle \sum^∞_{n=1}a_n$ converges, but the series $\displaystyle \sum^∞_{n=1}|a_n|$ diverges, the series $\displaystyle \sum^∞_{n=1}a_n$ is said to converge conditionally

conic section | a conic section is any curve formed by the intersection of a plane with a cone of two nappes

connected region | a region in which any two points can be connected by a path with a trace contained entirely inside the region

connected set | an open set $S$ that cannot be represented as the union of two or more disjoint, nonempty open subsets

conservative field | a vector field for which there exists a scalar function $f$ such that $\vecs ∇f=\vecs{F}$

constant multiple law for limits | the limit law $$\lim_{x→a}cf(x)=c⋅\lim_{x→a}f(x)=cL \nonumber$$

constant multiple rule | the derivative of a constant $c$ multiplied by a function $f$ is the same as the constant multiplied by the derivative: $\dfrac{d}{dx}\big(cf(x)\big)=cf′(x)$

constant rule | the derivative of a constant function is zero: $\dfrac{d}{dx}(c)=0$, where $c$ is a constant

constraint | an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem

continuity at a point | A function $f(x)$ is continuous at a point a if and only if the following three conditions are satisfied: (1) $f(a)$ is defined, (2) $\displaystyle \lim_{x→a}f(x)$ exists, and (3) $\displaystyle \lim{x→a}f(x)=f(a)$

continuity from the left | A function is continuous from the left at b if $\displaystyle \lim_{x→b^−}f(x)=f(b)$

continuity from the right | A function is continuous from the right at a if $\displaystyle \lim_{x→a^+}f(x)=f(a)$

continuity over an interval | a function that can be traced with a pencil without lifting the pencil; a function is continuous over an open interval if it is continuous at every point in the interval; a function $f(x)$ is continuous over a closed interval of the form [$a,b$] if it is continuous at every point in ($a,b$), and it is continuous from the right at $a$ and from the left at $b$

contour map | a plot of the various level curves of a given function $f(x,y)$

convergence of a series | a series converges if the sequence of partial sums for that series converges

convergent sequence | a convergent sequence is a sequence $\displaystyle {a_n}$ for which there exists a real number $\displaystyle L$ such that $\displaystyle a_n$ is arbitrarily close to $\displaystyle L$ as long as $\displaystyle n$ is sufficiently large

coordinate plane | a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the $xy$-plane, $xz$-plane, or the $yz$-plane

critical point | if $f'(c)=0$ or $f'(c)$ is undefined, we say that c is a critical point of $f$

critical point of a function of two variables | the point $(x_0,y_0)$ is called a critical point of $f(x,y)$ if one of the two following conditions holds: 1. $f_x(x_0,y_0)=f_y(x_0,y_0)=0$ 2. At least one of $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ do not exist

cross product | $\vecs u×\vecs v=(u_2v_3−u_3v_2)\mathbf{\hat i}−(u_1v_3−u_3v_1)\mathbf{\hat j}+(u_1v_2−u_2v_1)\mathbf{\hat k},$ where $\vecs u=⟨u_1,u_2,u_3⟩$ and $\vecs v=⟨v_1,v_2,v_3⟩$ determinant a real number associated with a square matrix parallelepiped a three-dimensional prism with six faces that are parallelograms torque the effect of a force that causes an object to rotate triple scalar product the dot product of a vector with the cross product of two other vectors: $\vecs u⋅(\vecs v×\vecs w)$ vector product the cross product of two vectors.

cross-section | the intersection of a plane and a solid object

cubic function | a polynomial of degree 3; that is, a function of the form $f(x)=ax^3+bx^2+cx+d$, where $a≠0$

curl | the curl of vector field $\vecs{F}=⟨P,Q,R⟩$, denoted $\vecs ∇× \vecs{F}$ is the “determinant” of the matrix $$\begin{vmatrix} \mathbf{\hat i} & \mathbf{\hat j} & \mathbf{\hat k} \ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \ P & Q & R \end{vmatrix}. \nonumber$$ and is given by the expression $(R_y−Q_z)\,\mathbf{\hat i} +(P_z−R_x)\,\mathbf{\hat j} +(Q_x−P_y)\,\mathbf{\hat k}$; it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point

curvature | the derivative of the unit tangent vector with respect to the arc-length parameter

cusp | a pointed end or part where two curves meet

cycloid | the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage

cylinder | a set of lines parallel to a given line passing through a given curve

cylindrical coordinate system | a way to describe a location in space with an ordered triple $(r,θ,z),$ where $(r,θ)$ represents the polar coordinates of the point’s projection in the $xy$-plane, and z represents the point’s projection onto the $z$-axis

decreasing on the interval $I$ | a function decreasing on the interval $I$ if, for all $x_1,\,x_2∈I,\;f(x_1)≥f(x_2)$ if $x_1<x_2$

definite integral | a primary operation of calculus; the area between the curve and the $x$-axis over a given interval is a definite integral

definite integral of a vector-valued function | the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function

degree | for a polynomial function, the value of the largest exponent of any term

density function | a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume

dependent variable | the output variable for a function

derivative | the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

derivative function | gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

derivative of a vector-valued function | the derivative of a vector-valued function $\vecs{r}(t)$ is $\vecs{r}′(t) = \lim \limits_{\Delta t \to 0} \frac{\vecs r(t+\Delta t)−\vecs r(t)}{ \Delta t}$, provided the limit exists

difference law for limits | the limit law $$\lim_{x→a}(f(x)−g(x))=\lim_{x→a}f(x)−\lim_{x→a}g(x)=L−M \nonumber$$

difference quotient | of a function $f(x)$ at $a$ is given by $\dfrac{f(a+h)−f(a)}{h}$ or $\dfrac{f(x)−f(a)}{x−a}$

difference rule | the derivative of the difference of a function $f$ and a function $g$ is the same as the difference of the derivative of $f$ and the derivative of $g$: $\dfrac{d}{dx}\big(f(x)−g(x)\big)=f′(x)−g′(x)$

differentiable | a function $f(x,y)$ is differentiable at $(x_0,y_0)$ if $f(x,y)$ can be expressed in the form $f(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x−x_0)+f_y(x_0,y_0)(y−y_0)+E(x,y),$ where the error term $E(x,y)$ satisfies $\lim_{(x,y)→(x_0,y_0)}\dfrac{E(x,y)}{\sqrt{(x−x_0)^2+(y−y_0)^2}}=0$

differentiable at $a$ | a function for which $f'(a)$ exists is differentiable at $a$

differentiable function | a function for which $f'(x)$ exists is a differentiable function

differentiable on $S$ | a function for which $f'(x)$ exists for each $x$ in the open set $S$ is differentiable on $S$

differential | the differential $dx$ is an independent variable that can be assigned any nonzero real number; the differential $dy$ is defined to be $dy=f'(x)\,dx$

differential calculus | the field of calculus concerned with the study of derivatives and their applications

differential equation | an equation involving a function $y=y(x)$ and one or more of its derivatives

differential form | given a differentiable function $y=f'(x),$ the equation $dy=f'(x)\,dx$ is the differential form of the derivative of $y$ with respect to $x$

differentiation | the process of taking a derivative

direction angles | the angles formed by a nonzero vector and the coordinate axes

direction cosines | the cosines of the angles formed by a nonzero vector and the coordinate axes

direction field (slope field) | a mathematical object used to graphically represent solutions to a first-order differential equation; at each point in a direction field, a line segment appears whose slope is equal to the slope of a solution to the differential equation passing through that point

direction vector | a vector parallel to a line that is used to describe the direction, or orientation, of the line in space

directional derivative | the derivative of a function in the direction of a given unit vector

directrix | a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two

discontinuity at a point | A function is discontinuous at a point or has a discontinuity at a point if it is not continuous at the point

discriminant | the value $4AC−B^2$, which is used to identify a conic when the equation contains a term involving $xy$, is called a discriminant

discriminant | the discriminant of the function $f(x,y)$ is given by the formula $D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)−(f_{xy}(x_0,y_0))^2$

disk method | a special case of the slicing method used with solids of revolution when the slices are disks

divergence | the divergence of a vector field $\vecs{F}=⟨P,Q,R⟩$, denoted $\vecs ∇× \vecs{F}$, is $P_x+Q_y+R_z$; it measures the “outflowing-ness” of a vector field

divergence of a series | a series diverges if the sequence of partial sums for that series diverges

divergence test | if $\displaystyle \lim_{n→∞}a_n≠0,$ then the series $\displaystyle \sum^∞_{n=1}a_n$ diverges

divergent sequence | a sequence that is not convergent is divergent

domain | the set of inputs for a function

dot product or scalar product | $\vecs{ u}⋅\vecs{ v}=u_1v_1+u_2v_2+u_3v_3$ where $\vecs{ u}=⟨u_1,u_2,u_3⟩$ and $\vecs{ v}=⟨v_1,v_2,v_3⟩$

double integral | of the function $f(x,y)$ over the region $R$ in the $xy$-plane is defined as the limit of a double Riemann sum, $$\iint_R f(x,y) \,dA = \lim_{m,n\rightarrow \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \,\Delta A. \nonumber$$

double Riemann sum | of the function $f(x,y)$ over a rectangular region $R$ is $$\sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \,\Delta A, \nonumber$$ where $R$ is divided into smaller subrectangles $R_{ij}$ and $(x_{ij}^*, y_{ij}^*)$ is an arbitrary point in $R_{ij}$

doubling time | if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by $(\ln 2)/k$

eccentricity | the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix

ellipsoid | a three-dimensional surface described by an equation of the form $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$; all traces of this surface are ellipses

elliptic cone | a three-dimensional surface described by an equation of the form $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=0$; traces of this surface include ellipses and intersecting lines

elliptic paraboloid | a three-dimensional surface described by an equation of the form $z=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}$; traces of this surface include ellipses and parabolas

end behavior | the behavior of a function as $x→∞$ and $x→−∞$

epsilon-delta definition of the limit | $\displaystyle \lim_{x→a}f(x)=L$ if for every $ε>0$, there exists a $δ>0$ such that if $0<|x−a|<δ$, then $|f(x)−L|<ε$

equilibrium solution | any solution to the differential equation of the form $y=c,$ where $c$ is a constant

equivalent vectors | vectors that have the same magnitude and the same direction

Euler’s Method | a numerical technique used to approximate solutions to an initial-value problem

even function | a function is even if $f(−x)=f(x)$ for all $x$ in the domain of $f$

explicit formula | a sequence may be defined by an explicit formula such that $\displaystyle a_n=f(n)$

exponent | the value $x$ in the expression $b^x$

exponential decay | systems that exhibit exponential decay follow a model of the form $y=y_0e^{−kt}$

exponential growth | systems that exhibit exponential growth follow a model of the form $y=y_0e^{kt}$

extreme value theorem | if $f$ is a continuous function over a finite, closed interval, then $f$ has an absolute maximum and an absolute minimum

Fermat’s theorem | if $f$ has a local extremum at $c$, then $c$ is a critical point of $f$

first derivative test | let $f$ be a continuous function over an interval $I$ containing a critical point $c$ such that $f$ is differentiable over $I$ except possibly at $c$; if $f'$ changes sign from positive to negative as $x$ increases through $c$, then $f$ has a local maximum at $c$; if $f'$ changes sign from negative to positive as $x$ increases through $c$, then $f$ has a local minimum at $c$; if $f'$ does not change sign as $x$ increases through $c$, then $f$ does not have a local extremum at $c$

flux | the rate of a fluid flowing across a curve in a vector field; the flux of vector field $\vecs F$ across plane curve $C$ is line integral $∫_C \vecs F·\frac{\vecs n(t)}{‖\vecs n(t)‖} \,ds$

flux integral | another name for a surface integral of a vector field; the preferred term in physics and engineering

focal parameter | the focal parameter is the distance from a focus of a conic section to the nearest directrix

focus | a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two

formal definition of an infinite limit | $\displaystyle \lim_{x→a}f(x)=\infty$ if for every $M>0$, there exists a $δ>0$ such that if $0<|x−a|<δ$, then $f(x)>M$ $\displaystyle \lim_{x→a}f(x)=-\infty$ if for every $M>0$, there exists a $δ>0$ such that if $0<|x−a|<δ$, then $f(x)<-M$

Frenet frame of reference | (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector

frustum | a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base

Fubini’s theorem | if $f(x,y)$ is a function of two variables that is continuous over a rectangular region $R = \big\{(x,y) \in \mathbb{R}^2 \,|\,a \leq x \leq b, \, c \leq y \leq d\big\}$, then the double integral of $f$ over the region equals an iterated integral, $$\displaystyle\iint_R f(x,y) \, dA = \int_a^b \int_c^d f(x,y) \,dx \, dy = \int_c^d \int_a^b f(x,y) \,dx \, dy \nonumber$$

function | a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

function of two variables | a function $z=f(x,y)$ that maps each ordered pair $(x,y)$ in a subset $D$ of $R^2$ to a unique real number $z$

Fundamental Theorem for Line Integrals | the value of line integral $\displaystyle \int_C\vecs ∇f⋅d\vecs r$ depends only on the value of $f$ at the endpoints of $C: \displaystyle \int_C \vecs ∇f⋅d\vecs r=f(\vecs r(b))−f(\vecs r(a))$

fundamental theorem of calculus | (also, evaluation theorem) we can evaluate a definite integral by evaluating the antiderivative of the integrand at the endpoints of the interval and subtracting

fundamental theorem of calculus | uses a definite integral to define an antiderivative of a function

fundamental theorem of calculus | the theorem, central to the entire development of calculus, that establishes the relationship between differentiation and integration

general form | an equation of a conic section written as a general second-degree equation

general form of the equation of a plane | an equation in the form $ax+by+cz+d=0,$ where $\vecs n=⟨a,b,c⟩$ is a normal vector of the plane, $P=(x_0,y_0,z_0)$ is a point on the plane, and $d=−ax_0−by_0−cz_0$

general solution (or family of solutions) | the entire set of solutions to a given differential equation

generalized chain rule | the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables

geometric sequence | a sequence $\displaystyle {a_n}$ in which the ratio $\displaystyle a_{n+1}/a_n$ is the same for all positive integers $\displaystyle n$ is called a geometric sequence

geometric series | a geometric series is a series that can be written in the form $\displaystyle \sum_{n=1}^∞ar^{n−1}=a+ar+ar^2+ar^3+⋯$

gradient | the gradient of the function $f(x,y)$ is defined to be $\vecs ∇f(x,y)=(∂f/∂x)\,\hat{\mathbf i}+(∂f/∂y)\,\hat{\mathbf j},$ which can be generalized to a function of any number of independent variables

gradient field | a vector field $\vecs{F}$ for which there exists a scalar function $f$ such that $\vecs ∇f=\vecs{F}$; in other words, a vector field that is the gradient of a function; such vector fields are also called conservative

graph of a function | the set of points $(x,y)$ such that $x$ is in the domain of $f$ and $y=f(x)$

graph of a function of two variables | a set of ordered triples $(x,y,z)$ that satisfies the equation $z=f(x,y)$ plotted in three-dimensional Cartesian space

Green’s theorem | relates the integral over a connected region to an integral over the boundary of the region

grid curves | curves on a surface that are parallel to grid lines in a coordinate plane

growth rate | the constant $r>0$ in the exponential growth function $P(t)=P_0e^{rt}$

half-life | if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by $(\ln 2)/k$

harmonic series | the harmonic series takes the form $\displaystyle \sum_{n=1}^∞\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+⋯$

heat flow | a vector field proportional to the negative temperature gradient in an object

helix | a three-dimensional curve in the shape of a spiral

higher-order derivative | a derivative of a derivative, from the second derivative to the $n^{\text{th}}$ derivative, is called a higher-order derivative

higher-order partial derivatives | second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives

homogeneous linear equation | a second-order differential equation that can be written in the form $a_2(x)y″+a_1(x)y′+a_0(x)y=r(x)$, but $r(x)=0$ for every value of $x$

Hooke’s law | this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, $F=kx$, where $k$ is a constant

horizontal asymptote | if $\displaystyle \lim_{x→∞}f(x)=L$ or $\displaystyle \lim_{x→−∞}f(x)=L$, then $y=L$ is a horizontal asymptote of $f$

horizontal line test | a function $f$ is one-to-one if and only if every horizontal line intersects the graph of $f$, at most, once

hydrostatic pressure | the pressure exerted by water on a submerged object

hyperbolic functions | the functions denoted $\sinh,\,\cosh,\,\operatorname{tanh},\,\operatorname{csch},\,\operatorname{sech},$ and $\coth$, which involve certain combinations of $e^x$ and $e^{−x}$

hyperboloid of one sheet | a three-dimensional surface described by an equation of the form $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=1;$ traces of this surface include ellipses and hyperbolas

hyperboloid of two sheets | a three-dimensional surface described by an equation of the form $\dfrac{z^2}{c^2}−\dfrac{x^2}{a^2}−\dfrac{y^2}{b^2}=1$; traces of this surface include ellipses and hyperbolas

implicit differentiation | is a technique for computing $\dfrac{dy}{dx}$ for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable $y$ as a function) and solving for $\dfrac{dy}{dx}$

improper double integral | a double integral over an unbounded region or of an unbounded function

improper integral | an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges

increasing on the interval $I$ | a function increasing on the interval $I$ if for all $x_1,\,x_2∈I,\;f(x_1)≤f(x_2)$ if $x_1<x_2$

indefinite integral | the most general antiderivative of $f(x)$ is the indefinite integral of $f$; we use the notation $\displaystyle \int f(x)\,dx$ to denote the indefinite integral of $f$

indefinite integral of a vector-valued function | a vector-valued function with a derivative that is equal to a given vector-valued function

independence of path | a vector field $\vecs{F}$ has path independence if $\displaystyle \int_{C_1} \vecs F⋅d\vecs r=\displaystyle \int_{C_2} \vecs F⋅d\vecs r$ for any curves $C_1$ and $C_2$ in the domain of $\vecs{F}$ with the same initial points and terminal points

independent variable | the input variable for a function

indeterminate forms | When evaluating a limit, the forms $\dfrac{0}{0}$,$∞/∞, 0⋅∞, ∞−∞, 0^0, ∞^0$, and $1^∞$ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is.

index variable | the subscript used to define the terms in a sequence is called the index

infinite discontinuity | An infinite discontinuity occurs at a point $a$ if $\displaystyle \lim_{x→a^−}f(x)=±∞$ or $\displaystyle \lim_{x→a^+}f(x)=±∞$

infinite limit | A function has an infinite limit at a point $a$ if it either increases or decreases without bound as it approaches $a$

infinite limit at infinity | a function that becomes arbitrarily large as $x$ becomes large

infinite series | an infinite series is an expression of the form $\displaystyle a_1+a_2+a_3+⋯=\sum_{n=1}^∞a_n$

inflection point | if $f$ is continuous at $c$ and $f$ changes concavity at $c$, the point $(c,f(c))$ is an inflection point of $f$

initial point | the starting point of a vector

initial population | the population at time $t=0$

initial value problem | a problem that requires finding a function $y$ that satisfies the differential equation $\dfrac{dy}{dx}=f(x)$ together with the initial condition $y(x_0)=y_0$

initial value(s) | a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable

initial velocity | the velocity at time $t=0$

initial-value problem | a differential equation together with an initial value or values

instantaneous rate of change | the rate of change of a function at any point along the function $a$, also called $f′(a)$, or the derivative of the function at $a$

instantaneous velocity | The instantaneous velocity of an object with a position function that is given by $s(t)$ is the value that the average velocities on intervals of the form [$t,a$] and [$a,t$] approach as the values of $t$ move closer to $a$, provided such a value exists

integrable function | a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as $n$ goes to infinity exists

integral calculus | the study of integrals and their applications

integral test | for a series $\displaystyle \sum^∞_{n=1}a_n$ with positive terms $a_n$, if there exists a continuous, decreasing function $f$ such that $f(n)=a_n$ for all positive integers $n$, then $$\sum_{n=1}^∞a_n \nonumber$$ and $$∫^∞_1f(x)\,dx \nonumber$$ either both converge or both diverge

integrand | the function to the right of the integration symbol; the integrand includes the function being integrated

integrating factor | any function $f(x)$ that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions

integration by parts | a technique of integration that allows the exchange of one integral for another using the formula $\displaystyle ∫​u\,dv=uv−∫​v\,du$

integration by substitution | a technique for integration that allows integration of functions that are the result of a chain-rule derivative

integration table | a table that lists integration formulas

interior point | a point $P_0$ of $\mathbb{R}$ is a boundary point if there is a $δ$ disk centered around $P_0$ contained completely in $\mathbb{R}$

Intermediate Value Theorem | Let $f$ be continuous over a closed bounded interval [$a,b$] if $z$ is any real number between $f(a)$ and $f(b)$, then there is a number c in [$a,b$] satisfying $f(c)=z$

intermediate variable | given a composition of functions (e.g., $\displaystyle f(x(t),y(t)))$, the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function $\displaystyle f(x(t),y(t)),$ the variables $\displaystyle x$ and $\displaystyle y$ are examples of intermediate variables

interval of convergence | the set of real numbers $x$ for which a power series converges

intuitive definition of the limit | If all values of the function $f(x)$ approach the real number $L$ as the values of $x(≠a)$ approach a, $f(x)$ approaches L

inverse function | for a function $f$, the inverse function $f^{−1}$ satisfies $f^{−1}(y)=x$ if $f(x)=y$

inverse hyperbolic functions | the inverses of the hyperbolic functions where $\cosh$ and $\operatorname{sech}$ are restricted to the domain $[0,∞)$;each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function

inverse trigonometric functions | the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions

iterated integral | for a function $f(x,y)$ over the region $R$ is a. $\displaystyle \int_a^b \int_c^d f(x,y) \,dx \, dy = \int_a^b \left[\int_c^d f(x,y) \, dy\right] \, dx,$ b. $\displaystyle \int_c^d \int_a^b f(x,y) \, dx \, dy = \int_c^d \left[\int_a^b f(x,y) \, dx\right] \, dy,$ where $a,b,c$, and $d$ are any real numbers and $R = [a,b] \times [c,d]$

iterative process | process in which a list of numbers $x_0,x_1,x_2,x_3…$ is generated by starting with a number $x_0$ and defining $x_n=F(x_{n−1})$ for $n≥1$

Jacobian | the Jacobian $J (u,v)$ in two variables is a $2 \times 2$ determinant: $$J(u,v) = \begin{vmatrix} \frac{\partial x}{\partial u} \frac{\partial y}{\partial u} \nonumber \ \frac{\partial x}{\partial v} \frac{\partial y}{\partial v} \end{vmatrix}; \nonumber$$ the Jacobian $J (u,v,w)$ in three variables is a $3 \times 3$ determinant: $$J(u,v,w) = \begin{vmatrix} \frac{\partial x}{\partial u} \frac{\partial y}{\partial u} \frac{\partial z}{\partial u} \nonumber \ \frac{\partial x}{\partial v} \frac{\partial y}{\partial v} \frac{\partial z}{\partial v} \nonumber \ \frac{\partial x}{\partial w} \frac{\partial y}{\partial w} \frac{\partial z}{\partial w}\end{vmatrix} \nonumber$$

jump discontinuity | A jump discontinuity occurs at a point $a$ if $\displaystyle \lim_{x→a^−}f(x)$ and $\displaystyle \lim_{x→a^+}f(x)$ both exist, but $\displaystyle \lim_{x→a^−}f(x)≠\lim_{x→a^+}f(x)$

Kepler’s laws of planetary motion | three laws governing the motion of planets, asteroids, and comets in orbit around the Sun

Lagrange multiplier | the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable $λ$

lamina | a thin sheet of material; laminas are thin enough that, for mathematical purposes, they can be treated as if they are two-dimensional

left-endpoint approximation | an approximation of the area under a curve computed by using the left endpoint of each subinterval to calculate the height of the vertical sides of each rectangle

level curve of a function of two variables | the set of points satisfying the equation $f(x,y)=c$ for some real number $c$ in the range of $f$

level surface of a function of three variables | the set of points satisfying the equation $f(x,y,z)=c$ for some real number $c$ in the range of $f$

limaçon | the graph of the equation $r=a+b\sin θ$ or $r=a+b\cos θ.$ If $a=b$ then the graph is a cardioid

limit | the process of letting x or t approach a in an expression; the limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ approaches as $x$ approaches $a$

limit at infinity | a function that approaches a limit value $L$ as $x$ becomes large

limit comparison test | Suppose $a_n,b_n≥0$ for all $n≥1$. If $\displaystyle \lim_{n→∞}a_n/b_n→L≠0$, then $\displaystyle \sum^∞_{n=1}a_n$ and $\displaystyle \sum^∞_{n=1}b_n$ both converge or both diverge; if $\displaystyle \lim_{n→∞}a_n/b_n→0$ and $\displaystyle \sum^∞_{n=1}b_n$ converges, then $\displaystyle \sum^∞_{n=1}a_n$ converges. If $\displaystyle \lim_{n→∞}a_n/b_n→∞$, and $\displaystyle \sum^∞_{n=1}b_n$ diverges, then $\displaystyle \sum^∞_{n=1}a_n$ diverges.

limit laws | the individual properties of limits; for each of the individual laws, let $f(x)$ and $g(x)$ be defined for all $x≠a$ over some open interval containing a; assume that L and M are real numbers so that $\lim_{x→a}f(x)=L$ and $\lim_{x→a}g(x)=M$; let c be a constant

limit of a sequence | the real number LL to which a sequence converges is called the limit of the sequence

limit of a vector-valued function | a vector-valued function $\vecs r(t)$ has a limit $\vecs L$ as $t$ approaches $a$ if $\lim \limits{t \to a} \left| \vecs r(t) - \vecs L \right| = 0$

limits of integration | these values appear near the top and bottom of the integral sign and define the interval over which the function should be integrated

line integral | the integral of a function along a curve in a plane or in space

linear | description of a first-order differential equation that can be written in the form $a(x)y′+b(x)y=c(x)$

linear approximation | the linear function $L(x)=f(a)+f'(a)(x−a)$ is the linear approximation of $f$ at $x=a$

linear approximation | given a function $f(x,y)$ and a tangent plane to the function at a point $(x_0,y_0)$, we can approximate $f(x,y)$ for points near $(x_0,y_0)$ using the tangent plane formula

linear function | a function that can be written in the form $f(x)=mx+b$

linearly dependent | a set of functions $f_1(x),f_2(x),…,f_n(x)$ for whichthere are constants $c_1,c_2,…c_n$, not all zero, such that $c_1f_1(x)+c_2f_2(x)+⋯+c_nf_n(x)=0$ for all \(x\) in the interval of interest

linearly independent | a set of functions $f_1(x),f_2(x),…,f_n(x)$ for which there are no constants $c_1,c_2,…c_n$, such that $c_1f_1(x)+c_2f_2(x)+⋯+c_nf_n(x)=0$ for all \(x\) in the interval of interest

local extremum | if $f$ has a local maximum or local minimum at $c$, we say $f$ has a local extremum at $c$

local maximum | if there exists an interval $I$ such that $f(c)≥f(x)$ for all $x∈I$, we say $f$ has a local maximum at $c$

local minimum | if there exists an interval $I$ such that $f(c)≤f(x)$ for all $x∈I$, we say $f$ has a local minimum at $c$

logarithmic differentiation | is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly

logarithmic function | a function of the form $f(x)=\log_b(x)$ for some base $b>0,\,b≠1$ such that $y=\log_b(x)$ if and only if $b^y=x$

logistic differential equation | a differential equation that incorporates the carrying capacity $K$ and growth rate rr into a population model

lower sum | a sum obtained by using the minimum value of $f(x)$ on each subinterval

L’Hôpital’s rule | If $f$ and $g$ are differentiable functions over an interval $a$, except possibly at $a$, and $\displaystyle \lim_{x→a}f(x)=0=\lim_{x→a}g(x)$ or $\displaystyle \lim_{x→a}f(x)$ and $\displaystyle \lim_{x→a}g(x)$ are infinite, then $\displaystyle \lim_{x→a}\dfrac{f(x)}{g(x)}=\lim_{x→a}\dfrac{f′(x)}{g′(x)}$, assuming the limit on the right exists or is $∞$ or $−∞$.

Maclaurin polynomial | a Taylor polynomial centered at $0$; the $n^{\text{th}}$-degree Taylor polynomial for $f$ at $0$ is the $n^{\text{th}}$-degree Maclaurin polynomial for $f$

Maclaurin series | a Taylor series for a function $f$ at $x=0$ is known as a Maclaurin series for $f$

magnitude | the length of a vector

major axis | the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis

marginal cost | is the derivative of the cost function, or the approximate cost of producing one more item

marginal profit | is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item

marginal revenue | is the derivative of the revenue function, or the approximate revenue obtained by selling one more item

mass flux | the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area

mathematical model | A method of simulating real-life situations with mathematical equations

mean value theorem | if $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$, then there exists $c∈(a,b)$ such that $f′(c)=\frac{f(b)−f(a)}{b−a}$

mean value theorem for integrals | guarantees that a point $c$ exists such that $f(c)$ is equal to the average value of the function

method of cylindrical shells | a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable

method of Lagrange multipliers | a method of solving an optimization problem subject to one or more constraints

method of undetermined coefficients | a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess

method of variation of parameters | a method that involves looking for particular solutions in the form $y_p(x)=u(x)y_1(x)+v(x)y_2(x)$, where $y_1$ and $y_2$ are linearly independent solutions to the complementary equations, and then solving a system of equations to find $u(x)$ and $v(x)$

midpoint rule | a rule that uses a Riemann sum of the form $\displaystyle M_n=\sum^n_{i=1}f(m_i)Δx$, where $m_i$ is the midpoint of the $i^{\text{th}}$ subinterval to approximate $\displaystyle ∫^b_af(x)\,dx$

minor axis | the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis

mixed partial derivatives | second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables

moment | if n masses are arranged on a number line, the moment of the system with respect to the origin is given by $\displaystyle M=\sum^n_{i=1}m_ix_i$; if, instead, we consider a region in the plane, bounded above by a function $f(x)$ over an interval $[a,b]$, then the moments of the region with respect to the $x$- and $y$-axes are given by $\displaystyle M_x=ρ∫^b_a\dfrac{[f(x)]^2}{2}\,dx$ and $\displaystyle M_y=ρ∫^b_axf(x)\,dx$, respectively

monotone sequence | an increasing or decreasing sequence

multivariable calculus | the study of the calculus of functions of two or more variables

nappe | a nappe is one half of a double cone

natural exponential function | the function $f(x)=e^x$

natural logarithm | the function $\ln x=\log_ex$

net change theorem | if we know the rate of change of a quantity, the net change theorem says the future quantity is equal to the initial quantity plus the integral of the rate of change of the quantity

net signed area | the area between a function and the $x$-axis such that the area below the $x$-axis is subtracted from the area above the $x$-axis; the result is the same as the definite integral of the function

Newton’s method | method for approximating roots of $f(x)=0;$ using an initial guess $x_0$; each subsequent approximation is defined by the equation $x_n=x_{n−1}−\frac{f(x_{n−1})}{f'(x_{n−1})}$

nonelementary integral | an integral for which the antiderivative of the integrand cannot be expressed as an elementary function

nonhomogeneous linear equation | a second-order differential equation that can be written in the form $a_2(x)y″+a_1(x)y′+a_0(x)y=r(x)$, but $r(x)≠0$ for some value of $x$

normal component of acceleration | the coefficient of the unit normal vector $\vecs N$ when the acceleration vector is written as a linear combination of $\vecs T$ and $\vecs N$

normal plane | a plane that is perpendicular to a curve at any point on the curve

normal vector | a vector perpendicular to a plane

normalization | using scalar multiplication to find a unit vector with a given direction

number e | as $m$ gets larger, the quantity $(1+(1/m)^m$ gets closer to some real number; we define that real number to be $e;$ the value of $e$ is approximately $2.718282$

numerical integration | the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule

objective function | the function that is to be maximized or minimized in an optimization problem

oblique asymptote | the line $y=mx+b$ if $f(x)$ approaches it as $x→∞$ or$x→−∞$

octants | the eight regions of space created by the coordinate planes

odd function | a function is odd if $f(−x)=−f(x)$ for all $x$ in the domain of $f$

one-sided limit | A one-sided limit of a function is a limit taken from either the left or the right

one-to-one function | a function $f$ is one-to-one if $f(x_1)≠f(x_2)$ if $x_1≠x_2$

one-to-one transformation | a transformation $T : G \rightarrow R$ defined as $T(u,v) = (x,y)$ is said to be one-to-one if no two points map to the same image point

open set | a set $S$ that contains none of its boundary points

optimization problem | calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers

optimization problems | problems that are solved by finding the maximum or minimum value of a function

order of a differential equation | the highest order of any derivative of the unknown function that appears in the equation

orientation | the direction that a point moves on a graph as the parameter increases

orientation of a curve | the orientation of a curve $C$ is a specified direction of $C$

orientation of a surface | if a surface has an “inner” side and an “outer” side, then an orientation is a choice of the inner or the outer side; the surface could also have “upward” and “downward” orientations

orthogonal vectors | vectors that form a right angle when placed in standard position

osculating circle | a circle that is tangent to a curve $C$ at a point $P$ and that shares the same curvature

osculating plane | the plane determined by the unit tangent and the unit normal vector

p-series | a series of the form $\displaystyle \sum^∞_{n=1}1/n^p$

parallelogram method | a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram

parameter | an independent variable that both $x$ and $y$ depend on in a parametric curve; usually represented by the variable $t$

parameter domain (parameter space) | the region of the $uv$-plane over which the parameters $u$ and $v$ vary for parameterization $\vecs r(u,v) = \langle x(u,v), \, y(u,v), \, z(u,v)\rangle$

parameterization of a curve | rewriting the equation of a curve defined by a function $y=f(x)$ as parametric equations

parameterized surface (parametric surface) | a surface given by a description of the form $\vecs r(u,v) = \langle x(u,v), \, y(u,v), \, z(u,v)\rangle$, where the parameters $u$ and $v$ vary over a parameter domain in the $uv$-plane

parametric curve | the graph of the parametric equations $x(t)$ and $y(t)$ over an interval $a≤t≤b$ combined with the equations

parametric equations | the equations $x=x(t)$ and $y=y(t)$ that define a parametric curve

parametric equations of a line | the set of equations $x=x_0+ta, y=y_0+tb,$ and $z=z_0+tc$ describing the line with direction vector $v=⟨a,b,c⟩$ passing through point $(x_0,y_0,z_0)$

partial derivative | a derivative of a function of more than one independent variable in which all the variables but one are held constant

partial differential equation | an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives

partial fraction decomposition | a technique used to break down a rational function into the sum of simple rational functions

partial sum | the $kth$ partial sum of the infinite series $\displaystyle \sum^∞_{n=1}a_n$ is the finite sum $\displaystyle S_k=\sum_{n=1}^ka_n=a_1+a_2+a_3+⋯+a_k$

particular solution | member of a family of solutions to a differential equation that satisfies a particular initial condition

particular solution | a solution $y_p(x)$ of a differential equation that contains no arbitrary constants

partition | a set of points that divides an interval into subintervals

percentage error | the relative error expressed as a percentage

periodic function | a function is periodic if it has a repeating pattern as the values of $x$ move from left to right

phase line | a visual representation of the behavior of solutions to an autonomous differential equation subject to various initial conditions

piecewise smooth curve | an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves

piecewise-defined function | a function that is defined differently on different parts of its domain

planar transformation | a function $T$ that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables

plane curve | the set of ordered pairs $(f(t),g(t))$ together with their defining parametric equations $x=f(t)$ and $y=g(t)$

point-slope equation | equation of a linear function indicating its slope and a point on the graph of the function

polar axis | the horizontal axis in the polar coordinate system corresponding to $r≥0$

polar coordinate system | a system for locating points in the plane. The coordinates are $r$, the radial coordinate, and $θ$, the angular coordinate

polar equation | an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system

polar rectangle | the region enclosed between the circles $r = a$ and $r = b$ and the angles $\theta = \alpha$ and $\theta = \beta$; it is described as $R = \{(r, \theta)\,|\,a \leq r \leq b, \, \alpha \leq \theta \leq \beta\}$

pole | the central point of the polar coordinate system, equivalent to the origin of a Cartesian system

polynomial function | a function of the form $f(x)=a_nx^n+a_{n−1}x^{n−1}+…+a_1x+a_0$

population growth rate | is the derivative of the population with respect to time

potential function | a scalar function $f$ such that $\vecs ∇f=\vecs{F}$

power function | a function of the form $f(x)=x^n$ for any positive integer $n≥1$

power law for limits | the limit law $$\lim_{x→a}(f(x))^n=(\lim_{x→a}f(x))^n=L^n \nonumber$$ for every positive integer n

power reduction formula | a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power

power rule | the derivative of a power function is a function in which the power on $x$ becomes the coefficient of the term and the power on $x$ in the derivative decreases by 1: If $n$ is an integer, then $\dfrac{d}{dx}\left(x^n\right)=nx^{n−1}$

power series | a series of the form $\sum_{n=0}^∞c_nx^n$ is a power series centered at $x=0$; a series of the form $\sum_{n=0}^∞c_n(x−a)^n$ is a power series centered at $x=a$

principal unit normal vector | a vector orthogonal to the unit tangent vector, given by the formula $\frac{\vecs T′(t)}{‖\vecs T′(t)‖}$

principal unit tangent vector | a unit vector tangent to a curve C

product law for limits | the limit law $$\lim_{x→a}(f(x)⋅g(x))=\lim_{x→a}f(x)⋅\lim_{x→a}g(x)=L⋅M \nonumber$$

product rule | the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: $\dfrac{d}{dx}\big(f(x)g(x)\big)=f′(x)g(x)+g′(x)f(x)$

projectile motion | motion of an object with an initial velocity but no force acting on it other than gravity

propagated error | the error that results in a calculated quantity $f(x)$ resulting from a measurement error $dx$

quadratic function | a polynomial of degree 2; that is, a function of the form $f(x)=ax^2+bx+c$ where $a≠0$

quadric surfaces | surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)

quotient law for limits | the limit law $\lim_{x→a}\dfrac{f(x)}{g(x)}=\dfrac{\lim_{x→a}f(x)}{\lim_{x→a}g(x)}=\dfrac{L}{M}$ for M≠0

quotient rule | the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: $\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f′(x)g(x)−g′(x)f(x)}{\big(g(x)\big)^2}$

radial coordinate | $r$ the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole

radial field | a vector field in which all vectors either point directly toward or directly away from the origin; the magnitude of any vector depends only on its distance from the origin

radians | for a circular arc of length $s$ on a circle of radius 1, the radian measure of the associated angle $θ$ is $s$

radius of convergence | if there exists a real number $R>0$ such that a power series centered at $x=a$ converges for $|x−a|<R$ and diverges for $|x−a|>R$, then $R$ is the radius of convergence; if the power series only converges at $x=a$, the radius of convergence is $R=0$; if the power series converges for all real numbers $x$, the radius of convergence is $R=∞$

radius of curvature | the reciprocal of the curvature

radius of gyration | the distance from an object’s center of mass to its axis of rotation

range | the set of outputs for a function

ratio test | for a series $\displaystyle \sum^∞_{n=1}a_n$ with nonzero terms, let $\displaystyle ρ=\lim_{n→∞}|a_{n+1}/a_n|$; if $0≤ρ<1$, the series converges absolutely; if $ρ>1$, the series diverges; if $ρ=1$, the test is inconclusive

rational function | a function of the form $f(x)=p(x)/q(x)$, where $p(x)$ and $q(x)$ are polynomials

recurrence relation | a recurrence relation is a relationship in which a term $a_n$ in a sequence is defined in terms of earlier terms in the sequence

region | an open, connected, nonempty subset of $\mathbb{R}^2$

regular parameterization | parameterization $\vecs r(u,v) = \langle x(u,v), \, y(u,v), \, z(u,v)\rangle$ such that $r_u \times r_v$ is not zero for point $(u,v)$ in the parameter domain

regular partition | a partition in which the subintervals all have the same width

related rates | are rates of change associated with two or more related quantities that are changing over time

relative error | given an absolute error $Δq$ for a particular quantity, $\frac{Δq}{q}$ is the relative error.

relative error | error as a percentage of the actual value, given by $$\text{relative error}=\left|\frac{A−B}{A}\right|⋅100\% \nonumber$$

remainder estimate | for a series $\displaystyle \sum^∞_{n=}1a_n$ with positive terms $a_n$ and a continuous, decreasing function $f$ such that $f(n)=a_n$ for all positive integers $n$, the remainder $\displaystyle R_N=\sum^∞_{n=1}a_n−\sum^N_{n=1}a_n$ satisfies the following estimate: $$∫^∞_{N+1}f(x)\,dx<R_N<∫^∞_Nf(x)\,dx \nonumber$$

removable discontinuity | A removable discontinuity occurs at a point $a$ if $f(x)$ is discontinuous at $a$, but $\displaystyle \lim_{x→a}f(x)$ exists

reparameterization | an alternative parameterization of a given vector-valued function

restricted domain | a subset of the domain of a function $f$

riemann sum | an estimate of the area under the curve of the form $A≈\displaystyle \sum_{i=1}^nf(x^∗_i)Δx$

right-endpoint approximation | the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of each rectangle

right-hand rule | a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the $z$-axis in such a way that the fingers curl from the positive $x$-axis to the positive $y$-axis, the thumb points in the direction of the positive $z$-axis

RLC series circuit | a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an RLC series circuit

rolle’s theorem | if $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$, and if $f(a)=f(b)$, then there exists $c∈(a,b)$ such that $f′(c)=0$

root function | a function of the form $f(x)=x^{1/n}$ for any integer $n≥2$

root law for limits | the limit law $\lim_{x→a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x→a}f(x)}=\sqrt[n]{L}$ for all L if n is odd and for $L≥0$ if n is even

root test | for a series $\displaystyle \sum^∞_{n=1}a_n,$ let $\displaystyle ρ=\lim_{n→∞}\sqrt[n]{|a_n|}$; if $0≤ρ<1$, the series converges absolutely; if $ρ>1$, the series diverges; if $ρ=1$, the test is inconclusive

rose | graph of the polar equation $r=a\cos 2θ$ or $r=a\sin 2θ$for a positive constant $a$

rotational field | a vector field in which the vector at point $(x,y)$ is tangent to a circle with radius $r=\sqrt{x^2+y^2}$; in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin

rulings | parallel lines that make up a cylindrical surface

saddle point | given the function $z=f(x,y),$ the point $(x_0,y_0,f(x_0,y_0))$ is a saddle point if both $f_x(x_0,y_0)=0$ and $f_y(x_0,y_0)=0$, but $f$ does not have a local extremum at $(x_0,y_0)$

scalar | a real number

scalar equation of a plane | the equation $a(x−x_0)+b(y−y_0)+c(z−z_0)=0$ used to describe a plane containing point $P=(x_0,y_0,z_0)$ with normal vector $n=⟨a,b,c⟩$ or its alternate form $ax+by+cz+d=0$, where $d=−ax_0−by_0−cz_0$

scalar line integral | the scalar line integral of a function $f$ along a curve $C$ with respect to arc length is the integral $\displaystyle \int_C f\,ds$, it is the integral of a scalar function $f$ along a curve in a plane or in space; such an integral is defined in terms of a Riemann sum, as is a single-variable integral

scalar multiplication | a vector operation that defines the product of a scalar and a vector

scalar projection | the magnitude of the vector projection of a vector

secant | A secant line to a function $f(x)$ at $a$ is a line through the point ($a,f(a)$) and another point on the function; the slope of the secant line is given by $m_{sec}=\dfrac{f(x)−f(a)}{x−a}$

second derivative test | suppose $f'(c)=0$ and $f'$' is continuous over an interval containing $c$; if $f''(c)>0$, then $f$ has a local minimum at $c$; if $f''(c)<0$, then $f$ has a local maximum at $c$; if $f''(c)=0$, then the test is inconclusive

separable differential equation | any equation that can be written in the form $y'=f(x)g(y)$

separation of variables | a method used to solve a separable differential equation

sequence | an ordered list of numbers of the form $\displaystyle a_1,a_2,a_3,…$ is a sequence

sigma notation | (also, summation notation) the Greek letter sigma ($Σ$) indicates addition of the values; the values of the index above and below the sigma indicate where to begin the summation and where to end it

simple curve | a curve that does not cross itself

simple harmonic motion | motion described by the equation $x(t)=c_1 \cos (ωt)+c_2 \sin (ωt)$, as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely

simply connected region | a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region

Simpson’s rule | a rule that approximates $\displaystyle ∫^b_af(x)\,dx$ using the area under a piecewise quadratic function. The approximation $S_n$ to $\displaystyle ∫^b_af(x)\,dx$ is given by $$S_n=\frac{Δx}{3}\big(f(x_0)+4\,f(x_1)+2\,f(x_2)+4\,f(x_3)+2\,f(x_4)+⋯+2\,f(x_{n−2})+4\,f(x_{n−1})+f(x_n)\big). \nonumber$$

skew lines | two lines that are not parallel but do not intersect

slicing method | a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume

slope | the change in $y$ for each unit change in $x$

slope-intercept form | equation of a linear function indicating its slope and $y$-intercept

smooth | curves where the vector-valued function $\vecs r(t)$ is differentiable with a non-zero derivative

solid of revolution | a solid generated by revolving a region in a plane around a line in that plane

solution curve | a curve graphed in a direction field that corresponds to the solution to the initial-value problem passing through a given point in the direction field

solution to a differential equation | a function $y=f(x)$ that satisfies a given differential equation

space curve | the set of ordered triples $(f(t),g(t),h(t))$ together with their defining parametric equations $x=f(t)$, $y=g(t)$ and $z=h(t)$

space-filling curve | a curve that completely occupies a two-dimensional subset of the real plane

speed | is the absolute value of velocity, that is, $|v(t)|$ is the speed of an object at time $t$ whose velocity is given by $v(t)$

sphere | the set of all points equidistant from a given point known as the center

spherical coordinate system | a way to describe a location in space with an ordered triple $(ρ,θ,φ),$ where $ρ$ is the distance between $P$ and the origin $(ρ≠0), θ$ is the same angle used to describe the location in cylindrical coordinates, and $φ$ is the angle formed by the positive $z$-axis and line segment $\bar{OP}$, where $O$ is the origin and $0≤φ≤π$

squeeze theorem | states that if $f(x)≤g(x)≤h(x)$ for all $x≠a$ over an open interval containing a and $\lim_{x→a}f(x)=L=\lim_ {x→a}h(x)$ where L is a real number, then $\lim_{x→a}g(x)=L$

standard equation of a sphere | $(x−a)^2+(y−b)^2+(z−c)^2=r^2$ describes a sphere with center $(a,b,c)$ and radius $r$

standard form | the form of a first-order linear differential equation obtained by writing the differential equation in the form $y'+p(x)y=q(x)$

standard form | an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes

standard unit vectors | unit vectors along the coordinate axes: $\hat{\mathbf i}=⟨1,0⟩,\, \hat{\mathbf j}=⟨0,1⟩$

standard-position vector | a vector with initial point $(0,0)$

steady-state solution | a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution

step size | the increment hh that is added to the xx value at each step in Euler’s Method

Stokes’ theorem | relates the flux integral over a surface $S$ to a line integral around the boundary $C$ of the surface $S$

stream function | if $\vecs F=⟨P,Q⟩$ is a source-free vector field, then stream function $g$ is a function such that $P=g_y$ and $Q=−g_x$

sum law for limits | The limit law $\lim_{x→a}(f(x)+g(x))=\lim_{x→a}f(x)+\lim_{x→a}g(x)=L+M$

sum rule | the derivative of the sum of a function $f$ and a function $g$ is the same as the sum of the derivative of $f$ and the derivative of $g$: $\dfrac{d}{dx}\big(f(x)+g(x)\big)=f′(x)+g′(x)$

surface | the graph of a function of two variables, $z=f(x,y)$

surface area | the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces

surface area | the area of surface $S$ given by the surface integral $$\iint_S \,dS \nonumber$$

surface independent | flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface

surface integral | an integral of a function over a surface

surface integral of a scalar-valued function | a surface integral in which the integrand is a scalar function

surface integral of a vector field | a surface integral in which the integrand is a vector field

symmetric equations of a line | the equations $\dfrac{x−x_0}{a}=\dfrac{y−y_0}{b}=\dfrac{z−z_0}{c}$ describing the line with direction vector $v=⟨a,b,c⟩$ passing through point $(x_0,y_0,z_0)$

symmetry about the origin | the graph of a function $f$ is symmetric about the origin if $(−x,−y)$ is on the graph of $f$ whenever $(x,y)$ is on the graph

symmetry about the $y$-axis | the graph of a function $f$ is symmetric about the $y$-axis if $(−x,y)$ is on the graph of $f$ whenever $(x,y)$ is on the graph

symmetry principle | the symmetry principle states that if a region $R$ is symmetric about a line $I$, then the centroid of $R$ lies on $I$

table of values | a table containing a list of inputs and their corresponding outputs

tangent | A tangent line to the graph of a function at a point ($a,f(a)$) is the line that secant lines through ($a,f(a)$) approach as they are taken through points on the function with $x$-values that approach $a$; the slope of the tangent line to a graph at $a$ measures the rate of change of the function at $a$

tangent line approximation (linearization) | since the linear approximation of $f$ at $x=a$ is defined using the equation of the tangent line, the linear approximation of $f$ at $x=a$ is also known as the tangent line approximation to $f$ at $x=a$

tangent plane | given a function $f(x,y)$ that is differentiable at a point $(x_0,y_0)$, the equation of the tangent plane to the surface $z=f(x,y)$ is given by $z=f(x_0,y_0)+f_x(x_0,y_0)(x−x_0)+f_y(x_0,y_0)(y−y_0)$

tangent vector | to $\vecs{r}(t)$ at $t=t_0$ any vector $\vecs v$ such that, when the tail of the vector is placed at point $\vecs r(t_0)$ on the graph, vector $\vecs{v}$ is tangent to curve C

tangential component of acceleration | the coefficient of the unit tangent vector $\vecs T$ when the acceleration vector is written as a linear combination of $\vecs T$ and $\vecs N$

Taylor polynomials | the $n^{\text{th}}$-degree Taylor polynomial for $f$ at $x=a$ is $p_n(x)=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n$

Taylor series | a power series at $a$ that converges to a function $f$ on some open interval containing $a$.

Taylor’s theorem with remainder | for a function $f$ and the $n^{\text{th}}$-degree Taylor polynomial for $f$ at $x=a$, the remainder $R_n(x)=f(x)−p_n(x)$ satisfies $R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1}$ for some$c$ between $x$ and $a$; if there exists an interval $I$ containing $a$ and a real number $M$ such that $∣f^{(n+1)}(x)∣≤M$ for all $x$ in $I$, then $|R_n(x)|≤\dfrac{M}{(n+1)!}|x−a|^{n+1}$

telescoping series | a telescoping series is one in which most of the terms cancel in each of the partial sums

term | the number $\displaystyle a_n$ in the sequence $\displaystyle {a_n}$ is called the $\displaystyle nth$ term of the sequence

term-by-term differentiation of a power series | a technique for evaluating the derivative of a power series $\displaystyle \sum_{n=0}^∞c_n(x−a)^n$ by evaluating the derivative of each term separately to create the new power series $\displaystyle \sum_{n=1}^∞nc_n(x−a)^{n−1}$

term-by-term integration of a power series | a technique for integrating a power series $\displaystyle \sum_{n=0}^∞c_n(x−a)^n$ by integrating each term separately to create the new power series $\displaystyle C+\sum_{n=0}^∞c_n\dfrac{(x−a)^{n+1}}{n+1}$

terminal point | the endpoint of a vector

theorem of Pappus for volume | this theorem states that the volume of a solid of revolution formed by revolving a region around an external axis is equal to the area of the region multiplied by the distance traveled by the centroid of the region

three-dimensional rectangular coordinate system | a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple $(x,y,z)$ that plots its location relative to the defining axes

threshold population | the minimum population that is necessary for a species to survive

total area | total area between a function and the $x$-axis is calculated by adding the area above the $x$-axis and the area below the $x$-axis; the result is the same as the definite integral of the absolute value of the function

total differential | the total differential of the function $f(x,y)$ at $(x_0,y_0)$ is given by the formula $dz=f_x(x_0,y_0)dx+fy(x_0,y_0)dy$

trace | the intersection of a three-dimensional surface with a coordinate plane

transcendental function | a function that cannot be expressed by a combination of basic arithmetic operations

transformation | a function that transforms a region GG in one plane into a region RR in another plane by a change of variables

transformation of a function | a shift, scaling, or reflection of a function

trapezoidal rule | a rule that approximates $\displaystyle ∫^b_af(x)\,dx$ using the area of trapezoids. The approximation $T_n$ to $\displaystyle ∫^b_af(x)\,dx$ is given by $$T_n=\frac{Δx}{2}\big(f(x_0)+2\, f(x_1)+2\, f(x_2)+⋯+2\, f(x_{n−1})+f(x_n)\big). \nonumber$$

tree diagram | illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for

triangle inequality | If $a$ and $b$ are any real numbers, then $|a+b|≤|a|+|b|$

triangle inequality | the length of any side of a triangle is less than the sum of the lengths of the other two sides

triangle method | a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector

trigonometric functions | functions of an angle defined as ratios of the lengths of the sides of a right triangle

trigonometric identity | an equation involving trigonometric functions that is true for all angles $θ$ for which the functions in the equation are defined

trigonometric integral | an integral involving powers and products of trigonometric functions

trigonometric substitution | an integration technique that converts an algebraic integral containing expressions of the form $\sqrt{a^2−x^2}$, $\sqrt{a^2+x^2}$, or $\sqrt{x^2−a^2}$ into a trigonometric integral

triple integral | the triple integral of a continuous function $f(x,y,z)$ over a rectangular solid box $B$ is the limit of a Riemann sum for a function of three variables, if this limit exists

triple integral in cylindrical coordinates | the limit of a triple Riemann sum, provided the following limit exists: $$lim_{l,m,n\rightarrow\infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(r_{ijk}^*, \theta_{ijk}^*, s_{ijk}^*) r_{ijk}^* \Delta r \Delta \theta \Delta z \nonumber$$

triple integral in spherical coordinates | the limit of a triple Riemann sum, provided the following limit exists: $$lim_{l,m,n\rightarrow\infty} \sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\rho_{ijk}^*, \theta_{ijk}^*, \varphi_{ijk}^*) (\rho_{ijk}^*)^2 \sin \, \varphi \Delta \rho \Delta \theta \Delta \varphi \nonumber$$

Type I | a region $D$ in the $xy$- plane is Type I if it lies between two vertical lines and the graphs of two continuous functions $g_1(x)$ and $g_2(x)$

Type II | a region $D$ in the $xy$-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions $h_1(y)$ and $h_2(h)$

unbounded sequence | a sequence that is not bounded is called unbounded

unit vector | a vector with magnitude $1$

unit vector field | a vector field in which the magnitude of every vector is 1

upper sum | a sum obtained by using the maximum value of $f(x)$ on each subinterval

variable of integration | indicates which variable you are integrating with respect to; if it is $x$, then the function in the integrand is followed by $dx$

vector | a mathematical object that has both magnitude and direction

vector addition | a vector operation that defines the sum of two vectors

vector difference | the vector difference $\vecs{v}−\vecs{w}$ is defined as $\vecs{v}+(−\vecs{w})=\vecs{v}+(−1)\vecs{w}$

vector equation of a line | the equation $\vecs r=\vecs r_0+t\vecs v$ used to describe a line with direction vector $\vecs v=⟨a,b,c⟩$ passing through point $P=(x_0,y_0,z_0)$, where $\vecs r_0=⟨x_0,y_0,z_0⟩$, is the position vector of point $P$

vector equation of a plane | the equation $\vecs n⋅\vecd{PQ}=0,$ where $P$ is a given point in the plane, $Q$ is any point in the plane, and $\vecs n$ is a normal vector of the plane

vector field | measured in $ℝ^2$, an assignment of a vector $\vecs{F}(x,y)$ to each point $(x,y)$ of a subset $D$ of $ℝ^2$; in $ℝ^3$, an assignment of a vector $\vecs{F}(x,y,z)$ to each point $(x,y,z)$ of a subset $D$ of $ℝ^3$

vector line integral | the vector line integral of vector field $\vecs F$ along curve $C$ is the integral of the dot product of $\vecs F$ with unit tangent vector $\vecs T$ of $C$ with respect to arc length, $∫_C \vecs F·\vecs T\, ds$; such an integral is defined in terms of a Riemann sum, similar to a single-variable integral

vector parameterization | any representation of a plane or space curve using a vector-valued function

vector projection | the component of a vector that follows a given direction

vector sum | the sum of two vectors, $\vecs{v}$ and $\vecs{w}$, can be constructed graphically by placing the initial point of $\vecs{w}$ at the terminal point of $\vecs{v}$; then the vector sum $\vecs{v}+\vecs{w}$ is the vector with an initial point that coincides with the initial point of $\vecs{v}$, and with a terminal point that coincides with the terminal point of $\vecs{w}$

vector-valued function | a function of the form $\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}$ or $\vecs r(t)=f(t)\hat{\mathbf{i}}+g(t)\hat{\mathbf{j}}+h(t)\hat{\mathbf{k}}$,where the component functions $f$, $g$, and $h$ are real-valued functions of the parameter $t$.

velocity vector | the derivative of the position vector

vertex | a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch

vertical asymptote | A function has a vertical asymptote at $x=a$ if the limit as $x$ approaches $a$ from the right or left is infinite

vertical line test | given the graph of a function, every vertical line intersects the graph, at most, once

vertical trace | the set of ordered triples $(c,y,z)$ that solves the equation $f(c,y)=z$ for a given constant $x=c$ or the set of ordered triples $(x,d,z)$ that solves the equation $f(x,d)=z$ for a given constant $y=d$

washer method | a special case of the slicing method used with solids of revolution when the slices are washers

work | the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance

work done by a force | work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector $\vecs{ F}$ and the displacement of an object by a vector $\vecs{ s}$, then the work done by the force is the dot product of $\vecs{ F}$ and $\vecs{ s}$.

zero vector | the vector with both initial point and terminal point $(0,0)$

zeros of a function | when a real number $x$ is a zero of a function $f,\;f(x)=0$

$δ$ ball | all points in $\mathbb{R}^3$ lying at a distance of less than $δ$ from $(x_0,y_0,z_0)$

$δ$ disk | an open disk of radius $δ$ centered at point $(a,b)$