Glossary of Terms

acceleration vector: the second derivative of the position vector

angular coordinate: $\theta$ 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

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

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 $\delta$ disk centered around $P_{0}$ contains points both inside and outside $R$

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

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 $r=a\left(1+\sin\theta \right)$ or $r=a\left(1+\cos\theta \right)$

characteristic equation: the equation $a\lambda^{2}+b\lambda+c=0$ for the differential equation $ay^{\prime\prime}+by^{\prime}+cy=0$

circulation: the tendency of a fluid to move in the direction of curve $C$ . If $C$ is a closed curve, then the circulation of ${\bf{F}}$ along $C$ is line integral $\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds$ , which we also denote $\displaystyle\oint_{C} {\bf{F}}\cdot{\bf{T}}ds$

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

closed curve: a curve for which there exists a parameterization ${\bf{r}}(t),a\le{t}\le{b}$ , such that ${\bf{r}}(a)={\bf{r}}(b)$ , and the curve is traversed exactly once

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

complementary equation: for the nonhomogeneous linear differential equation $a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)$ the associated homogeneous equation, called the complementary equation, is $a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=0$

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

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

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 $\nabla{f}={\bf{F}}$

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

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

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 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:

$f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0$
At least one of $f_{x}(x_{0},y_{0})$ and $f_{y}(x_{0},y_{0})$ do not exist

cross product: ${\bf{u}}\times{\bf{v}}=(u_{2}v_{3}-u_{3}v_{2}){\bf{i}}-(u_{1}v_{3}-u_{3}v_{1}){\bf{j}}+(u_{1}v_{2}-u_{2}v_{1}){\bf{k}}$ , where ${\bf{u}}=\langle{u_1,u_2,u_3}\rangle$ and ${\bf{v}}=\langle{v_1,v_2,v_3}\rangle$

curl: the curl of vector field ${\bf{F}}=\langle{P,Q,R}\rangle$ , denoted $\nabla\times{\bf{F}}$ is the “determinant” of the matrix $\begin{vmatrix}{\bf{i}} & {\bf{j}} & {\bf{k}}\\ \frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\P & Q & R\end{vmatrix}$ and is given by the expression $(R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}$ ; 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 $(r,\theta,z)$ , where $(r,\theta)$ represents the polar coordinates of the point’s projection in the $xy$ -plane, and $z$ represents the point’s projection onto the $z$ -axis

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

derivative of a vector-valued function: the derivative of a vector-valued function ${\bf{r}}(t)$ is ${\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}$ , provided the limit exists

determinant: a real number associated with a square matrix

differentiable: a function $f(x,y,z)$ 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 $\underset{(x,y)\to{(x_{0},y_{0})}}{\lim}\frac{E(x,y)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0$

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 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

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 $D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\left(f_{xy}(x_{0},y_{0})\right)^{2}$

divergence: the divergence of a vector field ${\bf{F}}=\langle{P,Q,R}\rangle$ , denoted $\nabla\times{\bf{F}}$ is $P_{x}+Q_{y}+R_{z}$ ; it measures the “outflowing-ness” of a vector field

divergence theorem: a theorem used to transform a difficult flux integral into an easier triple integral and vice versa

dot product or scalar product: ${\bf{u}}\cdot{\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}$ , where ${\bf{u}}=\langle{u_1,u_2,u_3}\rangle$ and ${\bf{v}}=\langle{v_1,v_2,v_3}\rangle$

double Riemann Sum: of the function $f(x,y)$ over a rectangular region $R$ is $\displaystyle\sum_{i=1}^{m} {} \displaystyle\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}$ where $R$ is divided into smaller sub rectangles $R_{ij}$ and $({x^{*}}_{i,j}, {y^{*}}_{i,j})$ is an arbitrary point in $R_{ij}$

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, $\underset{R}{\displaystyle\iint} f(x,y)dA=\underset{m,n\to{\infty}}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f(x_{ij}^{\ast},y_{ij}^{\ast})\Delta{A}$

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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ all traces of this surface are ellipses

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

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

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

flux: the rate of a fluid flowing across a curve in a vector field; the flux of vector field ${\bf{F}}$ across plane curve $C$ is line integral $\displaystyle\int_{C} {\bf{F}}\cdot{\frac{{\bf{n}}(t)}{\Arrowvert{\bf{n}}(t)\Arrowvert}}ds$

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

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

Fubini’s Theorem

$f(x,y)$ is a function of two variables that is continuous over a rectangular region

$R = \{(x,y)\in{\mathbb{R}}^{2}|a\leq x\leq b,c\leq y\leq d\}$ , then the double integral of

$f$ over the region equals an iterated integral,

$\underset{R}{\displaystyle\iint} f(x,y)dxdy={\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{c}^{d}}{\displaystyle\int_{a}^{b} {f(x,y){dx}{dy}}}$

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

Fundamental Theorem for Line Integrals: the value of the line integral $\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}$ depends only on the value of $f$ at the endpoints of $C$ : $\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))$

Gauss’ law: if $S$ is a piecewise, smooth closed surface in a vacuum and $Q$ is the total stationary charge inside of $S$ , then the flux of electrostatic field $\bf{E}$ across $S$ is $Q|{\varepsilon}_{0}$

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 ${\bf{n}}=\langle{a,b,c}\rangle$ is a normal vector of the plane, $P=(x_{0},y_{0},z_{0})$ is a point on the plane, and $d=-ax_{0}-by_{0}-cz_{0}$

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

gradient: the gradient of the function $f(x,y)$ is defined to be $\nabla f(x,y)=(\partial{f}{/}\partial{x}){\bf{i}}+(\partial{f}{/}\partial{y}){\bf{j}}$ which can be generalized to a function of any number of independent variables

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

graph of a function 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

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 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 $a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)$ but $r(x)=0$ for every value of $x$

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

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

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

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

independent of path (path independent): a vector field ${\bf{F}}$ has path independence if $\displaystyle\int_{C_{1}} {\bf{F}}\cdot{d{\bf{r}}}=\displaystyle\int_{C_{2}} {\bf{F}}\cdot{d{\bf{r}}}$ for any curves $C_{1}$ and $C_{2}$ in the domain of ${\bf{F}}$ with the same initial points and terminal points

initial point: the starting point of a vector

interior point: a point $P_{0}$ of $R$ is a boundary point if there is a $\delta$ disk centered around $P_{0}$ contained completely in $R$

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

inverse-square law: the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge

iterated Integral

for a function

$f(x,y)$ over the region

$\bf{R}$ is

${\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{a}^{b}}\left[{\displaystyle\int_{c}^{d} {f(x,y){dy}}}\right]{dx}$

${\displaystyle\int_{a}^{b}}{\displaystyle\int_{c}^{d} {f(x,y){dx}{dy}}}={\displaystyle\int_{c}^{d}}\left[{\displaystyle\int_{a}^{b} {f(x,y){dx}}}\right]{dy}$

Jacobian: the Jacobian $J(u ,v)$ in two variables is a $2{\times}2$ determinant:; $J(u,v) = \begin{vmatrix}\frac{dx}{du} & \frac{dy}{du}\\\frac{dx}{dv} & \frac{dy}{dv}\end{vmatrix}$; the Jacobian $J(u ,v, w)$ in three variables is a $3{\times}3$ determinant:; $J(u,v,w)=\begin{vmatrix}\frac{dx}{du} & \frac{dy}{du} & \frac{dz}{du}\\\frac{dx}{dv} & \frac{dy}{dv} & \frac{dz}{dv}\\\frac{dx}{dw} & \frac{dy}{dw} & \frac{dz}{dw}\end{vmatrix}$

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 $\lambda$

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 $r=a+b\sin\theta$ or $r=a+b\cos\theta$ . If $a=b$ then the graph is a cardioid

limit of a vector-valued function: a vector-valued function ${\bf{r}}(t)$ has a limit ${\bf{L}}$ as $t$ approaches $a$ if $\underset{t\to{a}}{\lim}|{\bf{r}}(t)-{\bf{L}}|=0$

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

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

linearly dependent: a set of function $f_{1}(x),f_{2}(x),\ldots f_{n}(x)$ for which there are constants $c_{1},c_{2},\ldots c_{n}$ , not all zero, such that $c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\cdots}+c_{n}f_{n}(x) = 0$ for all $x$ in the interval of interest

linearly independent: a set of function $f_{1}(x),f_{2}(x),\ldots f_{n}(x)$ for which there are no constants, such that $c_{1},c_{2},\ldots c_{n}$ , such that $c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\cdots}+c_{n}f_{n}(x) = 0$ for all $x$ in the interval of interest

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

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

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 $v(x)$ .

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

nappe: a nappe is one half of a double cone

nonhomogeneous linear equation: a second-order differential equation that can be written in the form $a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)$ but $r(x)\ne 0$ for some value of $x$

normal component of acceleration: the coefficient of the unit normal vector ${\bf{N}}$ when the acceleration vector is written as a linear combination of ${\bf{T}}$ and ${\bf{N}}$

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

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

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

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

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

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

parallelpiped: a three-dimensional prism with six faces that are parallelograms

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 ${\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}$

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

parametric curve: the graph of the parametric equations $x\left(t\right)$ and $y\left(t\right)$ over an interval $a\le t\le b$ combined with the equations

parametric equations: the equations $x=x\left(t\right)$ and $y=y\left(t\right)$ 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 ${\bf{v}}=\langle{a,b,c}\rangle$ passing through point $(x_{0},y_{0},z_{0})$

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

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

particular solution: a solution $y_{p}(x)$ of a differential equation that contains no arbitrary constants

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

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 $\left(f(t),g(t)\right)$ together with their defining parametric equations $x=f(t)$ and $y=g(t)$

polar axis: the horizontal axis in the polar coordinate system corresponding to $r\ge 0$

polar coordinate system: a system for locating points in the plane. The coordinates are $r$ , the radial coordinate, and $\theta$ , 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 ${\bf{R}}=\{(r,{\theta}) | a{\leq}r{\leq}b, {\alpha}{\leq}{\theta}{\leq}{\beta}\}$

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

potential function: a scalar function $f$ such that $\nabla{f}={\bf{F}}$

principal unit normal vector: a vector orthogonal to the unit tangent vector, given by the formula $\frac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}$

principal unit tangent vector: a unit vector tangent to a curve $C$

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

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

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

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

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

regular parameterization: parameterization ${\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}$ such that ${\bf{r}}_{u}{\times}{\bf{r}}_{v}$ is not zero for point $(u, v)$ in the parameter domain

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

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

rose: graph of the polar equation $r=a\cos{n}\theta$ or $r=a\sin{n}\theta$ for a positive constant $a$ and an integer $n \ge 2$

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 ${\bf{n}}=\langle{a,b,c}\rangle$ 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 $\displaystyle\int_C \! f\, \mathrm{d}s$ , 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

simple curve: a curve that does not cross itself

simple harmonic motion: motion described by the equation $x(t)=c_{1}\cos{(\omega{t})}+c_{2}\sin{(\omega{t})}$ 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

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

smooth: curves where the vector-valued function ${\bf{r}}(t)$ is differentiable with a non-zero derivative

space curve: the set of ordered triples $\left(f(t),g(t),h(t)\right)$ 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

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 $(\rho,\theta,\varphi)$ , where $\rho$ is the distance between $P$ and the origin $\rho \ne {0}$ , $\theta$ is the same angle used to describe the location in cylindrical coordinates, and $\varphi$ is the angle formed by the positive $z$ -axis and line segment $\overline{OP}$ where $O$ is the origin and $0\le\varphi\le\pi$

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: 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: ${\bf{i}}=\langle{1,0}\rangle$ , ${\bf{j}}=\langle{0,1}\rangle$

standard-position Vectors: 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

Stokes’ theorem: $S$ $C$ $S$

stream function: if ${\bf{F}} = {\langle}P, Q{\rangle}$ is a source-free vector field, then stream function g is a function such that $P = g_{y}$ , and $Q = -{g_{x}}$

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

surface area: the area of surface S given by the surface integral $\displaystyle{\int_{} {\int_{S} d{\bf{S}}}}$

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 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 $\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}$ describing the line with direction vector ${\bf{v}}=\langle{a,b,c}\rangle$ passing through point $(x_{0},y_{0},z_{0})$

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 ${\bf{r}}(t)$ at $t=t_{0}$ any vector ${\bf{v}}$ such that, when the tail of the vector is placed at point ${\bf{r}}(t_{0})$ on the graph, vector ${\bf{v}}$ is tangent to curve $C$

tangential component of acceleration: the coefficient of the unit tangent vector ${\bf{T}}$ when the acceleration vector is written as a linear combination of ${\bf{T}}$ and ${\bf{N}}$

terminal point: the endpoint of a vector

The Fundamental Theorem for Line Integrals: the value of the line integral $\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}$ depends only on the value of $f$ at the endpoints of $C$ : $\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))$

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

torque: the effect of a force that causes an object to rotate

total differential: the total differential of the function $f(x,y)$ at $(x_{0},y_{0})$ is given by the formula $dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy$

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

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

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

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

triple integral: the triple integral of a continuous function $f(x, y, z)$ over a rectangular solid box $\bf{B}$ 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: ${\displaystyle\lim_{l,m,n\to\infty}{\sum_{i=1}^{l}}{\displaystyle\sum_{j=1}^{m}}{\displaystyle\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\Delta}r{\Delta}{\theta}{\Delta}{z}}}$

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

triple scalar product: the dot product of a vector with the cross product of two other vectors: ${\bf{u}}\cdot({\bf{v}}\times{\bf{w}})$

Type I: a region $\bf{D}$ 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 $\bf{D}$ 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 $h_{2}(y)$

unit vector: a vector with magnitude $1$

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

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 ${\bf{v}}-{\bf{w}}$ is defined as ${\bf{v}}+(-{\bf{w}})={\bf{v}}+(-1){\bf{w}}$

vector equation of a line:: the equation ${\bf{r}} ={\bf{r}}_{0}+t{\bf{v}}$ used to describe a line with direction vector ${\bf{v}}=\langle{a,b,c}\rangle$ passing through point $P=(x_{0},y_{0},z_{0})$ , where ${\bf{r}}_{0}=\langle{x_{0},y_{0},z_{0}}\rangle$ is the position vector of point $P$

vector equation of a plane:: the equation ${\bf{n}}\cdot\overrightarrow{PQ}=0$ ,
where $P$ is a given point in the plane, $Q$ is any point in the plane, and ${\bf{n}}$ is a normal vector of the plane

vector field: measured in $\mathbb{R}^{2}$ , an assignment of a vector to each point of a subset of $\mathbb{R}^{2}$ ; in $\mathbb{R}^{3}$ , an assignment of a vector to each point of a subset of $\mathbb{R}^{3}$

vector line integral: the vector line integral of vector field ${\bf{F}}$ along curve $C$ is the integral of the dot product of ${\bf{F}}$ with unit tangent vector ${\bf{T}}$ of $C$ with respect to arc length, $\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds$ ; 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 product: the cross product of two vectors

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

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

vector-valued function: a function of the form ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}$ or ${\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}$ , 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 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 $y=d$

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 $\bf{F}$ and the displacement of an object by a vector $\bf{s}$ , then the work done by the force is the dot product of $\bf{F}$ and $\bf{s}$

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

$\delta$ ball: all points in $\mathbb{R}^{3}$ lying at a distance of less than $\delta$ from $(x_{0},y_{0},z_{0})$

$\delta$ disk: an open disk of radius $\delta$ centered at point $(a,b)$

Appendices

Candela Citations