Glossary of Terms

acceleration vector
the second derivative of the position vector
angular coordinate
[latex]\theta[/latex] 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 [latex]s(t)[/latex] that describes the arc length of curve [latex]C[/latex] as a function of [latex]t[/latex]
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 [latex]P_{0}[/latex] of [latex]R[/latex] is a boundary point if every [latex]\delta[/latex] disk centered around [latex]P_{0}[/latex] contains points both inside and outside [latex]R[/latex]
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 [latex]r=a\left(1+\sin\theta \right)[/latex] or [latex]r=a\left(1+\cos\theta \right)[/latex]
characteristic equation
the equation [latex]a\lambda^{2}+b\lambda+c=0[/latex] for the differential equation [latex]ay^{\prime\prime}+by^{\prime}+cy=0[/latex]
circulation
the tendency of a fluid to move in the direction of curve [latex]C[/latex]. If [latex]C[/latex] is a closed curve, then the circulation of [latex]{\bf{F}}[/latex] along [latex]C[/latex] is line integral [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex], which we also denote [latex]\displaystyle\oint_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex]
closed curve
a curve that begins and ends at the same point
closed curve
a curve for which there exists a parameterization [latex]{\bf{r}}(t),a\le{t}\le{b}[/latex], such that [latex]{\bf{r}}(a)={\bf{r}}(b)[/latex], and the curve is traversed exactly once
closed set
a set [latex]S[/latex] that contains all its boundary points
complementary equation
for the nonhomogeneous linear differential equation [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex] the associated homogeneous equation, called the complementary equation, is [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=0[/latex]
component
a scalar that describes either the vertical or horizontal direction of a vector
component functions
the component functions of the vector-valued function [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] are [latex]f(t)[/latex] and [latex]g(t)[/latex], and the component functions of the vector-valued function [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex] are [latex]f(t)[/latex], [latex]g(t)[/latex] and [latex]h(t)[/latex]
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 [latex]S[/latex] 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 [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex]
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 [latex]f(x,y)[/latex]
coordinate plane
a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the [latex]xy[/latex]-plane, [latex]xz[/latex]-plane, or the [latex]yz[/latex]-plane
critical point of a function of two variables
the point [latex](x_{0},y_{0})[/latex] is called a critical point of [latex]f(x,y)[/latex] if one of the two following conditions holds:

  1. [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[/latex]
  2. At least one of [latex]f_{x}(x_{0},y_{0})[/latex] and [latex]f_{y}(x_{0},y_{0})[/latex] do not exist
cross product
[latex]{\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}}[/latex], where [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex]
curl
the curl of vector field [latex]{\bf{F}}=\langle{P,Q,R}\rangle[/latex], denoted [latex]\nabla\times{\bf{F}}[/latex] is the “determinant” of the matrix [latex]\begin{vmatrix}{\bf{i}} & {\bf{j}} & {\bf{k}}\\ \frac{d}{dx} & \frac{d}{dy} & \frac{d}{dz}\\P & Q & R\end{vmatrix}[/latex] and is given by the expression [latex](R_{y}-Q_{z}){\bf{i}}+(P_{z}-R_{x}){\bf{j}}+(Q_{x}+P_{y}){\bf{k}}[/latex]; 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 [latex](r,\theta,z)[/latex], where [latex](r,\theta)[/latex] represents the polar coordinates of the point’s projection in the [latex]xy[/latex]-plane, and [latex]z[/latex] represents the point’s projection onto the [latex]z[/latex]-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 [latex]{\bf{r}}(t)[/latex] is [latex]{\bf{r}}^{\prime}(t)=\underset{\Delta{t}\to{0}}{\lim}\frac{{\bf{r}}(t+\Delta{t})-{\bf{r}}(t)}{\Delta{t}}[/latex], provided the limit exists
determinant
a real number associated with a square matrix
differentiable
a function [latex]f(x,y,z)[/latex] is differentiable at [latex](x_{0},y_{0})[/latex] if [latex]f(x,y)[/latex] can be expressed in the form [latex]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)[/latex], where the error term [latex]E(x,y)[/latex] satisfies [latex]\underset{(x,y)\to{(x_{0},y_{0})}}{\lim}\frac{E(x,y)}{\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[/latex]
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 [latex]4AC-{B}^{2}[/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[/latex], is called a discriminant
discriminant
the discriminant of the function [latex]f(x,y)[/latex] is given by the formula [latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\left(f_{xy}(x_{0},y_{0})\right)^{2}[/latex]
divergence
the divergence of a vector field [latex]{\bf{F}}=\langle{P,Q,R}\rangle[/latex], denoted [latex]\nabla\times{\bf{F}}[/latex] is [latex]P_{x}+Q_{y}+R_{z}[/latex]; 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
[latex]{\bf{u}}\cdot{\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[/latex], where [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and[latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex]
double Riemann Sum
of the function [latex]f(x,y)[/latex] over a rectangular region [latex]R[/latex] is [latex]\displaystyle\sum_{i=1}^{m} {} \displaystyle\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}[/latex] where [latex]R[/latex] is divided into smaller sub rectangles [latex]R_{ij}[/latex] and [latex]({x^{*}}_{i,j}, {y^{*}}_{i,j})[/latex] is an arbitrary point in [latex]R_{ij}[/latex]
double Integral
of the function [latex]f(x,y)[/latex] over the region [latex]R[/latex] in the [latex]xy[/latex]-plane is defined as the limit of a double Riemann sum, [latex]\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}[/latex]
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 [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/latex] all traces of this surface are ellipses
elliptic cone
a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0[/latex] traces of this surface include ellipses and intersecting lines
elliptic paraboloid
a three-dimensional surface described by an equation of the form [latex]z=\frac{x^2}{a^2}+\frac{y^2}{b^2}[/latex] 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 [latex]{\bf{F}}[/latex] across plane curve [latex]C[/latex] is line integral [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\frac{{\bf{n}}(t)}{\Arrowvert{\bf{n}}(t)\Arrowvert}}ds[/latex]
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
if [latex]f(x,y)[/latex] is a function of two variables that is continuous over a rectangular region [latex]R = \{(x,y)\in{\mathbb{R}}^{2}|a\leq x\leq b,c\leq y\leq d\}[/latex], then the double integral of [latex]f[/latex] over the region equals an iterated integral,
[latex]\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}}}[/latex]

function of two variables
a function [latex]z=f(x,y)[/latex] that maps each ordered pair [latex](x,y)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex] to a unique real number [latex]z[/latex]
Fundamental Theorem for Line Integrals
the value of the line integral [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}[/latex] depends only on the value of [latex]f[/latex] at the endpoints of [latex]C[/latex]: [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))[/latex]
Gauss’ law
if [latex]S[/latex] is a piecewise, smooth closed surface in a vacuum and [latex]Q[/latex] is the total stationary charge inside of [latex]S[/latex], then the flux of electrostatic field [latex]\bf{E}[/latex] across [latex]S[/latex] is [latex]Q|{\varepsilon}_{0}[/latex]
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 [latex]ax+by+cz+d=0[/latex], where [latex]{\bf{n}}=\langle{a,b,c}\rangle[/latex] is a normal vector of the plane, [latex]P=(x_{0},y_{0},z_{0})[/latex] is a point on the plane, and [latex]d=-ax_{0}-by_{0}-cz_{0}[/latex]
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 [latex]f(x,y)[/latex] is defined to be [latex]\nabla f(x,y)=(\partial{f}{/}\partial{x}){\bf{i}}+(\partial{f}{/}\partial{y}){\bf{j}}[/latex] which can be generalized to a function of any number of independent variables
gradient field
a vector field [latex]{\bf{F}}[/latex] for which there exists a scalar function [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex] 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 [latex](x,y,z)[/latex] that satisfies the equation [latex]z=f(x,y)[/latex] 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 [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex] but [latex]r(x)=0[/latex] for every value of [latex]x[/latex]
hyperboloid of one sheet
a three-dimensional surface described by an equation of the form [latex]\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1[/latex] traces of this surface include ellipses and parabolas
hyperboloid of two sheets
a three-dimensional surface described by an equation of the form [latex]\frac{z^2}{c^2}-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/latex] 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 [latex]{\bf{F}}[/latex] has path independence if [latex]\displaystyle\int_{C_{1}} {\bf{F}}\cdot{d{\bf{r}}}=\displaystyle\int_{C_{2}} {\bf{F}}\cdot{d{\bf{r}}}[/latex] for any curves [latex]C_{1}[/latex] and [latex]C_{2}[/latex] in the domain of [latex]{\bf{F}}[/latex] with the same initial points and terminal points
initial point
the starting point of a vector
interior point
a point [latex]P_{0}[/latex] of [latex]R[/latex] is a boundary point if there is a [latex]\delta[/latex] disk centered around [latex]P_{0}[/latex] contained completely in [latex]R[/latex]
intermediate variable
given a composition of functions (e.g., [latex]f\left(x(t),y(t)\right)[/latex]) the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function [latex]f\left(x(t),y(t)\right)[/latex] the variables [latex]x[/latex] and [latex]y[/latex] 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 [latex]f(x,y)[/latex] over the region [latex]\bf{R}[/latex] is

[latex]{\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}[/latex]

[latex]{\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}[/latex]

Jacobian
the Jacobian [latex]J(u ,v)[/latex] in two variables is a [latex]2{\times}2[/latex] determinant:
[latex]J(u,v) = \begin{vmatrix}\frac{dx}{du} & \frac{dy}{du}\\\frac{dx}{dv} & \frac{dy}{dv}\end{vmatrix}[/latex]
the Jacobian [latex]J(u ,v, w)[/latex] in three variables is a [latex]3{\times}3[/latex] determinant:
[latex]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}[/latex]
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 [latex]\lambda[/latex]
level curve of a function of two variables
the set of points satisfying the equation [latex]f(x,y)=c[/latex] for some real number [latex]c[/latex] in the range of [latex]f[/latex]
level surface of a function of three variables
the set of points satisfying the equation [latex]f(x,y,z)=c[/latex] for some real number [latex]c[/latex] in the range of [latex]f[/latex]
limaçon
the graph of the equation [latex]r=a+b\sin\theta[/latex] or [latex]r=a+b\cos\theta[/latex]. If [latex]a=b[/latex] then the graph is a cardioid
limit of a vector-valued function
a vector-valued function [latex]{\bf{r}}(t)[/latex] has a limit [latex]{\bf{L}}[/latex] as [latex]t[/latex] approaches [latex]a[/latex] if [latex]\underset{t\to{a}}{\lim}|{\bf{r}}(t)-{\bf{L}}|=0[/latex]
line integral
the integral of a function along a curve in a plane or in space
linear approximation
given a function [latex]f(x,y)[/latex] and a tangent plane to the function at a point [latex](x_{0},y_{0})[/latex] we can approximate [latex]f(x,y)[/latex] for points near [latex](x_{0},y_{0})[/latex] using the tangent plane formula
linearly dependent
a set of function [latex]f_{1}(x),f_{2}(x),\ldots f_{n}(x)[/latex] for which there are constants [latex]c_{1},c_{2},\ldots c_{n}[/latex], not all zero, such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\cdots}+c_{n}f_{n}(x) = 0[/latex] for all [latex]x[/latex] in the interval of interest
linearly independent
a set of function [latex]f_{1}(x),f_{2}(x),\ldots f_{n}(x)[/latex] for which there are no constants, such that [latex]c_{1},c_{2},\ldots c_{n}[/latex], such that [latex]c_{1}f_{1}(x) + c_{2}f_{2}(x) + {\cdots}+c_{n}f_{n}(x) = 0[/latex] for all [latex]x[/latex] 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 [latex]y_{p}(x)=u(x)y_{1}(x)+v(x)y_{2}(x)[/latex], where [latex]y_{1}[/latex] and [latex]y_{2}[/latex] are linearly independent solutions to the complementary equations, and then solving a system of equations to find [latex]u(x)[/latex] and [latex]v(x)[/latex].
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 [latex]a_{2}(x)y^{\prime\prime}+a_{1}(x)y^{\prime}+a_{0}(x)y=r(x)[/latex] but [latex]r(x)\ne 0[/latex] for some value of [latex]x[/latex]
normal component of acceleration
the coefficient of the unit normal vector [latex]{\bf{N}}[/latex] when the acceleration vector is written as a linear combination of [latex]{\bf{T}}[/latex] and [latex]{\bf{N}}[/latex]
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 [latex]T : G {\rightarrow} R[/latex] defined as [latex]T(u, v) = (x, y)[/latex] is said to be one-to-one if no two points map to the same image point
open set
a set [latex]S[/latex] 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 [latex]C[/latex] is a specified direction of [latex]C[/latex]
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 [latex]C[/latex] at a point [latex]P[/latex] 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 [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex]
parameterized surface
a surface given by a description of the form [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex] , where the parameters u and v vary over a parameter domain in the uv-plane
parametric curve
the graph of the parametric equations [latex]x\left(t\right)[/latex] and [latex]y\left(t\right)[/latex] over an interval [latex]a\le t\le b[/latex] combined with the equations
parametric equations
the equations [latex]x=x\left(t\right)[/latex] and [latex]y=y\left(t\right)[/latex] that define a parametric curve
parametric equations of a line:
the set of equations [latex]x=x_{0}+ta[/latex], [latex]y=y_{0}+tb[/latex], and [latex]z=z_{0}+tc[/latex] describing the line with direction vector [latex]{\bf{v}}=\langle{a,b,c}\rangle[/latex] passing through point [latex](x_{0},y_{0},z_{0})[/latex]
parameterization of a curve
rewriting the equation of a curve defined by a function [latex]y=f\left(x\right)[/latex] 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 [latex]y_{p}(x)[/latex] 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 [latex]T[/latex] that transforms a region [latex]G[/latex] in one plane into a region [latex]R[/latex] in another plane by a change of variables
plane curve
the set of ordered pairs [latex]\left(f(t),g(t)\right)[/latex] together with their defining parametric equations [latex]x=f(t)[/latex] and [latex]y=g(t)[/latex]
polar axis
the horizontal axis in the polar coordinate system corresponding to [latex]r\ge 0[/latex]
polar coordinate system
a system for locating points in the plane. The coordinates are [latex]r[/latex], the radial coordinate, and [latex]\theta[/latex], 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 [latex]r=a[/latex] and [latex]r=b[/latex] and the angles [latex]\theta = \alpha[/latex] and [latex]\theta = \beta[/latex]; it is described as [latex]{\bf{R}}=\{(r,{\theta}) | a{\leq}r{\leq}b, {\alpha}{\leq}{\theta}{\leq}{\beta}\}[/latex]
pole
the central point of the polar coordinate system, equivalent to the origin of a Cartesian system
potential function
a scalar function [latex]f[/latex] such that [latex]\nabla{f}={\bf{F}}[/latex]
principal unit normal vector
a vector orthogonal to the unit tangent vector, given by the formula [latex]\frac{{\bf{T}}^{\prime}(t)}{\parallel{\bf{T}}^{\prime}(t)\parallel}[/latex]
principal unit tangent vector
a unit vector tangent to a curve [latex]C[/latex]
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
[latex]r[/latex] 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 [latex]\mathbb{R}^{2}[/latex]
regular parameterization
parameterization [latex]{\bf{r}}(u, v) = {\langle} x (u, v), y (u, v), z (u, v) {\rangle}[/latex] such that [latex]{\bf{r}}_{u}{\times}{\bf{r}}_{v}[/latex] is not zero for point [latex](u, v)[/latex] 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 [latex]z[/latex]-axis in such a way that the fingers curl from the positive [latex]x[/latex]-axis to the positive [latex]y[/latex]-axis, the thumb points in the direction of the positive [latex]z[/latex]-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 [latex]r=a\cos{n}\theta[/latex] or [latex]r=a\sin{n}\theta[/latex] for a positive constant [latex]a[/latex] and an integer [latex]n \ge 2[/latex]
rotational field
a vector field in which the vector at point [latex](x,y)[/latex] is tangent to a circle with radius [latex]r=\sqrt{x^{2}+y^{2}}[/latex] 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 [latex]z=f(x,y)[/latex] the point [latex](x_{0},y_{0},f(x_{0},y_{0}))[/latex] is a saddle point if both [latex]f_{x}(x_{0},y_{0})=0[/latex] and [latex]f_{y}(x_{0},y_{0})=0[/latex], but [latex]f[/latex] does not have a local extremum at [latex](x_{0},y_{0})[/latex]
scalar
a real number
scalar equation of a plane:
the equation [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[/latex] used to describe a plane containing point [latex]P=(x_{0},y_{0},z_{0})[/latex] with normal vector [latex]{\bf{n}}=\langle{a,b,c}\rangle[/latex] or its alternate form [latex]ax+by+cz+d=0[/latex], where [latex]d=-ax_{0}-by_{0}-cz_{0}[/latex]
scalar line integral
the scalar line integral of a function [latex]f[/latex] along a curve [latex]C[/latex] with respect to arc length is the integral [latex]\displaystyle\int_C \! f\, \mathrm{d}s[/latex], it is the integral of a scalar function [latex]f[/latex] 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 [latex]x(t)=c_{1}\cos{(\omega{t})}+c_{2}\sin{(\omega{t})}[/latex] 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 [latex]{\bf{r}}(t)[/latex] is differentiable with a non-zero derivative
space curve
the set of ordered triples [latex]\left(f(t),g(t),h(t)\right)[/latex] together with their defining parametric equations [latex]x=f(t)[/latex], [latex]y=g(t)[/latex] and [latex]z=h(t)[/latex]
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 [latex](\rho,\theta,\varphi)[/latex], where [latex]\rho[/latex] is the distance between [latex]P[/latex] and the origin [latex]\rho \ne {0}[/latex], [latex]\theta[/latex] is the same angle used to describe the location in cylindrical coordinates, and [latex]\varphi[/latex] is the angle formed by the positive [latex]z[/latex]-axis and line segment [latex]\overline{OP}[/latex] where [latex]O[/latex] is the origin and [latex]0\le\varphi\le\pi[/latex]
standard equation of a sphere
[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[/latex] describes a sphere with center [latex](a,b,c)[/latex] and radius [latex]r[/latex]
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: [latex]{\bf{i}}=\langle{1,0}\rangle[/latex], [latex]{\bf{j}}=\langle{0,1}\rangle[/latex]
standard-position Vectors
a vector with initial point [latex](0,0)[/latex]
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
relates the flux integral over a surface [latex]S[/latex] to a line integral around the boundary [latex]C[/latex] of the surface [latex]S[/latex]
stream function
if [latex]{\bf{F}} = {\langle}P, Q{\rangle}[/latex] is a source-free vector field, then stream function g is a function such that [latex]P = g_{y}[/latex], and [latex]Q = -{g_{x}}[/latex]
surface
the graph of a function of two variables, [latex]z=f(x,y)[/latex]
surface area
the area of surface S given by the surface integral [latex]\displaystyle{\int_{} {\int_{S} d{\bf{S}}}}[/latex]
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 [latex]\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}[/latex] describing the line with direction vector [latex]{\bf{v}}=\langle{a,b,c}\rangle[/latex] passing through point [latex](x_{0},y_{0},z_{0})[/latex]
tangent plane
given a function [latex]f(x,y)[/latex] that is differentiable at a point [latex](x_{0},y_{0})[/latex] the equation of the tangent plane to the surface [latex]z=f(x,y)[/latex] is given by [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[/latex]
tangent vector
to [latex]{\bf{r}}(t)[/latex] at [latex]t=t_{0}[/latex] any vector [latex]{\bf{v}}[/latex] such that, when the tail of the vector is placed at point [latex]{\bf{r}}(t_{0})[/latex] on the graph, vector [latex]{\bf{v}}[/latex] is tangent to curve [latex]C[/latex]
tangential component of acceleration
the coefficient of the unit tangent vector [latex]{\bf{T}}[/latex] when the acceleration vector is written as a linear combination of [latex]{\bf{T}}[/latex] and [latex]{\bf{N}}[/latex]
terminal point
the endpoint of a vector
The Fundamental Theorem for Line Integrals
the value of the line integral [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}[/latex] depends only on the value of [latex]f[/latex] at the endpoints of [latex]C[/latex]: [latex]\displaystyle\int_{C} {\nabla}{f}\cdot{d{\bf{r}}}=f({\bf{r}}(b)))-f({\bf{r}}(a))[/latex]
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 [latex](x,y,z)[/latex] 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 [latex]f(x,y)[/latex] at [latex](x_{0},y_{0})[/latex] is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[/latex]
trace
the intersection of a three-dimensional surface with a coordinate plane
transformation
a function that transforms a region [latex]G[/latex] in one plane into a region [latex]R[/latex] 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 [latex]f(x, y, z)[/latex]over a rectangular solid box [latex]\bf{B}[/latex] 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:[latex]{\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}}}[/latex]
triple integral in spherical coordinates
the limit of a triple Riemann sum, provided the following limit exists: [latex]{\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}}}[/latex]
triple scalar product
the dot product of a vector with the cross product of two other vectors: [latex]{\bf{u}}\cdot({\bf{v}}\times{\bf{w}})[/latex]
Type I
a region [latex]\bf{D}[/latex] in the [latex]xy[/latex]-plane is Type I if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[/latex] and [latex]g_{2}(x)[/latex]
Type II
a region [latex]\bf{D}[/latex] in the [latex]xy[/latex]-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_{1}(y)[/latex] and [latex]h_{2}(y)[/latex]
unit vector
a vector with magnitude [latex]1[/latex]
unit vector field
a vector field in which the magnitude of every vector is [latex]1[/latex]
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 [latex]{\bf{v}}-{\bf{w}}[/latex] is defined as [latex]{\bf{v}}+(-{\bf{w}})={\bf{v}}+(-1){\bf{w}}[/latex]
vector equation of a line:
the equation [latex]{\bf{r}} ={\bf{r}}_{0}+t{\bf{v}}[/latex] used to describe a line with direction vector [latex]{\bf{v}}=\langle{a,b,c}\rangle[/latex] passing through point [latex]P=(x_{0},y_{0},z_{0})[/latex], where [latex]{\bf{r}}_{0}=\langle{x_{0},y_{0},z_{0}}\rangle[/latex] is the position vector of point [latex]P[/latex]
vector equation of a plane:
the equation [latex]{\bf{n}}\cdot\overrightarrow{PQ}=0[/latex],
where [latex]P[/latex] is a given point in the plane, [latex]Q[/latex] is any point in the plane, and [latex]{\bf{n}}[/latex] is a normal vector of the plane
vector field
measured in [latex]\mathbb{R}^{2}[/latex], an assignment of a vector [latex]{\bf{F}}(x,y)[/latex] to each point [latex](x,y)[/latex] of a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex]; in [latex]\mathbb{R}^{3}[/latex], an assignment of a vector [latex]{\bf{F}}(x,y,z)[/latex] to each point [latex](x,y,z)[/latex] of a subset [latex]D[/latex] of [latex]\mathbb{R}^{3}[/latex]
vector line integral
the vector line integral of vector field [latex]{\bf{F}}[/latex] along curve [latex]C[/latex] is the integral of the dot product of [latex]{\bf{F}}[/latex] with unit tangent vector [latex]{\bf{T}}[/latex] of [latex]C[/latex] with respect to arc length, [latex]\displaystyle\int_{C} {\bf{F}}\cdot{\bf{T}}ds[/latex]; 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, [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex]can be constructed graphically by placing the initial point of [latex]{\bf{w}}[/latex] at the terminal point of [latex]{\bf{v}}[/latex]; then the vector sum [latex]{\bf{v}}+{\bf{w}}[/latex] is the vector with an initial point that coincides with the initial point of [latex]{\bf{v}}[/latex], and with a terminal point that coincides with the terminal point of [latex]{\bf{w}}[/latex]
vector-valued function
a function of the form [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}[/latex] or [latex]{\bf{r}}(t)=f(t){\bf{i}}+g(t){\bf{j}}+h(t){\bf{k}}[/latex], where the component functions [latex]f[/latex], [latex]g[/latex], and [latex]h[/latex] are real-valued functions of the parameter [latex]t[/latex]
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 [latex](c,y,z)[/latex] that solves the equation [latex]f(c,y)=z[/latex] for a given constant [latex]x=c[/latex] or the set of ordered triples [latex](x,d,z)[/latex] that solves the equation [latex]f(x,d)=z[/latex] for a given constant [latex]y=d[/latex]
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 [latex]\bf{F}[/latex] and the displacement of an object by a vector [latex]\bf{s}[/latex], then the work done by the force is the dot product of [latex]\bf{F}[/latex] and [latex]\bf{s}[/latex]
zero vector
the vector with both initial point and terminal point [latex](0,0)[/latex]
[latex]\delta[/latex] ball
all points in [latex]\mathbb{R}^{3}[/latex] lying at a distance of less than [latex]\delta[/latex] from [latex](x_{0},y_{0},z_{0})[/latex]
[latex]\delta[/latex] disk
an open disk of radius [latex]\delta[/latex] centered at point [latex](a,b)[/latex]