- 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:
- [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[/latex]
- 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]
