{"id":3679,"date":"2022-03-14T17:28:54","date_gmt":"2022-03-14T17:28:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&p=3679"},"modified":"2022-11-01T23:23:47","modified_gmt":"2022-11-01T23:23:47","slug":"glossary-of-terms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/glossary-of-terms\/","title":{"raw":"Glossary of Terms","rendered":"Glossary of Terms"},"content":{"raw":"
\r\n \t
acceleration vector<\/dt>\r\n \t
the second derivative of the position vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
angular coordinate<\/dt>\r\n \t
[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<\/em>) axis, measured counterclockwise<\/dd>\r\n<\/dl>\r\n
\r\n \t
arc-length function<\/dt>\r\n \t
a function\u00a0[latex]s(t)[\/latex]\u00a0that describes the arc length of curve\u00a0[latex]C[\/latex]\u00a0as a function of\u00a0[latex]t[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
arc-length parameterization<\/dt>\r\n \t
a reparameterization of a vector-valued function in which the parameter is equal to the arc length<\/dd>\r\n<\/dl>\r\n
\r\n \t
binormal vector<\/dt>\r\n \t
a unit vector orthogonal to the unit tangent vector and the unit normal vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
boundary conditions<\/dt>\r\n \t
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<\/span><\/dd>\r\n<\/dl>\r\n
\r\n \t
boundary point<\/dt>\r\n \t
a point\u00a0[latex]P_{0}[\/latex]\u00a0of [latex]R[\/latex]\u00a0is a boundary point if every [latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contains points both inside and outside [latex]R[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
boundary-value problem<\/dt>\r\n \t
a differential equation with associated boundary conditions<\/span><\/dd>\r\n<\/dl>\r\n
\r\n \t
cardioid<\/dt>\r\n \t
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]<\/dd>\r\n<\/dl>\r\n
\r\n \t
characteristic equation<\/dt>\r\n \t
the equation [latex]a\\lambda^{2}+b\\lambda+c=0[\/latex]\u00a0<\/strong>for the differential equation [latex]ay^{\\prime\\prime}+by^{\\prime}+cy=0[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
circulation<\/dt>\r\n \t
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]\u00a0along [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex], which we also denote\u00a0[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
closed curve<\/dt>\r\n \t
a curve that begins and ends at the same point<\/span><\/dd>\r\n<\/dl>\r\n
\r\n \t
closed curve<\/dt>\r\n \t
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],\u00a0and the curve is traversed exactly once<\/dd>\r\n<\/dl>\r\n
\r\n \t
closed set<\/dt>\r\n \t
a set [latex]S[\/latex]\u00a0<\/strong>that contains all its boundary points<\/dd>\r\n<\/dl>\r\n
\r\n \t
complementary equation<\/dt>\r\n \t
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]\u00a0the associated homogeneous equation, called the complementary equation<\/em>, is\u00a0[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=0[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
component<\/dt>\r\n \t
a scalar that describes either the vertical or horizontal direction of a vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
component functions<\/dt>\r\n \t
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\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex] are\u00a0[latex]f(t)[\/latex],\u00a0[latex]g(t)[\/latex] and [latex]h(t)[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
conic section<\/dt>\r\n \t
a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\r\n<\/dl>\r\n
\r\n \t
connected region<\/dt>\r\n \t
a region in which any two points can be connected by a path with a trace contained entirely inside the region<\/span><\/dd>\r\n<\/dl>\r\n
\r\n \t
connected set<\/dt>\r\n \t
an open set [latex]S[\/latex]\u00a0that cannot be represented as the union of two or more disjoint, nonempty open subsets<\/dd>\r\n<\/dl>\r\n
\r\n \t
conservative field<\/dt>\r\n \t
a vector field for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
constraint<\/dt>\r\n \t
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<\/dd>\r\n<\/dl>\r\n
\r\n \t
contour map<\/dt>\r\n \t
a plot of the various level curves of a given function [latex]f(x,y)[\/latex]<\/dd>\r\n<\/dl>\r\n
\r\n \t
coordinate plane<\/dt>\r\n \t
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<\/span><\/dd>\r\n<\/dl>\r\n
\r\n \t
critical point of a function of two variables<\/dt>\r\n \t
the point\u00a0[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:\r\n
    \r\n \t
  1. [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex]<\/li>\r\n \t
  2. At least one of [latex]f_{x}(x_{0},y_{0})[\/latex] and [latex]f_{y}(x_{0},y_{0})[\/latex]\u00a0<\/strong>do not exist<\/li>\r\n<\/ol>\r\n<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    cross product<\/dt>\r\n \t
    [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\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    curl<\/dt>\r\n \t
    the curl of vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex]\u00a0is the \u201cdeterminant\u201d 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\u00a0[latex](R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex];\u00a0it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    curvature<\/dt>\r\n \t
    the derivative of the unit tangent vector with respect to the arc-length parameter<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    cusp<\/dt>\r\n \t
    a pointed end or part where two curves meet<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    cycloid<\/dt>\r\n \t
    the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    cylinder<\/dt>\r\n \t
    a set of lines parallel to a given line passing through a given curve<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    cylindrical coordinate system<\/dt>\r\n \t
    a way to describe a location in space with an ordered triple [latex](r,\\theta,z)[\/latex],\u00a0<\/strong>where\u00a0[latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane, and [latex]z[\/latex] represents the point's projection onto the [latex]z[\/latex]-axis<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    definite integral of a vector-valued function<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    derivative of a vector-valued function<\/dt>\r\n \t
    the derivative of a vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex]\u00a0<\/strong>is\u00a0[latex]{\\bf{r}}^{\\prime}(t)=\\underset{\\Delta{t}\\to{0}}{\\lim}\\frac{{\\bf{r}}(t+\\Delta{t})-{\\bf{r}}(t)}{\\Delta{t}}[\/latex],\u00a0<\/strong>provided the limit exists<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    determinant<\/dt>\r\n \t
    a real number associated with a square matrix<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    differentiable<\/dt>\r\n \t
    a function [latex]f(x,y,z)[\/latex]\u00a0is differentiable at\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0if [latex]f(x,y)[\/latex]\u00a0can 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]\u00a0satisfies [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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    direction angles<\/dt>\r\n \t
    the angles formed by a nonzero vector and the coordinate axes<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    direction cosines<\/dt>\r\n \t
    the cosines of the angles formed by a nonzero vector and the coordinate axes<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    direction vector<\/dt>\r\n \t
    a vector parallel to a line that is used to describe the direction, or orientation, of the line in space<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    directional derivative<\/dt>\r\n \t
    the derivative of a function in the direction of a given unit vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    directrix<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    discriminant<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    discriminant<\/dt>\r\n \t
    the discriminant of the function [latex]f(x,y)[\/latex]\u00a0is given by the formula\u00a0[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\\left(f_{xy}(x_{0},y_{0})\\right)^{2}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    divergence<\/dt>\r\n \t
    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 \u201coutflowing-ness\u201d of a vector field<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    divergence theorem<\/dt>\r\n \t
    a theorem used to transform a difficult flux integral into an easier triple integral and vice versa<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    dot product or scalar product<\/dt>\r\n \t
    [latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex]\u00a0and[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    double Riemann Sum<\/dt>\r\n \t
    of the function [latex]f(x,y)[\/latex]\u00a0over\u00a0a rectangular region [latex]R[\/latex]\u00a0is [latex]\\displaystyle\\sum_{i=1}^{m} {} \\displaystyle\\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}[\/latex] where [latex]R[\/latex]\u00a0is divided into smaller sub rectangles [latex]R_{ij}[\/latex] and [latex]({x^{*}}_{i,j}, {y^{*}}_{i,j})[\/latex]\u00a0is an arbitrary point in\u00a0[latex]R_{ij}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    double Integral<\/dt>\r\n \t
    of the function [latex]f(x,y)[\/latex]\u00a0<\/strong>over the region [latex]R[\/latex]\u00a0in the [latex]xy[\/latex]-plane\u00a0is defined as the limit of a double Riemann sum,\u00a0[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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    eccentricity<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    ellipsoid<\/dt>\r\n \t
    a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]\u00a0all traces of this surface are ellipses<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    elliptic cone<\/dt>\r\n \t
    a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0[\/latex]\u00a0traces of this surface include ellipses and intersecting lines<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    elliptic paraboloid<\/dt>\r\n \t
    a three-dimensional surface described by an equation of the form\u00a0[latex]z=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}[\/latex]\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    equivalent vectors<\/dt>\r\n \t
    vectors that have the same magnitude and the same direction<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    flux<\/dt>\r\n \t
    the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\\bf{F}}[\/latex]\u00a0across 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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    flux integral<\/dt>\r\n \t
    another name for a surface integral of a vector field; the preferred term in physics and engineering<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    focal parameter<\/dt>\r\n \t
    the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    focus<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Frenet frame of reference<\/dt>\r\n \t
    (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Fubini's Theorem<\/dt>\r\n \t
    if\u00a0[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,<\/dd>\r\n \t
    [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]\r\n
    <\/dl>\r\n<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    function of two variables<\/dt>\r\n \t
    a function [latex]z=f(x,y)[\/latex] that maps each ordered pair [latex](x,y)[\/latex]\u00a0<\/strong>in a subset [latex]D[\/latex]\u00a0of [latex]\\mathbb{R}^{2}[\/latex]\u00a0<\/strong>to a unique real number [latex]z[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Fundamental Theorem for Line Integrals<\/dt>\r\n \t
    the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Gauss' law<\/dt>\r\n \t
    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]\u00a0is [latex]Q|{\\varepsilon}_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    general form<\/dt>\r\n \t
    an equation of a conic section written as a general second-degree equation<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    general form of the equation of a plane<\/dt>\r\n \t
    an equation in the form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[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]\u00a0is a point on the plane, and\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    generalized chain rule<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    gradient<\/dt>\r\n \t
    the gradient of the function [latex]f(x,y)[\/latex]\u00a0is defined to be [latex]\\nabla f(x,y)=(\\partial{f}{\/}\\partial{x}){\\bf{i}}+(\\partial{f}{\/}\\partial{y}){\\bf{j}}[\/latex]\u00a0which can be generalized to a function of any number of independent variables<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    gradient field<\/dt>\r\n \t
    a vector field [latex]{\\bf{F}}[\/latex]\u00a0for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]\u00a0in other words, a vector field that is the gradient of a function; such vector fields are also called conservative<\/em><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    graph of a function of two variables<\/dt>\r\n \t
    a set of ordered triples\u00a0[latex](x,y,z)[\/latex]\u00a0that satisfies the equation [latex]z=f(x,y)[\/latex]\u00a0plotted in three-dimensional Cartesian space<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Green's theorem<\/dt>\r\n \t
    relates the integral over a connected region to an integral over the boundary of the region<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    grid curves<\/dt>\r\n \t
    curves on a surface that are parallel to grid lines in a coordinate plane<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    heat flow<\/dt>\r\n \t
    a vector field proportional to the negative temperature gradient in an object<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    helix<\/dt>\r\n \t
    a three-dimensional curve in the shape of a spiral<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    higher-order partial derivatives<\/dt>\r\n \t
    second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    homogeneous linear equation<\/dt>\r\n \t
    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]\u00a0but [latex]r(x)=0[\/latex]\u00a0for every value of [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    hyperboloid of one sheet<\/dt>\r\n \t
    a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    hyperboloid of two sheets<\/dt>\r\n \t
    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]\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    improper double integral<\/dt>\r\n \t
    a double integral over an unbounded region or of an unbounded function<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    indefinite integral of a vector-valued function<\/dt>\r\n \t
    a vector-valued function with a derivative that is equal to a given vector-valued function<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    independent of path (path independent)<\/dt>\r\n \t
    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]\u00a0for any curves [latex]C_{1}[\/latex]\u00a0and [latex]C_{2}[\/latex]\u00a0in the domain of [latex]{\\bf{F}}[\/latex] with the same initial points and terminal points<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    initial point<\/dt>\r\n \t
    the starting point of a vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    interior point<\/dt>\r\n \t
    a point\u00a0[latex]P_{0}[\/latex]\u00a0of\u00a0[latex]R[\/latex]\u00a0is a boundary point if there is a\u00a0[latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contained completely in\u00a0[latex]R[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    intermediate variable<\/dt>\r\n \t
    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\u00a0[latex]f\\left(x(t),y(t)\\right)[\/latex]<\/span>\u00a0the variables [latex]x[\/latex]\u00a0and\u00a0[latex]y[\/latex]\u00a0<\/strong>are examples of intermediate variables<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    inverse-square law<\/dt>\r\n \t
    the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    iterated Integral<\/dt>\r\n \t
    for a function [latex]f(x,y)[\/latex] over the region [latex]\\bf{R}[\/latex] is\r\n\r\n[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]<\/span>\r\n\r\n[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]<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Jacobian<\/dt>\r\n \t
    the Jacobian [latex]J(u ,v)[\/latex] in two variables is a [latex]2{\\times}2[\/latex] determinant:<\/dd>\r\n \t
    [latex]J(u,v) = \\begin{vmatrix}\\frac{dx}{du} & \\frac{dy}{du}\\\\\\frac{dx}{dv} & \\frac{dy}{dv}\\end{vmatrix}[\/latex]<\/dd>\r\n \t
    the Jacobian [latex]J(u ,v, w)[\/latex]\u00a0in three variables is a [latex]3{\\times}3[\/latex]\u00a0determinant:<\/dd>\r\n \t
    [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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Kepler's laws of planetary motion<\/dt>\r\n \t
    three laws governing the motion of planets, asteroids, and comets in orbit around the Sun<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Lagrange Multiplier<\/dt>\r\n \t
    the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable\u00a0[latex]\\lambda[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    level curve of a function of two variables<\/dt>\r\n \t
    the set of points satisfying the equation [latex]f(x,y)=c[\/latex]\u00a0<\/strong>for some real number [latex]c[\/latex]\u00a0in the range of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    level surface of a function of three variables<\/dt>\r\n \t
    the set of points satisfying the equation [latex]f(x,y,z)=c[\/latex]\u00a0<\/strong>for some real number [latex]c[\/latex] in the range of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    lima\u00e7on<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    limit of a vector-valued function<\/dt>\r\n \t
    a vector-valued function [latex]{\\bf{r}}(t)[\/latex] has a limit [latex]{\\bf{L}}[\/latex] as [latex]t[\/latex]\u00a0<\/em>approaches\u00a0[latex]a[\/latex]\u00a0if [latex]\\underset{t\\to{a}}{\\lim}|{\\bf{r}}(t)-{\\bf{L}}|=0[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    line integral<\/dt>\r\n \t
    the integral of a function along a curve in a plane or in space<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    linear approximation<\/dt>\r\n \t
    given a function [latex]f(x,y)[\/latex]\u00a0and a tangent plane to the function at a point [latex](x_{0},y_{0})[\/latex] we can approximate [latex]f(x,y)[\/latex]\u00a0for points near [latex](x_{0},y_{0})[\/latex] using the tangent plane formula<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    linearly dependent<\/dt>\r\n \t
    a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for 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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    linearly independent<\/dt>\r\n \t
    a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for which there are no constants, such that\u00a0[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]\u00a0<\/span>for all [latex]x[\/latex]<\/span>\u00a0in the interval of interest<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    magnitude<\/dt>\r\n \t
    the length of a vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    major axis<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    mass flux<\/dt>\r\n \t
    the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    method of Lagrange multipliers<\/dt>\r\n \t
    a method of solving an optimization problem subject to one or more constraints<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    method of undetermined coefficients<\/dt>\r\n \t
    a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    method of variation of parameters<\/dt>\r\n \t
    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]\u00a0and [latex]y_{2}[\/latex]\u00a0are linearly independent solutions to the complementary equations, and then solving a system of equations to find [latex]u(x)[\/latex]\u00a0and\u00a0[latex]v(x)[\/latex].<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    minor axis<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    mixed partial derivatives<\/dt>\r\n \t
    second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    nappe<\/dt>\r\n \t
    a nappe is one half of a double cone<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    nonhomogeneous linear equation<\/dt>\r\n \t
    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]\u00a0but [latex]r(x)\\ne 0[\/latex]\u00a0for some value of [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    normal component of acceleration<\/dt>\r\n \t
    the coefficient of the unit normal vector [latex]{\\bf{N}}[\/latex]\u00a0<\/span>when the acceleration vector is written as a linear combination of\u00a0[latex]{\\bf{T}}[\/latex]\u00a0and\u00a0[latex]{\\bf{N}}[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    normal plane<\/dt>\r\n \t
    a plane that is perpendicular to a curve at any point on the curve<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    normal vector<\/dt>\r\n \t
    a vector perpendicular to a plane<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    normalization<\/dt>\r\n \t
    using scalar multiplication to find a unit vector with a given direction<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    objective function<\/dt>\r\n \t
    the function that is to be maximized or minimized in an optimization problem<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    octants<\/dt>\r\n \t
    the eight regions of space created by the coordinate planes<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    one-to-one transformation<\/dt>\r\n \t
    a transformation [latex]T : G {\\rightarrow} R[\/latex]\u00a0defined as [latex]T(u, v) = (x, y)[\/latex]\u00a0is said to be one-to-one if no two points map to the same image point<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    open set<\/dt>\r\n \t
    a set\u00a0[latex]S[\/latex]\u00a0that contains none of its boundary points<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    optimization problem<\/dt>\r\n \t
    calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    orientation<\/dt>\r\n \t
    the direction that a point moves on a graph as the parameter increases<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    orientation of a curve<\/dt>\r\n \t
    the orientation of a curve [latex]C[\/latex]<\/span>\u00a0is a specified direction of [latex]C[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    orientation of a surface<\/dt>\r\n \t
    if a surface has an \u201cinner\u201d side and an \u201couter\u201d side, then an orientation is a choice of the inner or the outer side; the surface could also have \u201cupward\u201d and \u201cdownward\u201d orientations<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    orthogonal vectors<\/dt>\r\n \t
    vectors that form a right angle when placed in standard position<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    osculating circle<\/dt>\r\n \t
    a circle that is tangent to a curve\u00a0[latex]C[\/latex]\u00a0at a point [latex]P[\/latex]\u00a0<\/span>and that shares the same curvature<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    osculating plane<\/dt>\r\n \t
    the plane determined by the unit tangent and the unit normal vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parallelogram method<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parallelpiped<\/dt>\r\n \t
    a three-dimensional prism with six faces that are parallelograms<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parameter<\/dt>\r\n \t
    an independent variable that both x<\/em> and y<\/em> depend on in a parametric curve; usually represented by the variable t<\/em><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parameter domain (parameter space)<\/dt>\r\n \t
    the region of the uv<\/em> plane over which the parameters u<\/em> and v<\/em> vary for parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parameterized surface<\/dt>\r\n \t
    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] ,\u00a0where the parameters u<\/em> and v<\/em> vary over a parameter domain in the uv<\/em>-plane<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parametric curve<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parametric equations<\/dt>\r\n \t
    the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parametric equations of a line:<\/dt>\r\n \t
    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\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    parameterization of a curve<\/dt>\r\n \t
    rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    partial derivative<\/dt>\r\n \t
    a derivative of a function of more than one independent variable in which all the variables but one are held constant<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    partial differential equation<\/dt>\r\n \t
    an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    particular solution<\/dt>\r\n \t
    a solution [latex]y_{p}(x)[\/latex]\u00a0of a differential equation that contains no arbitrary constants<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    piecewise smooth curve<\/dt>\r\n \t
    an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    planar transformation<\/dt>\r\n \t
    a function [latex]T[\/latex]\u00a0that transforms a region [latex]G[\/latex]\u00a0in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    plane curve<\/dt>\r\n \t
    the set of ordered pairs [latex]\\left(f(t),g(t)\\right)[\/latex] together with their defining parametric equations\u00a0[latex]x=f(t)[\/latex] and [latex]y=g(t)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    polar axis<\/dt>\r\n \t
    the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    polar coordinate system<\/dt>\r\n \t
    a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta [\/latex], the angular coordinate<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    polar equation<\/dt>\r\n \t
    an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    polar rectangle<\/dt>\r\n \t
    the region enclosed between the circles [latex]r=a[\/latex] and [latex]r=b[\/latex]\u00a0and the angles [latex]\\theta = \\alpha[\/latex]\u00a0and [latex]\\theta = \\beta[\/latex]; it is described as\u00a0[latex]{\\bf{R}}=\\{(r,{\\theta}) | a{\\leq}r{\\leq}b, {\\alpha}{\\leq}{\\theta}{\\leq}{\\beta}\\}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    pole<\/dt>\r\n \t
    the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    potential function<\/dt>\r\n \t
    a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    principal unit normal vector<\/dt>\r\n \t
    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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    principal unit tangent vector<\/dt>\r\n \t
    a unit vector tangent to a curve [latex]C[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    projectile motion<\/dt>\r\n \t
    motion of an object with an initial velocity but no force acting on it other than gravity<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    quadric surfaces<\/dt>\r\n \t
    surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    radial coordinate<\/dt>\r\n \t
    [latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    radial field<\/dt>\r\n \t
    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<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    radius of curvature<\/dt>\r\n \t
    the reciprocal of the curvature<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    radius of gyration<\/dt>\r\n \t
    the distance from an object\u2019s center of mass to its axis of rotation<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    region<\/dt>\r\n \t
    an open, connected, nonempty subset of [latex]\\mathbb{R}^{2}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    regular parameterization<\/dt>\r\n \t
    parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]\u00a0such that [latex]{\\bf{r}}_{u}{\\times}{\\bf{r}}_{v}[\/latex] is not zero for point [latex](u, v)[\/latex] in the parameter domain<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    reparameterization<\/dt>\r\n \t
    an alternative parameterization of a given vector-valued function<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    right-hand rule<\/dt>\r\n \t
    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<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    RLC<\/em>\u00a0series circuit<\/dt>\r\n \t
    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 <\/span>RLC<\/em> series circuit<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    rose<\/dt>\r\n \t
    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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    rotational field<\/dt>\r\n \t
    a vector field in which the vector at point\u00a0[latex](x,y)[\/latex]<\/span>\u00a0is tangent to a circle with radius\u00a0[latex]r=\\sqrt{x^{2}+y^{2}}[\/latex]<\/span>\u00a0in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    rulings<\/dt>\r\n \t
    parallel lines that make up a cylindrical surface<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    saddle point<\/dt>\r\n \t
    given the function\u00a0[latex]z=f(x,y)[\/latex]\u00a0the point [latex](x_{0},y_{0},f(x_{0},y_{0}))[\/latex]\u00a0is 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]\u00a0does not have a local extremum at\u00a0[latex](x_{0},y_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    scalar<\/dt>\r\n \t
    <\/dt>\r\n \t
    a real number<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    scalar equation of a plane:<\/dt>\r\n \t
    the equation [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]\u00a0used to describe a plane containing point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0with normal vector\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0or its alternate form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    scalar line integral<\/dt>\r\n \t
    the scalar line integral of a function\u00a0[latex]f[\/latex]\u00a0along a curve [latex]C[\/latex] with respect to arc length is the integral [latex]\\displaystyle\\int_C \\! f\\, \\mathrm{d}s[\/latex],\u00a0it 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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    scalar multiplication<\/dt>\r\n \t
    a vector operation that defines the product of a scalar and a vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    scalar projection<\/dt>\r\n \t
    the magnitude of the vector projection of a vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    simple curve<\/dt>\r\n \t
    a curve that does not cross itself<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    simple harmonic motion<\/dt>\r\n \t
    motion described by the equation [latex]x(t)=c_{1}\\cos{(\\omega{t})}+c_{2}\\sin{(\\omega{t})}[\/latex]\u00a0as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    simply connected region<\/dt>\r\n \t
    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<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    skew lines:<\/dt>\r\n \t
    two lines that are not parallel but do not intersect<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    smooth<\/dt>\r\n \t
    curves where the vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex] is differentiable with a non-zero derivative<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    space curve<\/dt>\r\n \t
    the set of ordered triples\u00a0[latex]\\left(f(t),g(t),h(t)\\right)[\/latex]\u00a0together with their defining parametric equations [latex]x=f(t)[\/latex],\u00a0[latex]y=g(t)[\/latex] and [latex]z=h(t)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    space-filling curve<\/dt>\r\n \t
    a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    sphere<\/dt>\r\n \t
    the set of all points equidistant from a given point known as the center<\/em><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    spherical coordinate system<\/dt>\r\n \t
    a way to describe a location in space with an ordered triple\u00a0[latex](\\rho,\\theta,\\varphi)[\/latex],\u00a0<\/strong>where\u00a0[latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex]\u00a0and the origin [latex]\\rho \\ne\r\n{0}[\/latex],\u00a0[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]\u00a0<\/strong>where\u00a0[latex]O[\/latex]\u00a0is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    standard equation of a sphere<\/dt>\r\n \t
    [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex] describes a sphere with center [latex](a,b,c)[\/latex]\u00a0and radius [latex]r[\/latex]<\/span><\/span><\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    standard form<\/dt>\r\n \t
    an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    standard unit vectors<\/dt>\r\n \t
    unit vectors along the coordinate axes:\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex], [latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    standard-position Vectors<\/dt>\r\n \t
    a vector with initial point [latex](0,0)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    steady-state solution<\/dt>\r\n \t
    a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Stokes' theorem<\/dt>\r\n \t
    relates the flux integral over a surface <\/span>[latex]S[\/latex] to a line integral around the boundary <\/span>[latex]C[\/latex] of the surface <\/span>[latex]S[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    stream function<\/dt>\r\n \t
    if [latex]{\\bf{F}} = {\\langle}P, Q{\\rangle}[\/latex] is a source-free vector field, then stream function g<\/em> is a function such that [latex]P = g_{y}[\/latex], and\u00a0[latex]Q = -{g_{x}}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    surface<\/dt>\r\n \t
    the graph of a function of two variables, [latex]z=f(x,y)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    surface area<\/dt>\r\n \t
    the area of surface S<\/em> given by the surface integral [latex]\\displaystyle{\\int_{} {\\int_{S} d{\\bf{S}}}} [\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    surface independent<\/dt>\r\n \t
    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<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    surface integral of a scalar-valued function<\/dt>\r\n \t
    a surface integral in which the integrand is a scalar function<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    surface integral of a vector field<\/dt>\r\n \t
    a surface integral in which the integrand is a vector field<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    symmetric equations of a line:<\/dt>\r\n \t
    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\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n \t
    <\/dd>\r\n<\/dl>\r\n
    \r\n \t
    tangent plane<\/dt>\r\n \t
    given a function [latex]f(x,y)[\/latex]\u00a0that 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]\u00a0is 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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    tangent vector<\/dt>\r\n \t
    to [latex]{\\bf{r}}(t)[\/latex] at [latex]t=t_{0}[\/latex] any vector [latex]{\\bf{v}}[\/latex] such that, when the\u00a0tail of the vector is placed at point\u00a0[latex]{\\bf{r}}(t_{0})[\/latex]\u00a0on the graph, vector [latex]{\\bf{v}}[\/latex] is tangent to curve\u00a0[latex]C[\/latex]<\/dd>\r\n \t
    <\/dt>\r\n<\/dl>\r\n
    \r\n \t
    tangential component of acceleration<\/dt>\r\n \t
    the coefficient of the unit tangent vector [latex]{\\bf{T}}[\/latex] when the acceleration vector is written as a linear combination of\u00a0[latex]{\\bf{T}}[\/latex] and [latex]{\\bf{N}}[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    terminal point<\/dt>\r\n \t
    the endpoint of a vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    The Fundamental Theorem for Line Integrals<\/dt>\r\n \t
    the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    three-dimensional rectangular coordinate system<\/dt>\r\n \t
    a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple\u00a0[latex](x,y,z)[\/latex]\u00a0that plots\u00a0its\u00a0location relative to the defining axes<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    torque<\/dt>\r\n \t
    the effect of a force that causes an object to rotate<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    total differential<\/dt>\r\n \t
    the total differential of the function [latex]f(x,y)[\/latex]\u00a0at [latex](x_{0},y_{0})[\/latex]\u00a0is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    trace<\/dt>\r\n \t
    the intersection of a three-dimensional surface with a coordinate plane<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    transformation<\/dt>\r\n \t
    a function that transforms a region [latex]G[\/latex] in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    tree diagram<\/dt>\r\n \t
    illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    triangle inequality<\/dt>\r\n \t
    the length of any side of a triangle is less than the sum of the lengths of the other two sides<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    triangle method<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    triple integral<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    triple integral in cylindrical coordinates<\/dt>\r\n \t
    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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    triple integral in spherical coordinates<\/dt>\r\n \t
    the limit of a triple Riemann sum, provided the following limit exists:\u00a0[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]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    triple scalar product<\/dt>\r\n \t
    the dot product of a vector with the cross product of two other vectors: [latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Type I<\/dt>\r\n \t
    a region\u00a0[latex]\\bf{D}[\/latex] in the [latex]xy[\/latex]-plane is Type I\u00a0if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[\/latex] and\u00a0[latex]g_{2}(x)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    Type II<\/dt>\r\n \t
    a region [latex]\\bf{D}[\/latex]\u00a0<\/strong>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]\u00a0and\u00a0[latex]h_{2}(y)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    unit vector<\/dt>\r\n \t
    a vector with magnitude [latex]1[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    unit vector field<\/dt>\r\n \t
    a vector field in which the magnitude of every vector is [latex]1[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector<\/dt>\r\n \t
    a mathematical object that has both magnitude and direction<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector addition<\/dt>\r\n \t
    a vector operation that defines the sum of two vectors<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector difference<\/dt>\r\n \t
    the vector difference\u00a0[latex]{\\bf{v}}-{\\bf{w}}[\/latex]\u00a0is defined as\u00a0[latex]{\\bf{v}}+(-{\\bf{w}})={\\bf{v}}+(-1){\\bf{w}}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector equation of a line:<\/dt>\r\n \t
    the equation\u00a0[latex]{\\bf{r}} ={\\bf{r}}_{0}+t{\\bf{v}}[\/latex]\u00a0used 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],\u00a0<\/strong>where\u00a0[latex]{\\bf{r}}_{0}=\\langle{x_{0},y_{0},z_{0}}\\rangle[\/latex]\u00a0is the position vector of point [latex]P[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector equation of a plane:<\/dt>\r\n \t
    the equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex],\r\nwhere [latex]P[\/latex]\u00a0is a given point in the plane, [latex]Q[\/latex] is any point in the plane, and\u00a0[latex]{\\bf{n}}[\/latex] is a normal vector of the plane<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector field<\/dt>\r\n \t
    measured in [latex]\\mathbb{R}^{2}[\/latex],\u00a0an assignment of a vector [latex]{\\bf{F}}(x,y)[\/latex]<\/span>\u00a0to each point\u00a0[latex](x,y)[\/latex]<\/span>\u00a0of a subset\u00a0[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{2}[\/latex]; in [latex]\\mathbb{R}^{3}[\/latex],\u00a0an assignment of a vector [latex]{\\bf{F}}(x,y,z)[\/latex]<\/span> to each point\u00a0[latex](x,y,z)[\/latex]<\/span>\u00a0of a subset\u00a0[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{3}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector line integral<\/dt>\r\n \t
    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]\u00a0with unit tangent vector [latex]{\\bf{T}}[\/latex]\u00a0of [latex]C[\/latex] with respect to arc length, [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex];\u00a0such an integral is defined in terms of a Riemann sum, similar to a single-variable integral<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector parameterization<\/dt>\r\n \t
    any representation of a plane or space curve using a vector-valued function<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector product<\/dt>\r\n \t
    the cross product of two vectors<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector projection<\/dt>\r\n \t
    the component of a vector that follows a given direction<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector sum<\/dt>\r\n \t
    the sum of two vectors, [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex]can be constructed graphically by placing the initial point of\u00a0[latex]{\\bf{w}}[\/latex]\u00a0at the terminal point of [latex]{\\bf{v}}[\/latex]; then the vector sum\u00a0[latex]{\\bf{v}}+{\\bf{w}}[\/latex]\u00a0is 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\u00a0[latex]{\\bf{w}}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vector-valued function<\/dt>\r\n \t
    a function of the form [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex],\u00a0<\/strong>where the component functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are real-valued functions of the parameter [latex]t[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    velocity vector<\/dt>\r\n \t
    the derivative of the position vector<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vertex<\/dt>\r\n \t
    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<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    vertical trace<\/dt>\r\n \t
    the set of ordered triples [latex](c,y,z)[\/latex]\u00a0<\/strong>that solves the equation [latex]f(c,y)=z[\/latex]\u00a0for a given constant [latex]x=c[\/latex]\u00a0or the set of ordered triples [latex](x,d,z)[\/latex]\u00a0that solves the equation [latex]f(x,d)=z[\/latex]\u00a0for a given constant [latex]y=d[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    work done by a force<\/dt>\r\n \t
    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]\u00a0and [latex]\\bf{s}[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    zero vector<\/dt>\r\n \t
    the vector with both initial point and terminal point\u00a0[latex](0,0)[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    [latex]\\delta[\/latex]<\/strong>\u00a0ball<\/dt>\r\n \t
    all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than\u00a0[latex]\\delta[\/latex]\u00a0from [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n
    \r\n \t
    [latex]\\delta[\/latex] d<\/strong>isk<\/dt>\r\n \t
    an open disk of radius [latex]\\delta[\/latex]\u00a0centered at point\u00a0[latex](a,b)[\/latex]<\/dd>\r\n<\/dl>","rendered":"
    \n
    acceleration vector<\/dt>\n
    the second derivative of the position vector<\/dd>\n<\/dl>\n
    \n
    angular coordinate<\/dt>\n
    [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<\/em>) axis, measured counterclockwise<\/dd>\n<\/dl>\n
    \n
    arc-length function<\/dt>\n
    a function\u00a0[latex]s(t)[\/latex]\u00a0that describes the arc length of curve\u00a0[latex]C[\/latex]\u00a0as a function of\u00a0[latex]t[\/latex]<\/dd>\n<\/dl>\n
    \n
    arc-length parameterization<\/dt>\n
    a reparameterization of a vector-valued function in which the parameter is equal to the arc length<\/dd>\n<\/dl>\n
    \n
    binormal vector<\/dt>\n
    a unit vector orthogonal to the unit tangent vector and the unit normal vector<\/dd>\n<\/dl>\n
    \n
    boundary conditions<\/dt>\n
    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<\/span><\/dd>\n<\/dl>\n
    \n
    boundary point<\/dt>\n
    a point\u00a0[latex]P_{0}[\/latex]\u00a0of [latex]R[\/latex]\u00a0is a boundary point if every [latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contains points both inside and outside [latex]R[\/latex]<\/dd>\n<\/dl>\n
    \n
    boundary-value problem<\/dt>\n
    a differential equation with associated boundary conditions<\/span><\/dd>\n<\/dl>\n
    \n
    cardioid<\/dt>\n
    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]<\/dd>\n<\/dl>\n
    \n
    characteristic equation<\/dt>\n
    the equation [latex]a\\lambda^{2}+b\\lambda+c=0[\/latex]\u00a0<\/strong>for the differential equation [latex]ay^{\\prime\\prime}+by^{\\prime}+cy=0[\/latex]<\/dd>\n<\/dl>\n
    \n
    circulation<\/dt>\n
    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]\u00a0along [latex]C[\/latex] is line integral [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex], which we also denote\u00a0[latex]\\displaystyle\\oint_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex]<\/dd>\n<\/dl>\n
    \n
    closed curve<\/dt>\n
    a curve that begins and ends at the same point<\/span><\/dd>\n<\/dl>\n
    \n
    closed curve<\/dt>\n
    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],\u00a0and the curve is traversed exactly once<\/dd>\n<\/dl>\n
    \n
    closed set<\/dt>\n
    a set [latex]S[\/latex]\u00a0<\/strong>that contains all its boundary points<\/dd>\n<\/dl>\n
    \n
    complementary equation<\/dt>\n
    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]\u00a0the associated homogeneous equation, called the complementary equation<\/em>, is\u00a0[latex]a_{2}(x)y^{\\prime\\prime}+a_{1}(x)y^{\\prime}+a_{0}(x)y=0[\/latex]<\/dd>\n<\/dl>\n
    \n
    component<\/dt>\n
    a scalar that describes either the vertical or horizontal direction of a vector<\/dd>\n<\/dl>\n
    \n
    component functions<\/dt>\n
    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\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex] are\u00a0[latex]f(t)[\/latex],\u00a0[latex]g(t)[\/latex] and [latex]h(t)[\/latex]<\/dd>\n<\/dl>\n
    \n
    conic section<\/dt>\n
    a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\n<\/dl>\n
    \n
    connected region<\/dt>\n
    a region in which any two points can be connected by a path with a trace contained entirely inside the region<\/span><\/dd>\n<\/dl>\n
    \n
    connected set<\/dt>\n
    an open set [latex]S[\/latex]\u00a0that cannot be represented as the union of two or more disjoint, nonempty open subsets<\/dd>\n<\/dl>\n
    \n
    conservative field<\/dt>\n
    a vector field for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\n<\/dl>\n
    \n
    constraint<\/dt>\n
    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<\/dd>\n<\/dl>\n
    \n
    contour map<\/dt>\n
    a plot of the various level curves of a given function [latex]f(x,y)[\/latex]<\/dd>\n<\/dl>\n
    \n
    coordinate plane<\/dt>\n
    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<\/span><\/dd>\n<\/dl>\n
    \n
    critical point of a function of two variables<\/dt>\n
    the point\u00a0[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:<\/p>\n
      \n
    1. [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[\/latex]<\/li>\n
    2. At least one of [latex]f_{x}(x_{0},y_{0})[\/latex] and [latex]f_{y}(x_{0},y_{0})[\/latex]\u00a0<\/strong>do not exist<\/li>\n<\/ol>\n<\/dd>\n<\/dl>\n
      \n
      cross product<\/dt>\n
      [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\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\n<\/dl>\n
      \n
      curl<\/dt>\n
      the curl of vector field [latex]{\\bf{F}}=\\langle{P,Q,R}\\rangle[\/latex], denoted [latex]\\nabla\\times{\\bf{F}}[\/latex]\u00a0is the \u201cdeterminant\u201d 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\u00a0[latex](R_{y}-Q_{z}){\\bf{i}}+(P_{z}-R_{x}){\\bf{j}}+(Q_{x}+P_{y}){\\bf{k}}[\/latex];\u00a0it measures the tendency of particles at a point to rotate about the axis that points in the direction of the curl at the point<\/dd>\n<\/dl>\n
      \n
      curvature<\/dt>\n
      the derivative of the unit tangent vector with respect to the arc-length parameter<\/dd>\n<\/dl>\n
      \n
      cusp<\/dt>\n
      a pointed end or part where two curves meet<\/dd>\n<\/dl>\n
      \n
      cycloid<\/dt>\n
      the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\n<\/dl>\n
      \n
      cylinder<\/dt>\n
      a set of lines parallel to a given line passing through a given curve<\/dd>\n<\/dl>\n
      \n
      cylindrical coordinate system<\/dt>\n
      a way to describe a location in space with an ordered triple [latex](r,\\theta,z)[\/latex],\u00a0<\/strong>where\u00a0[latex](r,\\theta)[\/latex] represents the polar coordinates of the point\u2019s projection in the [latex]xy[\/latex]-plane, and [latex]z[\/latex] represents the point’s projection onto the [latex]z[\/latex]-axis<\/dd>\n<\/dl>\n
      \n
      definite integral of a vector-valued function<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      derivative of a vector-valued function<\/dt>\n
      the derivative of a vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex]\u00a0<\/strong>is\u00a0[latex]{\\bf{r}}^{\\prime}(t)=\\underset{\\Delta{t}\\to{0}}{\\lim}\\frac{{\\bf{r}}(t+\\Delta{t})-{\\bf{r}}(t)}{\\Delta{t}}[\/latex],\u00a0<\/strong>provided the limit exists<\/dd>\n<\/dl>\n
      \n
      determinant<\/dt>\n
      a real number associated with a square matrix<\/dd>\n<\/dl>\n
      \n
      differentiable<\/dt>\n
      a function [latex]f(x,y,z)[\/latex]\u00a0is differentiable at\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0if [latex]f(x,y)[\/latex]\u00a0can 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]\u00a0satisfies [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]<\/dd>\n<\/dl>\n
      \n
      direction angles<\/dt>\n
      the angles formed by a nonzero vector and the coordinate axes<\/dd>\n<\/dl>\n
      \n
      direction cosines<\/dt>\n
      the cosines of the angles formed by a nonzero vector and the coordinate axes<\/dd>\n<\/dl>\n
      \n
      direction vector<\/dt>\n
      a vector parallel to a line that is used to describe the direction, or orientation, of the line in space<\/dd>\n<\/dl>\n
      \n
      directional derivative<\/dt>\n
      the derivative of a function in the direction of a given unit vector<\/dd>\n<\/dl>\n
      \n
      directrix<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      discriminant<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      discriminant<\/dt>\n
      the discriminant of the function [latex]f(x,y)[\/latex]\u00a0is given by the formula\u00a0[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\\left(f_{xy}(x_{0},y_{0})\\right)^{2}[\/latex]<\/dd>\n<\/dl>\n
      \n
      divergence<\/dt>\n
      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 \u201coutflowing-ness\u201d of a vector field<\/dd>\n<\/dl>\n
      \n
      divergence theorem<\/dt>\n
      a theorem used to transform a difficult flux integral into an easier triple integral and vice versa<\/span><\/dd>\n<\/dl>\n
      \n
      dot product or scalar product<\/dt>\n
      [latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex]\u00a0and[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\n<\/dl>\n
      \n
      double Riemann Sum<\/dt>\n
      of the function [latex]f(x,y)[\/latex]\u00a0over\u00a0a rectangular region [latex]R[\/latex]\u00a0is [latex]\\displaystyle\\sum_{i=1}^{m} {} \\displaystyle\\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}[\/latex] where [latex]R[\/latex]\u00a0is divided into smaller sub rectangles [latex]R_{ij}[\/latex] and [latex]({x^{*}}_{i,j}, {y^{*}}_{i,j})[\/latex]\u00a0is an arbitrary point in\u00a0[latex]R_{ij}[\/latex]<\/dd>\n<\/dl>\n
      \n
      double Integral<\/dt>\n
      of the function [latex]f(x,y)[\/latex]\u00a0<\/strong>over the region [latex]R[\/latex]\u00a0in the [latex]xy[\/latex]-plane\u00a0is defined as the limit of a double Riemann sum,\u00a0[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]<\/dd>\n<\/dl>\n
      \n
      eccentricity<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      ellipsoid<\/dt>\n
      a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]\u00a0all traces of this surface are ellipses<\/dd>\n<\/dl>\n
      \n
      elliptic cone<\/dt>\n
      a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0[\/latex]\u00a0traces of this surface include ellipses and intersecting lines<\/dd>\n<\/dl>\n
      \n
      elliptic paraboloid<\/dt>\n
      a three-dimensional surface described by an equation of the form\u00a0[latex]z=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}[\/latex]\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n
      \n
      equivalent vectors<\/dt>\n
      vectors that have the same magnitude and the same direction<\/dd>\n<\/dl>\n
      \n
      flux<\/dt>\n
      the rate of a fluid flowing across a curve in a vector field; the flux of vector field [latex]{\\bf{F}}[\/latex]\u00a0across 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]<\/dd>\n<\/dl>\n
      \n
      flux integral<\/dt>\n
      another name for a surface integral of a vector field; the preferred term in physics and engineering<\/span><\/dd>\n<\/dl>\n
      \n
      focal parameter<\/dt>\n
      the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\n<\/dl>\n
      \n
      focus<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      Frenet frame of reference<\/dt>\n
      (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector<\/dd>\n<\/dl>\n
      \n
      Fubini’s Theorem<\/dt>\n
      if\u00a0[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,<\/dd>\n
      [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]<\/p>\n
      <\/dl>\n<\/dd>\n<\/dl>\n
      \n
      function of two variables<\/dt>\n
      a function [latex]z=f(x,y)[\/latex] that maps each ordered pair [latex](x,y)[\/latex]\u00a0<\/strong>in a subset [latex]D[\/latex]\u00a0of [latex]\\mathbb{R}^{2}[\/latex]\u00a0<\/strong>to a unique real number [latex]z[\/latex]<\/dd>\n<\/dl>\n
      \n
      Fundamental Theorem for Line Integrals<\/dt>\n
      the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\n<\/dl>\n
      \n
      Gauss’ law<\/dt>\n
      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]\u00a0is [latex]Q|{\\varepsilon}_{0}[\/latex]<\/dd>\n<\/dl>\n
      \n
      general form<\/dt>\n
      an equation of a conic section written as a general second-degree equation<\/dd>\n<\/dl>\n
      \n
      general form of the equation of a plane<\/dt>\n
      an equation in the form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[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]\u00a0is a point on the plane, and\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\n<\/dl>\n
      \n
      generalized chain rule<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      gradient<\/dt>\n
      the gradient of the function [latex]f(x,y)[\/latex]\u00a0is defined to be [latex]\\nabla f(x,y)=(\\partial{f}{\/}\\partial{x}){\\bf{i}}+(\\partial{f}{\/}\\partial{y}){\\bf{j}}[\/latex]\u00a0which can be generalized to a function of any number of independent variables<\/dd>\n<\/dl>\n
      \n
      gradient field<\/dt>\n
      a vector field [latex]{\\bf{F}}[\/latex]\u00a0for which there exists a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]\u00a0in other words, a vector field that is the gradient of a function; such vector fields are also called conservative<\/em><\/dd>\n<\/dl>\n
      \n
      graph of a function of two variables<\/dt>\n
      a set of ordered triples\u00a0[latex](x,y,z)[\/latex]\u00a0that satisfies the equation [latex]z=f(x,y)[\/latex]\u00a0plotted in three-dimensional Cartesian space<\/dd>\n<\/dl>\n
      \n
      Green’s theorem<\/dt>\n
      relates the integral over a connected region to an integral over the boundary of the region<\/span><\/dd>\n<\/dl>\n
      \n
      grid curves<\/dt>\n
      curves on a surface that are parallel to grid lines in a coordinate plane<\/span><\/dd>\n<\/dl>\n
      \n
      heat flow<\/dt>\n
      a vector field proportional to the negative temperature gradient in an object<\/span><\/dd>\n<\/dl>\n
      \n
      helix<\/dt>\n
      a three-dimensional curve in the shape of a spiral<\/dd>\n<\/dl>\n
      \n
      higher-order partial derivatives<\/dt>\n
      second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives<\/dd>\n<\/dl>\n
      \n
      homogeneous linear equation<\/dt>\n
      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]\u00a0but [latex]r(x)=0[\/latex]\u00a0for every value of [latex]x[\/latex]<\/dd>\n<\/dl>\n
      \n
      hyperboloid of one sheet<\/dt>\n
      a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n
      \n
      hyperboloid of two sheets<\/dt>\n
      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]\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n
      \n
      improper double integral<\/dt>\n
      a double integral over an unbounded region or of an unbounded function<\/span><\/dd>\n<\/dl>\n
      \n
      indefinite integral of a vector-valued function<\/dt>\n
      a vector-valued function with a derivative that is equal to a given vector-valued function<\/dd>\n<\/dl>\n
      \n
      independent of path (path independent)<\/dt>\n
      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]\u00a0for any curves [latex]C_{1}[\/latex]\u00a0and [latex]C_{2}[\/latex]\u00a0in the domain of [latex]{\\bf{F}}[\/latex] with the same initial points and terminal points<\/dd>\n<\/dl>\n
      \n
      initial point<\/dt>\n
      the starting point of a vector<\/dd>\n<\/dl>\n
      \n
      interior point<\/dt>\n
      a point\u00a0[latex]P_{0}[\/latex]\u00a0of\u00a0[latex]R[\/latex]\u00a0is a boundary point if there is a\u00a0[latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contained completely in\u00a0[latex]R[\/latex]<\/dd>\n<\/dl>\n
      \n
      intermediate variable<\/dt>\n
      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\u00a0[latex]f\\left(x(t),y(t)\\right)[\/latex]<\/span>\u00a0the variables [latex]x[\/latex]\u00a0and\u00a0[latex]y[\/latex]\u00a0<\/strong>are examples of intermediate variables<\/dd>\n<\/dl>\n
      \n
      inverse-square law<\/dt>\n
      the electrostatic force at a given point is inversely proportional to the square of the distance from the source of the charge<\/span><\/dd>\n<\/dl>\n
      \n
      iterated Integral<\/dt>\n
      for a function [latex]f(x,y)[\/latex] over the region [latex]\\bf{R}[\/latex] is<\/p>\n

      [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]<\/span><\/p>\n

      [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]<\/span><\/dd>\n<\/dl>\n

      \n
      Jacobian<\/dt>\n
      the Jacobian [latex]J(u ,v)[\/latex] in two variables is a [latex]2{\\times}2[\/latex] determinant:<\/dd>\n
      [latex]J(u,v) = \\begin{vmatrix}\\frac{dx}{du} & \\frac{dy}{du}\\\\\\frac{dx}{dv} & \\frac{dy}{dv}\\end{vmatrix}[\/latex]<\/dd>\n
      the Jacobian [latex]J(u ,v, w)[\/latex]\u00a0in three variables is a [latex]3{\\times}3[\/latex]\u00a0determinant:<\/dd>\n
      [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]<\/dd>\n<\/dl>\n
      \n
      Kepler’s laws of planetary motion<\/dt>\n
      three laws governing the motion of planets, asteroids, and comets in orbit around the Sun<\/dd>\n<\/dl>\n
      \n
      Lagrange Multiplier<\/dt>\n
      the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable\u00a0[latex]\\lambda[\/latex]<\/dd>\n<\/dl>\n
      \n
      level curve of a function of two variables<\/dt>\n
      the set of points satisfying the equation [latex]f(x,y)=c[\/latex]\u00a0<\/strong>for some real number [latex]c[\/latex]\u00a0in the range of [latex]f[\/latex]<\/dd>\n<\/dl>\n
      \n
      level surface of a function of three variables<\/dt>\n
      the set of points satisfying the equation [latex]f(x,y,z)=c[\/latex]\u00a0<\/strong>for some real number [latex]c[\/latex] in the range of [latex]f[\/latex]<\/dd>\n<\/dl>\n
      \n
      lima\u00e7on<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      limit of a vector-valued function<\/dt>\n
      a vector-valued function [latex]{\\bf{r}}(t)[\/latex] has a limit [latex]{\\bf{L}}[\/latex] as [latex]t[\/latex]\u00a0<\/em>approaches\u00a0[latex]a[\/latex]\u00a0if [latex]\\underset{t\\to{a}}{\\lim}|{\\bf{r}}(t)-{\\bf{L}}|=0[\/latex]<\/dd>\n<\/dl>\n
      \n
      line integral<\/dt>\n
      the integral of a function along a curve in a plane or in space<\/span><\/dd>\n<\/dl>\n
      \n
      linear approximation<\/dt>\n
      given a function [latex]f(x,y)[\/latex]\u00a0and a tangent plane to the function at a point [latex](x_{0},y_{0})[\/latex] we can approximate [latex]f(x,y)[\/latex]\u00a0for points near [latex](x_{0},y_{0})[\/latex] using the tangent plane formula<\/dd>\n<\/dl>\n
      \n
      linearly dependent<\/dt>\n
      a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for 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<\/dd>\n<\/dl>\n
      \n
      linearly independent<\/dt>\n
      a set of function [latex]f_{1}(x),f_{2}(x),\\ldots f_{n}(x)[\/latex]\u00a0for which there are no constants, such that\u00a0[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]\u00a0<\/span>for all [latex]x[\/latex]<\/span>\u00a0in the interval of interest<\/dd>\n<\/dl>\n
      \n
      magnitude<\/dt>\n
      the length of a vector<\/dd>\n<\/dl>\n
      \n
      major axis<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      mass flux<\/dt>\n
      the rate of mass flow of a fluid per unit area, measured in mass per unit time per unit area<\/span><\/dd>\n<\/dl>\n
      \n
      method of Lagrange multipliers<\/dt>\n
      a method of solving an optimization problem subject to one or more constraints<\/dd>\n<\/dl>\n
      \n
      method of undetermined coefficients<\/dt>\n
      a method that involves making a guess about the form of the particular solution, then solving for the coefficients in the guess<\/span><\/dd>\n<\/dl>\n
      \n
      method of variation of parameters<\/dt>\n
      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]\u00a0and [latex]y_{2}[\/latex]\u00a0are linearly independent solutions to the complementary equations, and then solving a system of equations to find [latex]u(x)[\/latex]\u00a0and\u00a0[latex]v(x)[\/latex].<\/dd>\n<\/dl>\n
      \n
      minor axis<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      mixed partial derivatives<\/dt>\n
      second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables<\/dd>\n<\/dl>\n
      \n
      nappe<\/dt>\n
      a nappe is one half of a double cone<\/dd>\n<\/dl>\n
      \n
      nonhomogeneous linear equation<\/dt>\n
      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]\u00a0but [latex]r(x)\\ne 0[\/latex]\u00a0for some value of [latex]x[\/latex]<\/dd>\n<\/dl>\n
      \n
      normal component of acceleration<\/dt>\n
      the coefficient of the unit normal vector [latex]{\\bf{N}}[\/latex]\u00a0<\/span>when the acceleration vector is written as a linear combination of\u00a0[latex]{\\bf{T}}[\/latex]\u00a0and\u00a0[latex]{\\bf{N}}[\/latex]<\/span><\/dd>\n<\/dl>\n
      \n
      normal plane<\/dt>\n
      a plane that is perpendicular to a curve at any point on the curve<\/dd>\n<\/dl>\n
      \n
      normal vector<\/dt>\n
      a vector perpendicular to a plane<\/dd>\n<\/dl>\n
      \n
      normalization<\/dt>\n
      using scalar multiplication to find a unit vector with a given direction<\/dd>\n<\/dl>\n
      \n
      objective function<\/dt>\n
      the function that is to be maximized or minimized in an optimization problem<\/dd>\n<\/dl>\n
      \n
      octants<\/dt>\n
      the eight regions of space created by the coordinate planes<\/dd>\n<\/dl>\n
      \n
      one-to-one transformation<\/dt>\n
      a transformation [latex]T : G {\\rightarrow} R[\/latex]\u00a0defined as [latex]T(u, v) = (x, y)[\/latex]\u00a0is said to be one-to-one if no two points map to the same image point<\/dd>\n<\/dl>\n
      \n
      open set<\/dt>\n
      a set\u00a0[latex]S[\/latex]\u00a0that contains none of its boundary points<\/dd>\n<\/dl>\n
      \n
      optimization problem<\/dt>\n
      calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers<\/dd>\n<\/dl>\n
      \n
      orientation<\/dt>\n
      the direction that a point moves on a graph as the parameter increases<\/dd>\n<\/dl>\n
      \n
      orientation of a curve<\/dt>\n
      the orientation of a curve [latex]C[\/latex]<\/span>\u00a0is a specified direction of [latex]C[\/latex]<\/span><\/dd>\n<\/dl>\n
      \n
      orientation of a surface<\/dt>\n
      if a surface has an \u201cinner\u201d side and an \u201couter\u201d side, then an orientation is a choice of the inner or the outer side; the surface could also have \u201cupward\u201d and \u201cdownward\u201d orientations<\/span><\/dd>\n<\/dl>\n
      \n
      orthogonal vectors<\/dt>\n
      vectors that form a right angle when placed in standard position<\/dd>\n<\/dl>\n
      \n
      osculating circle<\/dt>\n
      a circle that is tangent to a curve\u00a0[latex]C[\/latex]\u00a0at a point [latex]P[\/latex]\u00a0<\/span>and that shares the same curvature<\/span><\/dd>\n<\/dl>\n
      \n
      osculating plane<\/dt>\n
      the plane determined by the unit tangent and the unit normal vector<\/dd>\n<\/dl>\n
      \n
      parallelogram method<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      parallelpiped<\/dt>\n
      a three-dimensional prism with six faces that are parallelograms<\/dd>\n<\/dl>\n
      \n
      parameter<\/dt>\n
      an independent variable that both x<\/em> and y<\/em> depend on in a parametric curve; usually represented by the variable t<\/em><\/dd>\n<\/dl>\n
      \n
      parameter domain (parameter space)<\/dt>\n
      the region of the uv<\/em> plane over which the parameters u<\/em> and v<\/em> vary for parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]<\/dd>\n<\/dl>\n
      \n
      parameterized surface<\/dt>\n
      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] ,\u00a0where the parameters u<\/em> and v<\/em> vary over a parameter domain in the uv<\/em>-plane<\/dd>\n<\/dl>\n
      \n
      parametric curve<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      parametric equations<\/dt>\n
      the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\n<\/dl>\n
      \n
      parametric equations of a line:<\/dt>\n
      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\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n<\/dl>\n
      \n
      parameterization of a curve<\/dt>\n
      rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\n<\/dl>\n
      \n
      partial derivative<\/dt>\n
      a derivative of a function of more than one independent variable in which all the variables but one are held constant<\/dd>\n<\/dl>\n
      \n
      partial differential equation<\/dt>\n
      an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives<\/dd>\n<\/dl>\n
      \n
      particular solution<\/dt>\n
      a solution [latex]y_{p}(x)[\/latex]\u00a0of a differential equation that contains no arbitrary constants<\/dd>\n<\/dl>\n
      \n
      piecewise smooth curve<\/dt>\n
      an oriented curve that is not smooth, but can be written as the union of finitely many smooth curves<\/span><\/dd>\n<\/dl>\n
      \n
      planar transformation<\/dt>\n
      a function [latex]T[\/latex]\u00a0that transforms a region [latex]G[\/latex]\u00a0in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\n<\/dl>\n
      \n
      plane curve<\/dt>\n
      the set of ordered pairs [latex]\\left(f(t),g(t)\\right)[\/latex] together with their defining parametric equations\u00a0[latex]x=f(t)[\/latex] and [latex]y=g(t)[\/latex]<\/dd>\n<\/dl>\n
      \n
      polar axis<\/dt>\n
      the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\n<\/dl>\n
      \n
      polar coordinate system<\/dt>\n
      a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta[\/latex], the angular coordinate<\/dd>\n<\/dl>\n
      \n
      polar equation<\/dt>\n
      an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\n<\/dl>\n
      \n
      polar rectangle<\/dt>\n
      the region enclosed between the circles [latex]r=a[\/latex] and [latex]r=b[\/latex]\u00a0and the angles [latex]\\theta = \\alpha[\/latex]\u00a0and [latex]\\theta = \\beta[\/latex]; it is described as\u00a0[latex]{\\bf{R}}=\\{(r,{\\theta}) | a{\\leq}r{\\leq}b, {\\alpha}{\\leq}{\\theta}{\\leq}{\\beta}\\}[\/latex]<\/dd>\n<\/dl>\n
      \n
      pole<\/dt>\n
      the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\n<\/dl>\n
      \n
      potential function<\/dt>\n
      a scalar function [latex]f[\/latex]\u00a0such that [latex]\\nabla{f}={\\bf{F}}[\/latex]<\/dd>\n<\/dl>\n
      \n
      principal unit normal vector<\/dt>\n
      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]<\/dd>\n<\/dl>\n
      \n
      principal unit tangent vector<\/dt>\n
      a unit vector tangent to a curve [latex]C[\/latex]<\/dd>\n<\/dl>\n
      \n
      projectile motion<\/dt>\n
      motion of an object with an initial velocity but no force acting on it other than gravity<\/dd>\n<\/dl>\n
      \n
      quadric surfaces<\/dt>\n
      surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)<\/dd>\n<\/dl>\n
      \n
      radial coordinate<\/dt>\n
      [latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\n<\/dl>\n
      \n
      radial field<\/dt>\n
      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<\/span><\/dd>\n<\/dl>\n
      \n
      radius of curvature<\/dt>\n
      the reciprocal of the curvature<\/dd>\n<\/dl>\n
      \n
      radius of gyration<\/dt>\n
      the distance from an object\u2019s center of mass to its axis of rotation<\/span><\/dd>\n<\/dl>\n
      \n
      region<\/dt>\n
      an open, connected, nonempty subset of [latex]\\mathbb{R}^{2}[\/latex]<\/dd>\n<\/dl>\n
      \n
      regular parameterization<\/dt>\n
      parameterization [latex]{\\bf{r}}(u, v) = {\\langle} x (u, v), y (u, v), z (u, v) {\\rangle}[\/latex]\u00a0such that [latex]{\\bf{r}}_{u}{\\times}{\\bf{r}}_{v}[\/latex] is not zero for point [latex](u, v)[\/latex] in the parameter domain<\/dd>\n<\/dl>\n
      \n
      reparameterization<\/dt>\n
      an alternative parameterization of a given vector-valued function<\/dd>\n<\/dl>\n
      \n
      right-hand rule<\/dt>\n
      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<\/span><\/dd>\n<\/dl>\n
      \n
      RLC<\/em>\u00a0series circuit<\/dt>\n
      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 <\/span>RLC<\/em> series circuit<\/span><\/dd>\n<\/dl>\n
      \n
      rose<\/dt>\n
      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]<\/dd>\n<\/dl>\n
      \n
      rotational field<\/dt>\n
      a vector field in which the vector at point\u00a0[latex](x,y)[\/latex]<\/span>\u00a0is tangent to a circle with radius\u00a0[latex]r=\\sqrt{x^{2}+y^{2}}[\/latex]<\/span>\u00a0in a rotational field, all vectors flow either clockwise or counterclockwise, and the magnitude of a vector depends only on its distance from the origin<\/dd>\n<\/dl>\n
      \n
      rulings<\/dt>\n
      parallel lines that make up a cylindrical surface<\/dd>\n<\/dl>\n
      \n
      saddle point<\/dt>\n
      given the function\u00a0[latex]z=f(x,y)[\/latex]\u00a0the point [latex](x_{0},y_{0},f(x_{0},y_{0}))[\/latex]\u00a0is 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]\u00a0does not have a local extremum at\u00a0[latex](x_{0},y_{0})[\/latex]<\/dd>\n<\/dl>\n
      \n
      scalar<\/dt>\n
      <\/dt>\n
      a real number<\/dd>\n<\/dl>\n
      \n
      scalar equation of a plane:<\/dt>\n
      the equation [latex]a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0[\/latex]\u00a0used to describe a plane containing point\u00a0[latex]P=(x_{0},y_{0},z_{0})[\/latex]\u00a0with normal vector\u00a0[latex]{\\bf{n}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0or its alternate form\u00a0[latex]ax+by+cz+d=0[\/latex], where\u00a0[latex]d=-ax_{0}-by_{0}-cz_{0}[\/latex]<\/dd>\n<\/dl>\n
      \n
      scalar line integral<\/dt>\n
      the scalar line integral of a function\u00a0[latex]f[\/latex]\u00a0along a curve [latex]C[\/latex] with respect to arc length is the integral [latex]\\displaystyle\\int_C \\! f\\, \\mathrm{d}s[\/latex],\u00a0it 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<\/dd>\n<\/dl>\n
      \n
      scalar multiplication<\/dt>\n
      a vector operation that defines the product of a scalar and a vector<\/dd>\n<\/dl>\n
      \n
      scalar projection<\/dt>\n
      the magnitude of the vector projection of a vector<\/dd>\n<\/dl>\n
      \n
      simple curve<\/dt>\n
      a curve that does not cross itself<\/span><\/dd>\n<\/dl>\n
      \n
      simple harmonic motion<\/dt>\n
      motion described by the equation [latex]x(t)=c_{1}\\cos{(\\omega{t})}+c_{2}\\sin{(\\omega{t})}[\/latex]\u00a0as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely<\/dd>\n<\/dl>\n
      \n
      simply connected region<\/dt>\n
      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<\/span><\/dd>\n<\/dl>\n
      \n
      skew lines:<\/dt>\n
      two lines that are not parallel but do not intersect<\/dd>\n<\/dl>\n
      \n
      smooth<\/dt>\n
      curves where the vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex] is differentiable with a non-zero derivative<\/dd>\n<\/dl>\n
      \n
      space curve<\/dt>\n
      the set of ordered triples\u00a0[latex]\\left(f(t),g(t),h(t)\\right)[\/latex]\u00a0together with their defining parametric equations [latex]x=f(t)[\/latex],\u00a0[latex]y=g(t)[\/latex] and [latex]z=h(t)[\/latex]<\/dd>\n<\/dl>\n
      \n
      space-filling curve<\/dt>\n
      a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\n<\/dl>\n
      \n
      sphere<\/dt>\n
      the set of all points equidistant from a given point known as the center<\/em><\/dd>\n<\/dl>\n
      \n
      spherical coordinate system<\/dt>\n
      a way to describe a location in space with an ordered triple\u00a0[latex](\\rho,\\theta,\\varphi)[\/latex],\u00a0<\/strong>where\u00a0[latex]\\rho[\/latex]\u00a0is the distance between\u00a0[latex]P[\/latex]\u00a0and the origin [latex]\\rho \\ne {0}[\/latex],\u00a0[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]\u00a0<\/strong>where\u00a0[latex]O[\/latex]\u00a0is the origin and [latex]0\\le\\varphi\\le\\pi[\/latex]<\/dd>\n<\/dl>\n
      \n
      standard equation of a sphere<\/dt>\n
      [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex] describes a sphere with center [latex](a,b,c)[\/latex]\u00a0and radius [latex]r[\/latex]<\/span><\/span><\/span><\/dd>\n<\/dl>\n
      \n
      standard form<\/dt>\n
      an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\n<\/dl>\n
      \n
      standard unit vectors<\/dt>\n
      unit vectors along the coordinate axes:\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex], [latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]<\/dd>\n<\/dl>\n
      \n
      standard-position Vectors<\/dt>\n
      a vector with initial point [latex](0,0)[\/latex]<\/dd>\n<\/dl>\n
      \n
      steady-state solution<\/dt>\n
      a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution<\/span><\/dd>\n<\/dl>\n
      \n
      Stokes’ theorem<\/dt>\n
      relates the flux integral over a surface <\/span>[latex]S[\/latex] to a line integral around the boundary <\/span>[latex]C[\/latex] of the surface <\/span>[latex]S[\/latex]<\/dd>\n<\/dl>\n
      \n
      stream function<\/dt>\n
      if [latex]{\\bf{F}} = {\\langle}P, Q{\\rangle}[\/latex] is a source-free vector field, then stream function g<\/em> is a function such that [latex]P = g_{y}[\/latex], and\u00a0[latex]Q = -{g_{x}}[\/latex]<\/dd>\n<\/dl>\n
      \n
      surface<\/dt>\n
      the graph of a function of two variables, [latex]z=f(x,y)[\/latex]<\/dd>\n<\/dl>\n
      \n
      surface area<\/dt>\n
      the area of surface S<\/em> given by the surface integral [latex]\\displaystyle{\\int_{} {\\int_{S} d{\\bf{S}}}}[\/latex]<\/dd>\n<\/dl>\n
      \n
      surface independent<\/dt>\n
      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<\/span><\/dd>\n<\/dl>\n
      \n
      surface integral of a scalar-valued function<\/dt>\n
      a surface integral in which the integrand is a scalar function<\/span><\/dd>\n<\/dl>\n
      \n
      surface integral of a vector field<\/dt>\n
      a surface integral in which the integrand is a vector field<\/span><\/dd>\n<\/dl>\n
      \n
      symmetric equations of a line:<\/dt>\n
      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\u00a0[latex]{\\bf{v}}=\\langle{a,b,c}\\rangle[\/latex]\u00a0passing through point [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n
      <\/dd>\n<\/dl>\n
      \n
      tangent plane<\/dt>\n
      given a function [latex]f(x,y)[\/latex]\u00a0that 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]\u00a0is 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]<\/dd>\n<\/dl>\n
      \n
      tangent vector<\/dt>\n
      to [latex]{\\bf{r}}(t)[\/latex] at [latex]t=t_{0}[\/latex] any vector [latex]{\\bf{v}}[\/latex] such that, when the\u00a0tail of the vector is placed at point\u00a0[latex]{\\bf{r}}(t_{0})[\/latex]\u00a0on the graph, vector [latex]{\\bf{v}}[\/latex] is tangent to curve\u00a0[latex]C[\/latex]<\/dd>\n
      <\/dt>\n<\/dl>\n
      \n
      tangential component of acceleration<\/dt>\n
      the coefficient of the unit tangent vector [latex]{\\bf{T}}[\/latex] when the acceleration vector is written as a linear combination of\u00a0[latex]{\\bf{T}}[\/latex] and [latex]{\\bf{N}}[\/latex]<\/span><\/dd>\n<\/dl>\n
      \n
      terminal point<\/dt>\n
      the endpoint of a vector<\/dd>\n<\/dl>\n
      \n
      The Fundamental Theorem for Line Integrals<\/dt>\n
      the value of the line integral [latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}[\/latex]\u00a0depends only on the value of [latex]f[\/latex]\u00a0at the endpoints of [latex]C[\/latex]:\u00a0[latex]\\displaystyle\\int_{C} {\\nabla}{f}\\cdot{d{\\bf{r}}}=f({\\bf{r}}(b)))-f({\\bf{r}}(a))[\/latex]<\/dd>\n<\/dl>\n
      \n
      three-dimensional rectangular coordinate system<\/dt>\n
      a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple\u00a0[latex](x,y,z)[\/latex]\u00a0that plots\u00a0its\u00a0location relative to the defining axes<\/span><\/dd>\n<\/dl>\n
      \n
      torque<\/dt>\n
      the effect of a force that causes an object to rotate<\/dd>\n<\/dl>\n
      \n
      total differential<\/dt>\n
      the total differential of the function [latex]f(x,y)[\/latex]\u00a0at [latex](x_{0},y_{0})[\/latex]\u00a0is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[\/latex]<\/dd>\n<\/dl>\n
      \n
      trace<\/dt>\n
      the intersection of a three-dimensional surface with a coordinate plane<\/dd>\n<\/dl>\n
      \n
      transformation<\/dt>\n
      a function that transforms a region [latex]G[\/latex] in one plane into a region [latex]R[\/latex]\u00a0in another plane by a change of variables<\/dd>\n<\/dl>\n
      \n
      tree diagram<\/dt>\n
      illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for<\/dd>\n<\/dl>\n
      \n
      triangle inequality<\/dt>\n
      the length of any side of a triangle is less than the sum of the lengths of the other two sides<\/dd>\n<\/dl>\n
      \n
      triangle method<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      triple integral<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      triple integral in cylindrical coordinates<\/dt>\n
      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]<\/dd>\n<\/dl>\n
      \n
      triple integral in spherical coordinates<\/dt>\n
      the limit of a triple Riemann sum, provided the following limit exists:\u00a0[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]<\/dd>\n<\/dl>\n
      \n
      triple scalar product<\/dt>\n
      the dot product of a vector with the cross product of two other vectors: [latex]{\\bf{u}}\\cdot({\\bf{v}}\\times{\\bf{w}})[\/latex]<\/dd>\n<\/dl>\n
      \n
      Type I<\/dt>\n
      a region\u00a0[latex]\\bf{D}[\/latex] in the [latex]xy[\/latex]-plane is Type I\u00a0if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[\/latex] and\u00a0[latex]g_{2}(x)[\/latex]<\/dd>\n<\/dl>\n
      \n
      Type II<\/dt>\n
      a region [latex]\\bf{D}[\/latex]\u00a0<\/strong>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]\u00a0and\u00a0[latex]h_{2}(y)[\/latex]<\/dd>\n<\/dl>\n
      \n
      unit vector<\/dt>\n
      a vector with magnitude [latex]1[\/latex]<\/dd>\n<\/dl>\n
      \n
      unit vector field<\/dt>\n
      a vector field in which the magnitude of every vector is [latex]1[\/latex]<\/span><\/dd>\n<\/dl>\n
      \n
      vector<\/dt>\n
      a mathematical object that has both magnitude and direction<\/dd>\n<\/dl>\n
      \n
      vector addition<\/dt>\n
      a vector operation that defines the sum of two vectors<\/dd>\n<\/dl>\n
      \n
      vector difference<\/dt>\n
      the vector difference\u00a0[latex]{\\bf{v}}-{\\bf{w}}[\/latex]\u00a0is defined as\u00a0[latex]{\\bf{v}}+(-{\\bf{w}})={\\bf{v}}+(-1){\\bf{w}}[\/latex]<\/dd>\n<\/dl>\n
      \n
      vector equation of a line:<\/dt>\n
      the equation\u00a0[latex]{\\bf{r}} ={\\bf{r}}_{0}+t{\\bf{v}}[\/latex]\u00a0used 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],\u00a0<\/strong>where\u00a0[latex]{\\bf{r}}_{0}=\\langle{x_{0},y_{0},z_{0}}\\rangle[\/latex]\u00a0is the position vector of point [latex]P[\/latex]<\/dd>\n<\/dl>\n
      \n
      vector equation of a plane:<\/dt>\n
      the equation\u00a0[latex]{\\bf{n}}\\cdot\\overrightarrow{PQ}=0[\/latex],
      \nwhere [latex]P[\/latex]\u00a0is a given point in the plane, [latex]Q[\/latex] is any point in the plane, and\u00a0[latex]{\\bf{n}}[\/latex] is a normal vector of the plane<\/dd>\n<\/dl>\n
      \n
      vector field<\/dt>\n
      measured in [latex]\\mathbb{R}^{2}[\/latex],\u00a0an assignment of a vector [latex]{\\bf{F}}(x,y)[\/latex]<\/span>\u00a0to each point\u00a0[latex](x,y)[\/latex]<\/span>\u00a0of a subset\u00a0[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{2}[\/latex]; in [latex]\\mathbb{R}^{3}[\/latex],\u00a0an assignment of a vector [latex]{\\bf{F}}(x,y,z)[\/latex]<\/span> to each point\u00a0[latex](x,y,z)[\/latex]<\/span>\u00a0of a subset\u00a0[latex]D[\/latex]<\/span>\u00a0of [latex]\\mathbb{R}^{3}[\/latex]<\/dd>\n<\/dl>\n
      \n
      vector line integral<\/dt>\n
      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]\u00a0with unit tangent vector [latex]{\\bf{T}}[\/latex]\u00a0of [latex]C[\/latex] with respect to arc length, [latex]\\displaystyle\\int_{C} {\\bf{F}}\\cdot{\\bf{T}}ds[\/latex];\u00a0such an integral is defined in terms of a Riemann sum, similar to a single-variable integral<\/dd>\n<\/dl>\n
      \n
      vector parameterization<\/dt>\n
      any representation of a plane or space curve using a vector-valued function<\/dd>\n<\/dl>\n
      \n
      vector product<\/dt>\n
      the cross product of two vectors<\/dd>\n<\/dl>\n
      \n
      vector projection<\/dt>\n
      the component of a vector that follows a given direction<\/dd>\n<\/dl>\n
      \n
      vector sum<\/dt>\n
      the sum of two vectors, [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex]can be constructed graphically by placing the initial point of\u00a0[latex]{\\bf{w}}[\/latex]\u00a0at the terminal point of [latex]{\\bf{v}}[\/latex]; then the vector sum\u00a0[latex]{\\bf{v}}+{\\bf{w}}[\/latex]\u00a0is 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\u00a0[latex]{\\bf{w}}[\/latex]<\/dd>\n<\/dl>\n
      \n
      vector-valued function<\/dt>\n
      a function of the form [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] or\u00a0[latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex],\u00a0<\/strong>where the component functions [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] are real-valued functions of the parameter [latex]t[\/latex]<\/dd>\n<\/dl>\n
      \n
      velocity vector<\/dt>\n
      the derivative of the position vector<\/dd>\n<\/dl>\n
      \n
      vertex<\/dt>\n
      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<\/dd>\n<\/dl>\n
      \n
      vertical trace<\/dt>\n
      the set of ordered triples [latex](c,y,z)[\/latex]\u00a0<\/strong>that solves the equation [latex]f(c,y)=z[\/latex]\u00a0for a given constant [latex]x=c[\/latex]\u00a0or the set of ordered triples [latex](x,d,z)[\/latex]\u00a0that solves the equation [latex]f(x,d)=z[\/latex]\u00a0for a given constant [latex]y=d[\/latex]<\/dd>\n<\/dl>\n
      \n
      work done by a force<\/dt>\n
      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]\u00a0and [latex]\\bf{s}[\/latex]<\/dd>\n<\/dl>\n
      \n
      zero vector<\/dt>\n
      the vector with both initial point and terminal point\u00a0[latex](0,0)[\/latex]<\/dd>\n<\/dl>\n
      \n
      [latex]\\delta[\/latex]<\/strong>\u00a0ball<\/dt>\n
      all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than\u00a0[latex]\\delta[\/latex]\u00a0from [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n<\/dl>\n
      \n
      [latex]\\delta[\/latex] d<\/strong>isk<\/dt>\n
      an open disk of radius [latex]\\delta[\/latex]\u00a0centered at point\u00a0[latex](a,b)[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t
      \n\t\t\t

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