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κ = â T â² ( t ) â â r â² ( t ) â or κ = â r â² ( t ) à r â³ ( t ) â â r â² ( t ) â 3 or κ = | y â³ | [ 1 + ( y â² ) 2 ] 3 / 2 κ = â T â² ( t ) â â r â² ( t ) â or κ = â r â² ( t ) à r â³ ( t ) â â r â² ( t ) â 3 or κ = | y â³ | [ 1 + ( y â² ) 2 ] 3 / 2 | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
N ( t ) = T â² ( t ) â T â² ( t ) â N ( t ) = T â² ( t ) â T â² ( t ) â | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
B ( t ) = T ( t ) Ã N ( t ) B ( t ) = T ( t ) Ã N ( t ) | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
v ( t ) = r â² ( t ) v ( t ) = r â² ( t ) | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
a ( t ) = v â² ( t ) = râ³ ( t ) a ( t ) = v â² ( t ) = râ³ ( t ) | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
v ( t ) = â v ( t ) â = â r â² ( t ) â = d s d t v ( t ) = â v ( t ) â = â r â² ( t ) â = d s d t | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
a T = a · T = v · a â v â a T = a · T = v · a â v â | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
a N = a · N = â v à a â â v â = â a â 2 â a T 2 a N = a · N = â v à a â â v â = â a â 2 â a T 2 | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
acceleration vector : the second derivative of the position vector | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
arc-length function : a functions(t)s(t)that describes the arc length of curveCas a function oft | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
arc-length parameterization : a reparameterization of a vector-valued function in which the parameter is equal to the arc length | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
binormal vector : a unit vector orthogonal to the unit tangent vector and the unit normal vector | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
component functions : the component functions of the vector-valued functionr(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)jaref(t)f(t)andg(t),g(t),and the component functions of the vector-valued functionr(t)=f(t)i+g(t)j+h(t)kr(t)=f(t)i+g(t)j+h(t)karef(t),f(t),g(t)g(t)andh(t)h(t) | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
curvature : the derivative of the unit tangent vector with respect to the arc-length parameter | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
definite integral of a vector-valued function : the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
derivative of a vector-valued function : the derivative of a vector-valued functionr(t)r(t)isrâ²(t)=limÎtâ0r(t+Ît)âr(t)Ît,râ²(t)=limÎtâ0r(t+Ît)âr(t)Ît,provided the limit exists | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
Frenet frame of reference : (TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
helix : a three-dimensional curve in the shape of a spiral | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
indefinite integral of a vector-valued function : a vector-valued function with a derivative that is equal to a given vector-valued function | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
Keplerâs laws of planetary motion : three laws governing the motion of planets, asteroids, and comets in orbit around the Sun | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
limit of a vector-valued function : a vector-valued functionr(t)r(t)has a limitLastapproachesaiflimtâa|r(t)âL|=0limtâa|r(t)âL|=0 | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
normal component of acceleration : the coefficient of the unit normal vectorNwhen the acceleration vector is written as a linear combination ofTTandNN | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
normal plane : a plane that is perpendicular to a curve at any point on the curve | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
osculating circle : a circle that is tangent to a curveCat a pointPand that shares the same curvature | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
osculating plane : the plane determined by the unit tangent and the unit normal vector | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
plane curve : the set of ordered pairs(f(t),g(t))(f(t),g(t))together with their defining parametric equationsx=f(t)x=f(t)andy=g(t)y=g(t) | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
principal unit normal vector : a vector orthogonal to the unit tangent vector, given by the formulaTâ²(t)âTâ²(t)âTâ²(t)âTâ²(t)â | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
principal unit tangent vector : a unit vector tangent to a curveC | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
projectile motion : motion of an object with an initial velocity but no force acting on it other than gravity | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
radius of curvature : the reciprocal of the curvature | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
reparameterization : an alternative parameterization of a given vector-valued function | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
smooth : curves where the vector-valued functionr(t)r(t)is differentiable with a non-zero derivative | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
space curve : the set of ordered triples(f(t),g(t),h(t))(f(t),g(t),h(t))together with their defining parametric equationsx=f(t),x=f(t),y=g(t)y=g(t)andz=h(t)z=h(t) | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
tangent vector : tor(t)r(t)att=t0t=t0any vectorvsuch that, when the tail of the vector is placed at pointr(t0)r(t0)on the graph, vectorvis tangent to curveC | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
tangential component of acceleration : the coefficient of the unit tangent vectorTwhen the acceleration vector is written as a linear combination ofTTandNN | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
vector parameterization : any representation of a plane or space curve using a vector-valued function | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
vector-valued function : a function of the formr(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)jorr(t)=f(t)i+g(t)j+h(t)k,r(t)=f(t)i+g(t)j+h(t)k,where the component functionsf, g,andhare real-valued functions of the parametert | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
velocity vector : the derivative of the position vector | https://openstax.org/books/calculus-volume-3/pages/3-key-terms |
The graph of a function of two variables is a surface inâ3â3and can be studied using level curves and vertical traces. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A set of level curves is called a contour map. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
To study limits and continuity for functions of two variables, we use aδδdisk centered around a given point. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A function of several variables has a limit if for any point in aδδball centered at a pointP,P,the value of the function at that point is arbitrarily close to a fixed value (the limit value). | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
The limit laws established for a function of one variable have natural extensions to functions of more than one variable. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A partial derivative is a derivative involving a function of more than one independent variable. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
Higher-order partial derivatives can be calculated in the same way as higher-order derivatives. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
Tangent planes can be used to approximate values of functions near known values. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A function is differentiable at a point if it is âsmoothâ at that point (i.e., no corners or discontinuities exist at that point). | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
The total differential can be used to approximate the change in a functionz=f(x0,y0)z=f(x0,y0)at the point(x0,y0)(x0,y0)for given values ofÎxÎxandÎy.Îy. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A directional derivative represents a rate of change of a function in any given direction. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
The gradient can be used in a formula to calculate the directional derivative. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
The gradient indicates the direction of greatest change of a function of more than one variable. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A critical point of the functionf(x,y)f(x,y)is any point(x0,y0)(x0,y0)where eitherfx(x0,y0)=fy(x0,y0)=0,fx(x0,y0)=fy(x0,y0)=0,or at least one offx(x0,y0)fx(x0,y0)andfy(x0,y0)fy(x0,y0)do not exist. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
A saddle point is a point(x0,y0)(x0,y0)wherefx(x0,y0)=fy(x0,y0)=0,fx(x0,y0)=fy(x0,y0)=0,but(x0,y0)(x0,y0)is neither a maximum nor a minimum at that point. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
An objective function combined with one or more constraints is an example of an optimization problem. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy. | https://openstax.org/books/calculus-volume-3/pages/4-key-concepts |
f ( a , y ) = z f ( a , y ) = z for x = a x = a or f ( x , b ) = z f ( x , b ) = z for y = b y = b | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
f ( x , y , z ) = c f ( x , y , z ) = c | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â f â x = lim h â 0 f ( x + h , y ) â f ( x , y ) h â f â x = lim h â 0 f ( x + h , y ) â f ( x , y ) h | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â f â y = lim k â 0 f ( x , y + k ) â f ( x , y ) k â f â y = lim k â 0 f ( x , y + k ) â f ( x , y ) k | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x â x 0 ) + f y ( x 0 , y 0 ) ( y â y 0 ) z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x â x 0 ) + f y ( x 0 , y 0 ) ( y â y 0 ) | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
L ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x â x 0 ) + f y ( x 0 , y 0 ) ( y â y 0 ) L ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x â x 0 ) + f y ( x 0 , y 0 ) ( y â y 0 ) | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
d z = f x ( x 0 , y 0 ) d x + f y ( x 0 , y 0 ) d y . d z = f x ( x 0 , y 0 ) d x + f y ( x 0 , y 0 ) d y . | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
f ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x â x 0 ) + f y ( x 0 , y 0 ) ( y â y 0 ) + E ( x , y ) , f ( x , y ) = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x â x 0 ) + f y ( x 0 , y 0 ) ( y â y 0 ) + E ( x , y ) , where the error term E E satisfies lim ( x , y ) â ( x 0 , y 0 ) E ( x , y ) ( x â x 0 )... | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
f ( x , y ) = f ( x 0 , y 0 , z 0 ) + f x ( x 0 , y 0 , z 0 ) ( x â x 0 ) + f y ( x 0 , y 0 , z 0 ) ( y â y 0 ) + f z ( x 0 , y 0 , z 0 ) ( z â z 0 ) + E ( x , y , z ) , f ( x , y ) = f ( x 0 , y 0 , z 0 ) + f x ( x 0 , y 0 , z 0 ) ( x â x 0 ) + f y ( x 0 , y 0 , z 0 ) ( y â y 0 ) + f z ( x 0 , y 0 , z 0 ) ( ... | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
d z d t = â z â x · d x d t + â z â y · d y d t d z d t = â z â x · d x d t + â z â y · d y d t | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
d z d u = â z â x â x â u + â z â y â x â u d z d u = â z â x â x â u + â z â y â x â u d z d v = â z â x â x â v + â z â y â y â v d z d v = â z â x â x â v + â z â y â y â v | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â w â t j = â w â x 1 â x 1 â t j + â w â x 2 â x 1 â t j + ⯠+ â w â x m â x m â t j â w â t j = â w â x 1 â x 1 â t j + â w â x 2 â x 1 â t j + ⯠+ â w â x m â x m â t j | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
D u f ( a , b ) = lim h â 0 f ( a + h cos θ , b + h sin θ ) â f ( a , b ) h D u f ( a , b ) = lim h â 0 f ( a + h cos θ , b + h sin θ ) â f ( a , b ) h or D u f ( x , y ) = f x ( x , y ) cos θ + f y ( x , y ) sin θ D u f ( x , y ) = f x ( x , y ) cos θ + f y ( x , y ) sin θ | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â f ( x , y ) = f x ( x , y ) i + f y ( x , y ) j â f ( x , y ) = f x ( x , y ) i + f y ( x , y ) j | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â f ( x , y , z ) = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k â f ( x , y , z ) = f x ( x , y , z ) i + f y ( x , y , z ) j + f z ( x , y , z ) k | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
D u f ( x , y , z ) = â f ( x , y , z ) · u = f x ( x , y , z ) cos α + f y ( x , y , z ) cos β + f x ( x , y , z ) cos γ D u f ( x , y , z ) = â f ( x , y , z ) · u = f x ( x , y , z ) cos α + f y ( x , y , z ) cos β + f x ( x , y , z ) cos γ | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
D = f x x ( x 0 , y 0 ) f y y ( x 0 , y 0 ) â ( f x y ( x 0 , y 0 ) ) 2 D = f x x ( x 0 , y 0 ) f y y ( x 0 , y 0 ) â ( f x y ( x 0 , y 0 ) ) 2 | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â f ( x 0 , y 0 ) = λ â g ( x 0 , y 0 ) g ( x 0 , y 0 ) = 0 â f ( x 0 , y 0 ) = λ â g ( x 0 , y 0 ) g ( x 0 , y 0 ) = 0 | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
â f ( x 0 , y 0 , z 0 ) = λ 1 â g ( x 0 , y 0 , z 0 ) + λ 2 â h ( x 0 , y 0 , z 0 ) g ( x 0 , y 0 , z 0 ) = 0 h ( x 0 , y 0 , z 0 ) = 0 â f ( x 0 , y 0 , z 0 ) = λ 1 â g ( x 0 , y 0 , z 0 ) + λ 2 â h ( x 0 , y 0 , z 0 ) g ( x 0 , y 0 , z 0 ) = 0 h ( x 0 , y 0 , z 0 ) = 0 | https://openstax.org/books/calculus-volume-3/pages/4-key-equations |
boundary point : a pointP0P0ofRRis a boundary point if everyδδdisk centered aroundP0P0contains points both inside and outsideRR | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
closed set : a setSSthat contains all its boundary points | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
connected set : an open setSSthat cannot be represented as the union of two or more disjoint, nonempty open subsets | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
constraint : an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
contour map : a plot of the various level curves of a given functionf(x,y)f(x,y) | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
critical point of a function of two variables : the point(x0,y0)(x0,y0)is called a critical point off(x,y)f(x,y)if one of the two following conditions holds:fx(x0,y0)=fy(x0,y0)=0fx(x0,y0)=fy(x0,y0)=0At least one offx(x0,y0)fx(x0,y0)andfy(x0,y0)fy(x0,y0)do not exist | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
differentiable : a functionf(x,y,z)f(x,y,z)is differentiable at(x0,y0)(x0,y0)iff(x,y)f(x,y)can be expressed in the formf(x,y)=f(x0,y0)+fx(x0,y0)(xâx0)+fy(x0,y0)(yây0)+E(x,y),f(x,y)=f(x0,y0)+fx(x0,y0)(xâx0)+fy(x0,y0)(yây0)+E(x,y),where the error termE(x,y)E(x,y)satisfieslim(x,y)â(x0,y0)E(x,y)(xâx0)2+(yây0)... | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
directional derivative : the derivative of a function in the direction of a given unit vector | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
discriminant : the discriminant of the functionf(x,y)f(x,y)is given by the formulaD=fxx(x0,y0)fyy(x0,y0)â(fxy(x0,y0))2D=fxx(x0,y0)fyy(x0,y0)â(fxy(x0,y0))2 | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
function of two variables : a functionz=f(x,y)z=f(x,y)that maps each ordered pair(x,y)(x,y)in a subsetDDofâ2â2to a unique real numberzz | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
generalized chain rule : the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
gradient : the gradient of the functionf(x,y)f(x,y)is defined to beâf(x,y)=(âf/âx)i+(âf/ây)j,âf(x,y)=(âf/âx)i+(âf/ây)j,which can be generalized to a function of any number of independent variables | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
graph of a function of two variables : a set of ordered triples(x,y,z)(x,y,z)that satisfies the equationz=f(x,y)z=f(x,y)plotted in three-dimensional Cartesian space | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
higher-order partial derivatives : second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
interior point : a pointP0P0ofRRis a boundary point if there is aδδdisk centered aroundP0P0contained completely inRR | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
intermediate variable : given a composition of functions (e.g.,f(x(t),y(t))),f(x(t),y(t))),the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the functionf(x(t),y(t)),f(x(t),y(t)),the variablesxandyxandyare examples of intermediate variab... | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
Lagrange multiplier : the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variableλλ | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
level curve of a function of two variables : the set of points satisfying the equationf(x,y)=cf(x,y)=cfor some real numberccin the range offf | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
level surface of a function of three variables : the set of points satisfying the equationf(x,y,z)=cf(x,y,z)=cfor some real numberccin the range offf | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
linear approximation : given a functionf(x,y)f(x,y)and a tangent plane to the function at a point(x0,y0),(x0,y0),we can approximatef(x,y)f(x,y)for points near(x0,y0)(x0,y0)using the tangent plane formula | https://openstax.org/books/calculus-volume-3/pages/4-key-terms |
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