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A = â« a b y ( t ) x â² ( t ) d t A = â« a b y ( t ) x â² ( t ) d t | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
s = â« t 1 t 2 ( d x d t ) 2 + ( d y d t ) 2 d t s = â« t 1 t 2 ( d x d t ) 2 + ( d y d t ) 2 d t | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
S = 2 Ï â« a b y ( t ) ( x â² ( t ) ) 2 + ( y â² ( t ) ) 2 d t S = 2 Ï â« a b y ( t ) ( x â² ( t ) ) 2 + ( y â² ( t ) ) 2 d t | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
A = 1 2 ⫠α β [ f ( θ ) ] 2 d θ = 1 2 ⫠α β r 2 d θ A = 1 2 ⫠α β [ f ( θ ) ] 2 d θ = 1 2 ⫠α β r 2 d θ | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
L = ⫠α β [ f ( θ ) ] 2 + [ f Ⲡ( θ ) ] 2 d θ = ⫠α β r 2 + ( d r d θ ) 2 d θ L = ⫠α β [ f ( θ ) ] 2 + [ f Ⲡ( θ ) ] 2 d θ = ⫠α β r 2 + ( d r d θ ) 2 d θ | https://openstax.org/books/calculus-volume-3/pages/1-key-equations |
angular coordinate : θθthe angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
cardioid : a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid isr=a(1+sinθ)r=a(1+sinθ)orr=a(1+cosθ)r=a(1+cosθ) | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
conic section : a conic section is any curve formed by the intersection of a plane with a cone of two nappes | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
cusp : a pointed end or part where two curves meet | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
cycloid : the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
directrix : a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
discriminant : the value4ACâB2,4ACâB2,which is used to identify a conic when the equation contains a term involvingxy,xy,is called a discriminant | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
eccentricity : the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
focal parameter : the focal parameter is the distance from a focus of a conic section to the nearest directrix | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
focus : a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
general form : an equation of a conic section written as a general second-degree equation | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
limaçon : the graph of the equationr=a+bsinθr=a+bsinθorr=a+bcosθ.r=a+bcosθ.Ifa=ba=bthen the graph is a cardioid | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
major axis : the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
minor axis : the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
nappe : a nappe is one half of a double cone | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
orientation : the direction that a point moves on a graph as the parameter increases | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
parameter : an independent variable that bothxandydepend on in a parametric curve; usually represented by the variablet | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
parameterization of a curve : rewriting the equation of a curve defined by a functiony=f(x)y=f(x)as parametric equations | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
parametric curve : the graph of the parametric equationsx(t)x(t)andy(t)y(t)over an intervalaâ¤tâ¤baâ¤tâ¤bcombined with the equations | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
parametric equations : the equationsx=x(t)x=x(t)andy=y(t)y=y(t)that define a parametric curve | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
polar axis : the horizontal axis in the polar coordinate system corresponding torâ¥0râ¥0 | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
polar coordinate system : a system for locating points in the plane. The coordinates arer,r,the radial coordinate, andθ,θ,the angular coordinate | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
polar equation : an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
pole : the central point of the polar coordinate system, equivalent to the origin of a Cartesian system | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
radial coordinate : rrthe coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
rose : graph of the polar equationr=acos2θr=acos2θorr=asin2θr=asin2θfor a positive constanta | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
space-filling curve : a curve that completely occupies a two-dimensional subset of the real plane | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
standard form : an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
vertex : a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch | https://openstax.org/books/calculus-volume-3/pages/1-key-terms |
Vectors are used to represent quantities that have both magnitude and direction. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Subtraction of vectors is defined in terms of adding the negative of the vector. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
A vector is written in component form asv=â©x,yâª.v=â©x,yâª. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The magnitude of a vector is a scalar:âvâ=x2+y2.âvâ=x2+y2. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
A unit vectoruuhas magnitude11and can be found by dividing a vector by its magnitude:u=1âvâv.u=1âvâv.The standard unit vectors arei=â©1,0âªandj=â©0,1âª.i=â©1,0âªandj=â©0,1âª.A vectorv=â©x,yâªv=â©x,yâªcan be expressed in terms of the standard unit vectors asv=xi+yj.v=xi+yj. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Vectors are often used in physics and engineering to represent forces and velocities, among other quantities. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples(x,y,z)(x,y,z)are used to describe the location of a point in space. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The distanceddbetween points(x1,y1,z1)(x1,y1,z1)and(x2,y2,z2)(x2,y2,z2)is given by the formulad=(x2âx1)2+(y2ây1)2+(z2âz1)2.d=(x2âx1)2+(y2ây1)2+(z2âz1)2. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
In three dimensions, the equationsx=a,y=b,andz=cx=a,y=b,andz=cdescribe planes that are parallel to the coordinate planes. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The standard equation of a sphere with center(a,b,c)(a,b,c)and radiusrris(xâa)2+(yâb)2+(zâc)2=r2.(xâa)2+(yâb)2+(zâc)2=r2. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
In three dimensions, as in two, vectors are commonly expressed in component form,v=â©x,y,zâª,v=â©x,y,zâª,or in terms of the standard unit vectors,xi+yj+zk.xi+yj+zk. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Properties of vectors in space are a natural extension of the properties for vectors in a plane. Letv=â©x1,y1,z1âªv=â©x1,y1,z1âªandw=â©x2,y2,z2âªw=â©x2,y2,z2âªbe vectors, and letkkbe a scalar.Scalar multiplication:kv=â©kx1,ky1,kz1âªkv=â©kx1,ky1,kz1âªVector addition:v+w=â©x1,y1,z1âª+â©x2,y2,z2âª=â©x1+x2... | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Scalar multiplication:kv=â©kx1,ky1,kz1âªkv=â©kx1,ky1,kz1⪠| https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Vector addition:v+w=â©x1,y1,z1âª+â©x2,y2,z2âª=â©x1+x2,y1+y2,z1+z2âªv+w=â©x1,y1,z1âª+â©x2,y2,z2âª=â©x1+x2,y1+y2,z1+z2⪠| https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Vector subtraction:vâw=â©x1,y1,z1âªââ©x2,y2,z2âª=â©x1âx2,y1ây2,z1âz2âªvâw=â©x1,y1,z1âªââ©x2,y2,z2âª=â©x1âx2,y1ây2,z1âz2⪠| https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Vector magnitude:âvâ=x12+y12+z12âvâ=x12+y12+z12 | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Unit vector in the direction of v:vâvâ=1âvââ©x1,y1,z1âª=â©x1âvâ,y1âvâ,z1âvââª,vâvâ=1âvââ©x1,y1,z1âª=â©x1âvâ,y1âvâ,z1âvââª,vâ0vâ0 | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The dot product, or scalar product, of two vectorsu=â©u1,u2,u3âªu=â©u1,u2,u3âªandv=â©v1,v2,v3âªv=â©v1,v2,v3âªisu·v=u1v1+u2v2+u3v3.u·v=u1v1+u2v2+u3v3. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The dot product satisfies the following properties:u·v=v·uu·v=v·uu·(v+w)=u·v+u·wu·(v+w)=u·v+u·wc(u·v)=(cu)·v=u·(cv)c(u·v)=(cu)·v=u·(cv)v·v=âvâ2v·v=âvâ2 | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
u·v=v·uu·v=v·u | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
u·(v+w)=u·v+u·wu·(v+w)=u·v+u·w | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
c(u·v)=(cu)·v=u·(cv)c(u·v)=(cu)·v=u·(cv) | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
v·v=âvâ2v·v=âvâ2 | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The dot product of two vectors can be expressed, alternatively, asu·v=âuââvâcosθ.u·v=âuââvâcosθ.This form of the dot product is useful for finding the measure of the angle formed by two vectors. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Vectorsuuandvvare orthogonal ifu·v=0.u·v=0. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The angles formed by a nonzero vector and the coordinate axes are called thedirection anglesfor the vector. The cosines of these angles are known as thedirection cosines. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The vector projection ofvvontouuis the vectorprojuv=u·vâuâ2u.projuv=u·vâuâ2u.The magnitude of this vector is known as thescalar projectionofvvontouu, given bycompuv=u·vâuâ.compuv=u·vâuâ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Work is done when a force is applied to an object, causing displacement. When the force is represented by the vectorFand the displacement is represented by the vectors, then the work doneWis given by the formulaW=F·s=âFââsâcosθ.W=F·s=âFââsâcosθ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The cross productuÃvuÃvof two vectorsu=â©u1,u2,u3âªu=â©u1,u2,u3âªandv=â©v1,v2,v3âªv=â©v1,v2,v3âªis a vector orthogonal to bothuuandv.v.Its length is given byâuÃvâ=âuâ·âvâ·sinθ,âuÃvâ=âuâ·âvâ·sinθ,whereθθis the angle betweenuuandv.v.Its direction is given by the right-hand rule... | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The algebraic formula for calculating the cross product of two vectors,u=â©u1,u2,u3âªandv=â©v1,v2,v3âª,u=â©u1,u2,u3âªandv=â©v1,v2,v3âª,isuÃv=(u2v3âu3v2)iâ(u1v3âu3v1)j+(u1v2âu2v1)k.uÃv=(u2v3âu3v2)iâ(u1v3âu3v1)j+(u1v2âu2v1)k. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The cross product satisfies the following properties for vectorsu,v,andw,u,v,andw,and scalarc:c:uÃv=â(vÃu)uÃv=â(vÃu)uÃ(v+w)=uÃv+uÃwuÃ(v+w)=uÃv+uÃwc(uÃv)=(cu)Ãv=uÃ(cv)c(uÃv)=(cu)Ãv=uÃ(cv)uÃ0=0Ãu=0uÃ0=0Ãu=0vÃv=0vÃv=0u·(vÃw)=(uÃv)·wu·(vÃw)=(uÃv)·w | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
uÃv=â(vÃu)uÃv=â(vÃu) | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
uÃ(v+w)=uÃv+uÃwuÃ(v+w)=uÃv+uÃw | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
c(uÃv)=(cu)Ãv=uÃ(cv)c(uÃv)=(cu)Ãv=uÃ(cv) | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
uÃ0=0Ãu=0uÃ0=0Ãu=0 | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
vÃv=0vÃv=0 | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
u·(vÃw)=(uÃv)·wu·(vÃw)=(uÃv)·w | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The cross product of vectorsu=â©u1,u2,u3âªu=â©u1,u2,u3âªandv=â©v1,v2,v3âªv=â©v1,v2,v3âªis the determinant|ijku1u2u3v1v2v3|.|ijku1u2u3v1v2v3|. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
If vectorsuuandvvform adjacent sides of a parallelogram, then the area of the parallelogram is given byâuÃvâ.âuÃvâ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The triple scalar product of vectorsu,u,v,v,andwwisu·(vÃw).u·(vÃw). | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The volume of a parallelepiped with adjacent edges given by vectorsu,v,andwu,v,andwisV=|u·(vÃw)|.V=|u·(vÃw)|. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
If the triple scalar product of vectorsu,v,andwu,v,andwis zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The cross product can be used to identify a vector orthogonal to two given vectors or to a plane. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
TorqueÏÏmeasures the tendency of a force to produce rotation about an axis of rotation. If forceFFis acting at a distancerrfrom the axis, then torque is equal to the cross product ofrrandF:F:Ï=rÃF.Ï=rÃF. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vectorv=â©a,b,câªv=â©a,b,câªpassing through pointP=(x0,y0,z0)P=(x0,y0,z0)isr=r0+tv,r=r0+tv,wherer0=â©x0,y0,z0âªr0=â©x0,y0,z0âªis the position vector of pointP.P.This equation can be rewr... | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
LetLLbe a line in space passing through pointPPwith direction vectorv.v.IfQQis any point not onL,L,then the distance fromQQtoLLisd=âPQâÃvââvâ.d=âPQâÃvââvâ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Given a pointPPand vectorn,n,the set of all pointsQQsatisfying equationn·PQâ=0n·PQâ=0forms a plane. Equationn·PQâ=0n·PQâ=0is known as thevector equation of a plane. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The scalar equation of a plane containing pointP=(x0,y0,z0)P=(x0,y0,z0)with normal vectorn=â©a,b,câªn=â©a,b,câªisa(xâx0)+b(yây0)+c(zâz0)=0.a(xâx0)+b(yây0)+c(zâz0)=0.This equation can be expressed asax+by+cz+d=0,ax+by+cz+d=0,whered=âax0âby0âcz0.d=âax0âby0âcz0.This form of the equation is some... | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Suppose a plane with normal vectornpasses through pointQ.Q.The distanceDDfrom the plane to pointPPnot in the plane is given byD=âprojnQPââ=|compnQPâ|=|QPâ·n|ânâ.D=âprojnQPââ=|compnQPâ|=|QPâ·n|ânâ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The normal vectors of parallel planes are parallel. When two planes intersect, they form a line. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The measure of the angleθθbetween two intersecting planes can be found using the equation:cosθ=|n1·n2|ân1âân2â,cosθ=|n1·n2|ân1âân2â,wheren1n1andn2n2are normal vectors to the planes. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The distanceDDfrom point(x0,y0,z0)(x0,y0,z0)to planeax+by+cz+d=0ax+by+cz+d=0is given byD=|a(x0âx1)+b(y0ây1)+c(z0âz1)|a2+b2+c2=|ax0+by0+cz0+d|a2+b2+c2.D=|a(x0âx1)+b(y0ây1)+c(z0âz1)|a2+b2+c2=|ax0+by0+cz0+d|a2+b2+c2. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
A set of lines parallel to a given line passing through a given curve is called acylinder, or acylindrical surface. The parallel lines are calledrulings. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
The intersection of a three-dimensional surface and a plane is called atrace. To find the trace in thexy-,yz-, orxz-planes, setz=0,x=0,ory=0,z=0,x=0,ory=0,respectively. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the formAx2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
Important quadric surfaces are summarized inFigure 2.87andFigure 2.88. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
In the cylindrical coordinate system, a point in space is represented by the ordered triple(r,θ,z),(r,θ,z),where(r,θ)(r,θ)represents the polar coordinates of the pointâs projection in thexy-plane andzzrepresents the pointâs projection onto thez-axis. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
To convert a point from cylindrical coordinates to Cartesian coordinates, use equationsx=rcosθ,x=rcosθ,y=rsinθ,y=rsinθ,andz=z.z=z. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
To convert a point from Cartesian coordinates to cylindrical coordinates, use equationsr2=x2+y2,r2=x2+y2,tanθ=yx,tanθ=yx,andz=z.z=z. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
In the spherical coordinate system, a pointPPin space is represented by the ordered triple(Ï,θ,Ï),(Ï,θ,Ï),whereÏÏis the distance betweenPPand the origin(Ïâ0),(Ïâ0),θθis the same angle used to describe the location in cylindrical coordinates, andÏÏis the angle formed by the positivez-axis and line segmen... | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
To convert a point from spherical coordinates to Cartesian coordinates, use equationsx=ÏsinÏcosθ,x=ÏsinÏcosθ,y=ÏsinÏsinθ,y=ÏsinÏsinθ,andz=ÏcosÏ.z=ÏcosÏ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
To convert a point from Cartesian coordinates to spherical coordinates, use equationsÏ2=x2+y2+z2,Ï2=x2+y2+z2,tanθ=yx,tanθ=yx,andÏ=arccos(zx2+y2+z2).Ï=arccos(zx2+y2+z2). | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
To convert a point from spherical coordinates to cylindrical coordinates, use equationsr=ÏsinÏ,r=ÏsinÏ,θ=θ,θ=θ,andz=ÏcosÏ.z=ÏcosÏ. | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
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