text stringlengths 2 2.33k | source stringclasses 826
values |
|---|---|
To convert a point from cylindrical coordinates to spherical coordinates, use equationsÏ=r2+z2,Ï=r2+z2,θ=θ,θ=θ,andÏ=arccos(zr2+z2).Ï=arccos(zr2+z2). | https://openstax.org/books/calculus-volume-3/pages/2-key-concepts |
d = ( x 2 â x 1 ) 2 + ( y 2 â y 1 ) 2 + ( z 2 â z 1 ) 2 d = ( x 2 â x 1 ) 2 + ( y 2 â y 1 ) 2 + ( z 2 â z 1 ) 2 | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
( x â a ) 2 + ( y â b ) 2 + ( z â c ) 2 = r 2 ( x â a ) 2 + ( y â b ) 2 + ( z â c ) 2 = r 2 | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 = â u â â v â cos θ u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 = â u â â v â cos θ | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
cos θ = u · v â u â â v â cos θ = u · v â u â â v â | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
proj u v = u · v â u â 2 u proj u v = u · v â u â 2 u | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
comp u v = u · v â u â comp u v = u · v â u â | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
W = F · P Q â = â F â â P Q â â cos θ W = F · P Q â = â F â â P Q â â cos θ | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
u à v = ( u 2 v 3 â u 3 v 2 ) i â ( u 1 v 3 â u 3 v 1 ) j + ( u 1 v 2 â u 2 v 1 ) k u à v = ( u 2 v 3 â u 3 v 2 ) i â ( u 1 v 3 â u 3 v 1 ) j + ( u 1 v 2 â u 2 v 1 ) k | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
r = r 0 + t v r = r 0 + t v | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
x â x 0 a = y â y 0 b = z â z 0 c x â x 0 a = y â y 0 b = z â z 0 c | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
n · P Q â = 0 n · P Q â = 0 | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
a ( x â x 0 ) + b ( y â y 0 ) + c ( z â z 0 ) = 0 a ( x â x 0 ) + b ( y â y 0 ) + c ( z â z 0 ) = 0 | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
d = â proj n Q P â â = | comp n Q P â | = | Q P â · n | â n â d = â proj n Q P â â = | comp n Q P â | = | Q P â · n | â n â | https://openstax.org/books/calculus-volume-3/pages/2-key-equations |
component : a scalar that describes either the vertical or horizontal direction of a vector | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
coordinate plane : a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: thexy-plane,xz-plane, or theyz-plane | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
cross product : uÃv=(u2v3âu3v2)iâ(u1v3âu3v1)j+(u1v2âu2v1)k,uÃv=(u2v3âu3v2)iâ(u1v3âu3v1)j+(u1v2âu2v1)k,whereu=â©u1,u2,u3âªu=â©u1,u2,u3âªandv=â©v1,v2,v3âªv=â©v1,v2,v3⪠| https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
cylinder : a set of lines parallel to a given line passing through a given curve | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
cylindrical coordinate system : a way to describe a location in space with an 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-terms |
determinant : a real number associated with a square matrix | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
direction angles : the angles formed by a nonzero vector and the coordinate axes | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
direction cosines : the cosines of the angles formed by a nonzero vector and the coordinate axes | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
direction vector : a vector parallel to a line that is used to describe the direction, or orientation, of the line in space | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
dot product or scalar product : u·v=u1v1+u2v2+u3v3u·v=u1v1+u2v2+u3v3whereu=â©u1,u2,u3âªu=â©u1,u2,u3âªandv=â©v1,v2,v3âªv=â©v1,v2,v3⪠| https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
ellipsoid : a three-dimensional surface described by an equation of the formx2a2+y2b2+z2c2=1;x2a2+y2b2+z2c2=1;all traces of this surface are ellipses | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
elliptic cone : a three-dimensional surface described by an equation of the formx2a2+y2b2âz2c2=0;x2a2+y2b2âz2c2=0;traces of this surface include ellipses and intersecting lines | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
elliptic paraboloid : a three-dimensional surface described by an equation of the formz=x2a2+y2b2;z=x2a2+y2b2;traces of this surface include ellipses and parabolas | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
equivalent vectors : vectors that have the same magnitude and the same direction | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
general form of the equation of a plane : an equation in the formax+by+cz+d=0,ax+by+cz+d=0,wheren=â©a,b,câªn=â©a,b,câªis a normal vector of the plane,P=(x0,y0,z0)P=(x0,y0,z0)is a point on the plane, andd=âax0âby0âcz0d=âax0âby0âcz0 | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
hyperboloid of one sheet : a three-dimensional surface described by an equation of the formx2a2+y2b2âz2c2=1;x2a2+y2b2âz2c2=1;traces of this surface include ellipses and hyperbolas | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
hyperboloid of two sheets : a three-dimensional surface described by an equation of the formz2c2âx2a2ây2b2=1;z2c2âx2a2ây2b2=1;traces of this surface include ellipses and hyperbolas | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
initial point : the starting point of a vector | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
magnitude : the length of a vector | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
normal vector : a vector perpendicular to a plane | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
normalization : using scalar multiplication to find a unit vector with a given direction | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
octants : the eight regions of space created by the coordinate planes | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
orthogonal vectors : vectors that form a right angle when placed in standard position | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
parallelepiped : a three-dimensional prism with six faces that are parallelograms | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
parallelogram method : a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
parametric equations of a line : the set of equationsx=x0+ta,x=x0+ta,y=y0+tb,y=y0+tb,andz=z0+tcz=z0+tcdescribing the line with direction vectorv=â©a,b,câªv=â©a,b,câªpassing through point(x0,y0,z0)(x0,y0,z0) | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
quadric surfaces : surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas) | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
right-hand rule : a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around thez-axis in such a way that the fingers curl from the positivex-axis to the positivey-axis, the thumb points in the direction of the positivez-axis | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
rulings : parallel lines that make up a cylindrical surface | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
scalar : a real number | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
scalar equation of a plane : the equationa(xâx0)+b(yây0)+c(zâz0)=0a(xâx0)+b(yây0)+c(zâz0)=0used to describe a plane containing pointP=(x0,y0,z0)P=(x0,y0,z0)with normal vectorn=â©a,b,câªn=â©a,b,câªor its alternate formax+by+cz+d=0,ax+by+cz+d=0,whered=âax0âby0âcz0d=âax0âby0âcz0 | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
scalar multiplication : a vector operation that defines the product of a scalar and a vector | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
scalar projection : the magnitude of the vector projection of a vector | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
skew lines : two lines that are not parallel but do not intersect | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
sphere : the set of all points equidistant from a given point known as thecenter | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
spherical coordinate system : a way to describe a location in space with an 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-terms |
standard equation of a sphere : (xâa)2+(yâb)2+(zâc)2=r2(xâa)2+(yâb)2+(zâc)2=r2describes a sphere with center(a,b,c)(a,b,c)and radiusrr | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
standard unit vectors : unit vectors along the coordinate axes:i=â©1,0âª,j=â©0,1âªi=â©1,0âª,j=â©0,1⪠| https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
standard-position vector : a vector with initial point(0,0)(0,0) | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
symmetric equations ofaaline : the equationsxâx0a=yây0b=zâz0cxâx0a=yây0b=zâz0cdescribing the line with direction vectorv=â©a,b,câªv=â©a,b,câªpassing through point(x0,y0,z0)(x0,y0,z0) | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
terminal point : the endpoint of a vector | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
three-dimensional rectangular coordinate system : a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple(x,y,z)(x,y,z)that plots its location relative to the defining axes | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
torque : the effect of a force that causes an object to rotate | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
trace : the intersection of a three-dimensional surface with a coordinate plane | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
triangle inequality : the length of any side of a triangle is less than the sum of the lengths of the other two sides | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
triangle method : a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of... | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
triple scalar product : the dot product of a vector with the cross product of two other vectors:u·(vÃw)u·(vÃw) | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
unit vector : a vector with margnitude11 | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector : a mathematical object that has both magnitude and direction | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector addition : a vector operation that defines the sum of two vectors | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector difference : the vector differencevâwvâwis defined asv+(âw)=v+(â1)wv+(âw)=v+(â1)w | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector equation of a line : the equationr=r0+tvr=r0+tvused to describe a line with direction vectorv=â©a,b,câªv=â©a,b,câªpassing through pointP=(x0,y0,z0),P=(x0,y0,z0),wherer0=â©x0,y0,z0âª,r0=â©x0,y0,z0âª,is the position vector of pointPP | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector equation of a plane : the equationn·PQâ=0,n·PQâ=0,wherePPis a given point in the plane,QQis any point in the plane, andnnis a normal vector of the plane | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector product : the cross product of two vectors | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector projection : the component of a vector that follows a given direction | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
vector sum : the sum of two vectors,vvandw,w,can be constructed graphically by placing the initial point ofwwat the terminal point ofv;v;then the vector sumv+wv+wis the vector with an initial point that coincides with the initial point ofv,v,and with a terminal point that coincides with the terminal point ofww | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
work done by a force : work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vectorFand the displacement of an object by a vectors, then the work done by the force is the dot product ofFands. | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
zero vector : the vector with both initial point and terminal point(0,0)(0,0) | https://openstax.org/books/calculus-volume-3/pages/2-key-terms |
A vector-valued function is 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-concepts |
The graph of a vector-valued function of the formr(t)=f(t)i+g(t)jr(t)=f(t)i+g(t)jis called aplane curve. The graph of a vector-valued function of the formr(t)=f(t)i+g(t)j+h(t)kr(t)=f(t)i+g(t)j+h(t)kis called aspace curve. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
It is possible to represent an arbitrary plane curve by a vector-valued function. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
To calculate the limit of a vector-valued function, calculate the limits of the component functions separately. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
Many of the properties of differentiation from theIntroduction to Derivativesalso apply to vector-valued functions. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The derivative of a vector-valued functionr(t)r(t)is also a tangent vector to the curve. The unit tangent vectorT(t)T(t)is calculated by dividing the derivative of a vector-valued function by its magnitude. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The arc-length function for a vector-valued function is calculated using the integral formulas(t)=â«atârâ²(u)âdu.s(t)=â«atârâ²(u)âdu.This formula is valid in both two and three dimensions. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The principal unit normal vector attis defined to beN(t)=Tâ²(t)âTâ²(t)â.N(t)=Tâ²(t)âTâ²(t)â. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The binormal vector attis defined asB(t)=T(t)ÃN(t),B(t)=T(t)ÃN(t),whereT(t)T(t)is the unit tangent vector. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
Ifr(t)r(t)represents the position of an object at timet, thenrâ²(t)râ²(t)represents the velocity andrâ³(t)râ³(t)represents the acceleration of the object at timet.The magnitude of the velocity vector is speed. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
The acceleration vector always points toward the concave side of the curve defined byr(t).r(t).The tangential and normal components of accelerationaTaTandaNaNare the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
Keplerâs three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Keplerâs three laws. | https://openstax.org/books/calculus-volume-3/pages/3-key-concepts |
r ( t ) = f ( t ) i + g ( t ) j or r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , or r ( t ) = ⩠f ( t ) , g ( t ) ⪠or r ( t ) = ⩠f ( t ) , g ( t ) , h ( t ) ⪠r ( t ) = f ( t ) i + g ( t ) j or r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , or r ( t ) = ⩠f ( t ) , g ( t ) ⪠or r ( t ) = ⩠f ( t ) , g ( ... | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
lim t â a r ( t ) = [ lim t â a f ( t ) ] i + [ lim t â a g ( t ) ] j or lim t â a r ( t ) = [ lim t â a f ( t ) ] i + [ lim t â a g ( t ) ] j + [ lim t â a h ( t ) ] k lim t â a r ( t ) = [ lim t â a f ( t ) ] i + [ lim t â a g ( t ) ] j or lim t â a r ( t ) = [ lim t â a f ( t ) ] i + [ lim t ... | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
r â² ( t ) = lim Î t â 0 r ( t + Î t ) â r ( t ) Î t r â² ( t ) = lim Î t â 0 r ( t + Î t ) â r ( t ) Î t | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
T ( t ) = r â² ( t ) â r â² ( t ) â T ( t ) = r â² ( t ) â r â² ( t ) â | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
â« [ f ( t ) i + g ( t ) j + h ( t ) k ] d t = [ â« f ( t ) d t ] i + [ â« g ( t ) d t ] j + [ â« h ( t ) d t ] k â« [ f ( t ) i + g ( t ) j + h ( t ) k ] d t = [ â« f ( t ) d t ] i + [ â« g ( t ) d t ] j + [ â« h ( t ) d t ] k | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
â« a b [ f ( t ) i + g ( t ) j + h ( t ) k ] d t = [ â« a b f ( t ) d t ] i + [ â« a b g ( t ) d t ] j + [ â« a b h ( t ) d t ] k â« a b [ f ( t ) i + g ( t ) j + h ( t ) k ] d t = [ â« a b f ( t ) d t ] i + [ â« a b g ( t ) d t ] j + [ â« a b h ( t ) d t ] k | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
s = â« a b [ f â² ( t ) ] 2 + [ g â² ( t ) ] 2 + [ h â² ( t ) ] 2 d t = â« a b â r â² ( t ) â d t s = â« a b [ f â² ( t ) ] 2 + [ g â² ( t ) ] 2 + [ h â² ( t ) ] 2 d t = â« a b â r â² ( t ) â d t | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
s ( t ) = â« a t ( f â² ( u ) ) 2 + ( g â² ( u ) ) 2 + ( h â² ( u ) ) 2 d u or s ( t ) = â« a t â r â² ( u ) â d u s ( t ) = â« a t ( f â² ( u ) ) 2 + ( g â² ( u ) ) 2 + ( h â² ( u ) ) 2 d u or s ( t ) = â« a t â r â² ( u ) â d u | https://openstax.org/books/calculus-volume-3/pages/3-key-equations |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.