Calculus Volume 3

# Key Terms

component
a scalar that describes either the vertical or horizontal direction of a vector
coordinate plane
a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the xy-plane, xz-plane, or the yz-plane
cross product
$u×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k,u×v=(u2v3−u3v2)i−(u1v3−u3v1)j+(u1v2−u2v1)k,$ where $u=〈u1,u2,u3〉u=〈u1,u2,u3〉$ and $v=〈v1,v2,v3〉v=〈v1,v2,v3〉$
cylinder
a set of lines parallel to a given line passing through a given curve
cylindrical coordinate system
a way to describe a location in space with an ordered triple $(r,θ,z),(r,θ,z),$ where $(r,θ)(r,θ)$ represents the polar coordinates of the point’s projection in the xy-plane, and $zz$ represents the point’s projection onto the z-axis
determinant
a real number associated with a square matrix
direction angles
the angles formed by a nonzero vector and the coordinate axes
direction cosines
the cosines of the angles formed by a nonzero vector and the coordinate axes
direction vector
a vector parallel to a line that is used to describe the direction, or orientation, of the line in space
dot product or scalar product
$u·v=u1v1+u2v2+u3v3u·v=u1v1+u2v2+u3v3$ where $u=〈u1,u2,u3〉u=〈u1,u2,u3〉$ and $v=〈v1,v2,v3〉v=〈v1,v2,v3〉$
ellipsoid
a three-dimensional surface described by an equation of the form $x2a2+y2b2+z2c2=1;x2a2+y2b2+z2c2=1;$ all traces of this surface are ellipses
elliptic cone
a three-dimensional surface described by an equation of the form $x2a2+y2b2−z2c2=0;x2a2+y2b2−z2c2=0;$ traces of this surface include ellipses and intersecting lines
elliptic paraboloid
a three-dimensional surface described by an equation of the form $z=x2a2+y2b2;z=x2a2+y2b2;$ traces of this surface include ellipses and parabolas
equivalent vectors
vectors that have the same magnitude and the same direction
general form of the equation of a plane
an equation in the form $ax+by+cz+d=0,ax+by+cz+d=0,$ where $n=〈a,b,c〉n=〈a,b,c〉$ is a normal vector of the plane, $P=(x0,y0,z0)P=(x0,y0,z0)$ is a point on the plane, and $d=−ax0−by0−cz0d=−ax0−by0−cz0$
hyperboloid of one sheet
a three-dimensional surface described by an equation of the form $x2a2+y2b2−z2c2=1;x2a2+y2b2−z2c2=1;$ traces of this surface include ellipses and hyperbolas
hyperboloid of two sheets
a three-dimensional surface described by an equation of the form $z2c2−x2a2−y2b2=1;z2c2−x2a2−y2b2=1;$ traces of this surface include ellipses and hyperbolas
initial point
the starting point of a vector
magnitude
the length of a vector
normal vector
a vector perpendicular to a plane
normalization
using scalar multiplication to find a unit vector with a given direction
octants
the eight regions of space created by the coordinate planes
orthogonal vectors
vectors that form a right angle when placed in standard position
parallelepiped
a three-dimensional prism with six faces that are parallelograms
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
parametric equations of a line
the set of equations $x=x0+ta,x=x0+ta,$ $y=y0+tb,y=y0+tb,$ and $z=z0+tcz=z0+tc$ describing the line with direction vector $v=〈a,b,c〉v=〈a,b,c〉$ passing through point $(x0,y0,z0)(x0,y0,z0)$
surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)
right-hand rule
a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the z-axis in such a way that the fingers curl from the positive x-axis to the positive y-axis, the thumb points in the direction of the positive z-axis
rulings
parallel lines that make up a cylindrical surface
scalar
a real number
scalar equation of a plane
the equation $a(x−x0)+b(y−y0)+c(z−z0)=0a(x−x0)+b(y−y0)+c(z−z0)=0$ used to describe a plane containing point $P=(x0,y0,z0)P=(x0,y0,z0)$ with normal vector $n=〈a,b,c〉n=〈a,b,c〉$ or its alternate form $ax+by+cz+d=0,ax+by+cz+d=0,$ where $d=−ax0−by0−cz0d=−ax0−by0−cz0$
scalar multiplication
a vector operation that defines the product of a scalar and a vector
scalar projection
the magnitude of the vector projection of a vector
skew lines
two lines that are not parallel but do not intersect
sphere
the set of all points equidistant from a given point known as the center
spherical coordinate system
a way to describe a location in space with an ordered triple $(ρ,θ,φ),(ρ,θ,φ),$ where $ρρ$ is the distance between $PP$ and the origin $(ρ≠0),(ρ≠0),$ $θθ$ is the same angle used to describe the location in cylindrical coordinates, and $φφ$ is the angle formed by the positive z-axis and line segment $OP—,OP—,$ where $OO$ is the origin and $0≤φ≤π0≤φ≤π$
standard equation of a sphere
$(x−a)2+(y−b)2+(z−c)2=r2(x−a)2+(y−b)2+(z−c)2=r2$ describes a sphere with center $(a,b,c)(a,b,c)$ and radius $rr$
standard unit vectors
unit vectors along the coordinate axes: $i=〈1,0〉,j=〈0,1〉i=〈1,0〉,j=〈0,1〉$
standard-position vector
a vector with initial point $(0,0)(0,0)$
symmetric equations of $aa$ line
the equations $x−x0a=y−y0b=z−z0cx−x0a=y−y0b=z−z0c$ describing the line with direction vector $v=〈a,b,c〉v=〈a,b,c〉$ passing through point $(x0,y0,z0)(x0,y0,z0)$
terminal point
the endpoint of a vector
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
torque
the effect of a force that causes an object to rotate
trace
the intersection of a three-dimensional surface with a coordinate plane
triangle inequality
the length of any side of a triangle is less than the sum of the lengths of the other two sides
triangle method
a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector
triple scalar product
the dot product of a vector with the cross product of two other vectors: $u·(v×w)u·(v×w)$
unit vector
a vector with margnitude $11$
vector
a mathematical object that has both magnitude and direction
a vector operation that defines the sum of two vectors
vector difference
the vector difference $v−wv−w$ is defined as $v+(−w)=v+(−1)wv+(−w)=v+(−1)w$
vector equation of a line
the equation $r=r0+tvr=r0+tv$ used to describe a line with direction vector $v=〈a,b,c〉v=〈a,b,c〉$ passing through point $P=(x0,y0,z0),P=(x0,y0,z0),$ where $r0=〈x0,y0,z0〉,r0=〈x0,y0,z0〉,$ is the position vector of point $PP$
vector equation of a plane
the equation $n·PQ→=0,n·PQ→=0,$ where $PP$ is a given point in the plane, $QQ$ is any point in the plane, and $nn$ is a normal vector of the plane
vector product
the cross product of two vectors
vector projection
the component of a vector that follows a given direction
vector sum
the sum of two vectors, $vv$ and $w,w,$ can be constructed graphically by placing the initial point of $ww$ at the terminal point of $v;v;$ then the vector sum $v+wv+w$ is the vector with an initial point that coincides with the initial point of $v,v,$ and with a terminal point that coincides with the terminal point of $ww$
work done by a force
work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector F and the displacement of an object by a vector s, then the work done by the force is the dot product of F and s.
zero vector
the vector with both initial point and terminal point $(0,0)(0,0)$