MTH 209 Exam 2 Vocab

Three Dimensional Coordinate System

A coordinate system with horizontal, vertical, and depth axes.

Coordinate Plane

A plane formed by two coordinate axes

Octant

One of the eight regions created by the three coordinate planes.

First Octant

The region where all three coordinates are positive

Ordered Triple

Three values used to identify a point in space

Sphere

The set of points a fixed distance from a center

Center of a Sphere

The fixed point equally distant from every point on the sphere

Radius of a Sphere

The fixed distance from the center to the surface

Midpoint in Space

The point halfway between two three dimensional points

Three Dimensional Vector

A vector with horizontal, vertical, and depth components

Zero Vector

A vector with no magnitude.

Standard Unit Vectors

The basic unit vectors that point along the three positive coordinate axes.

Parallel Vectors

Vectors that are scalar multiples of each other.

Dot Product

A vector operation that produces a scalar from matching components.

Orthogonal Vectors

Vectors that are perpendicular.

Direction Angles

The angles between a vector and the coordinate axes.

Direction Cosines

The cosines of a vector’s direction angles.

Projection

The part of one vector that points in the direction of another vector.

Orthogonal Component

The part of a vector perpendicular to another vector.

Work

The result of a force acting through a displacement.

Cross Product

A three dimensional vector operation that produces a perpendicular vector.

Right Hand Rule

A method for determining cross product direction.

Triple Scalar Product

A scalar operation involving three vectors that can measure volume.

Parallelepiped

A solid with six parallelogram shaped faces.

Direction Vector

A vector parallel to a line that gives its direction.

Parametric Equation of a Line

An equation that uses a parameter to describe every point on a line

Symmetric Equation of a Line

A line equation that relates the coordinate changes to direction components

Plane

A flat two dimensional surface extending without end in space

Normal Vector

A vector perpendicular to a plane.

Trace of a Plane

The line where a plane intersects a coordinate plane.

Distance from a Point to a Plane

The shortest distance measured perpendicular to the plane.

Distance from a Point to a Line

The shortest distance measured perpendicular to the line.

Cylindrical Surface

A surface created by moving parallel lines along a curve.

Generating Curve

The curve that forms a cylindrical surface.

Directrix

Another name for the generating curve of a cylinder.

Rulings

The parallel lines that form a cylindrical surface.

Right Cylinder

A cylinder whose rulings are perpendicular to its base plane.

Quadric Surface

A three dimensional surface described by a second degree equation

Ellipsoid

A closed, oval shaped quadric surface

Hyperboloid of One Sheet

A connected hourglass shaped quadric surface

Hyperboloid of Two Sheets

A quadric surface made of two separate pieces.

Elliptic Cone

A cone with elliptical cross sections

Elliptic Paraboloid

A bowl shaped quadric surface

Hyperbolic Paraboloid

A saddle shaped quadric surface.

Trace of a Surface

The curve created where a surface intersects a plane.

Surface of Revolution

A surface created by rotating a curve around an axis.

Cylindrical Coordinates

A three dimensional coordinate system using radial distance, angle, and height

Spherical Coordinates

A three dimensional coordinate system using distance from the origin and two angles.

Pole

The origin in a cylindrical coordinate system.

Radial Distance

The distance from the vertical axis in cylindrical coordinates.

Azimuthal Angle

The angle around the vertical axis.

Polar Angle

The angle measured downward from the positive vertical axis in spherical coordinates.

Central Symmetry

A property in which a shape is arranged symmetrically around a center point.

Vector Valued Function

A function that assigns a vector to each input.

Space Curve

A curve in three dimensional space.

Scalar Function

A function that produces a single real number.

Component Function

One scalar function that makes up a vector valued function

Continuity of a Vector Function

The condition that all component functions are continuous.

Derivative of a Vector Function

A vector found by differentiating every component function

Integral of a Vector Function

A vector found by integrating every component function

Smooth Curve

A curve with a continuous and nonzero velocity vector.

Position Vector

A vector that gives an object’s location.

Velocity Vector

A vector that gives the rate and direction of position change.

Acceleration Vector

A vector that gives the rate and direction of velocity change.

Speed

The magnitude of velocity.

Projectile

An object moving under the force of gravity.

Unit Tangent Vector

A unit vector pointing in the direction of motion along a curve.

Tangent Line

A line that touches a curve and points in its instantaneous direction.

Principal Unit Normal Vector

A unit vector pointing toward the direction a curve turns

Tangential Acceleration

The part of acceleration that changes speed.

Normal Acceleration

The part of acceleration that changes direction

Centripetal Acceleration

Another name for normal acceleration

Arc Length

The distance measured along a curve.

Arc Length Parameter

A parameter that measures distance traveled along a curve

Curvature

A measure of how sharply a curve bends

Circle of Curvature

The circle that best approximates a curve at a point

Radius of Curvature

The radius of the circle of curvature

Center of Curvature

The center of the circle of curvature.

Function of Two Variables

A function that assigns one output to each allowed pair of inputs

Independent Variables

The input variables of a function

Dependent Variable

The output variable of a function

Domain

The set of allowed input values.

Range

The set of possible output values.

Polynomial Function of Two Variables

A function made from sums of variable terms with nonnegative whole number powers.

Rational Function

A quotient of two polynomial functions.

Scalar Field

A numerical value assigned to every point in a region.

Level Curve

A curve connecting points where a function has the same value.

Contour Line

Another name for a level curve.

Level Surface

A surface connecting points where a three variable function has the same value.

Delta Neighborhood

A disk centered at a point with a chosen radius.

Open Disk

A disk that does not include its boundary.

Closed Disk

A disk that includes its boundary.

Interior Point

A point with a neighborhood entirely inside a region.

Open Region

A region in which every point is an interior point.

Boundary Point

A point where every nearby neighborhood contains points inside and outside a region.

Closed Region

A region that includes all of its boundary points.

Limit of a Function of Two Variables

The value a function approaches from every possible path toward a point.

Path Dependence

A situation where different paths toward a point give different function values.

Continuity

The condition where a function’s value equals its limit at a point

Open Sphere

A three dimensional region around a point that does not include its boundary.