Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called

Image:Coord XYZ.svg,

^{3} form a

y, ''F''z">'F''_{x}, ''F''_{y}, ''F''_{z}
:$\backslash begin\; \backslash mathbf\; \&\; \backslash mathbf\; \&\; \backslash mathbf\; \backslash \backslash \; \backslash \backslash \; \&\; \&\; \backslash \backslash \; \backslash \backslash \; F\_x\; \&\; F\_y\; \&\; F\_z\; \backslash end$
where i, j, and k are the

^{''n''} → R, the line integral along a ^{''n''} → R^{''n''}, the line integral along a ^{3} of a

Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry

Keith Matthews from University of Queensland, 1991 {{Dimension topics Euclidean solid geometry, * Analytic geometry Multi-dimensional geometry Three-dimensional coordinate systems 3 (number)

parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

s) are required to determine the position of an element (i.e., point
Point or points may refer to:
Places
* Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point
Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

). This is the informal meaning of the term dimension
In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...

.
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of numbers
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduc ...

can be understood as the Cartesian coordinates
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted $\backslash R^n,$ and can be identified to the -dimensional Euclidean space.
When , this space is called ''three-dimensional Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

'' (or simply Euclidean space when the context is clear), and serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called 3-manifold
. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.
In mathematics
Mathematics (from ...

s. In this classical example, when the three values refer to measurements in different directions (coordinates
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

), any three directions can be chosen, provided that vectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...

in these directions do not all lie in the same 2-space (plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse
*Plane (Magic: Th ...

). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms ''width
Length is a measure of distance
Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two countie ...

/breadth'', ''height
200px, A cuboid demonstrating the dimensions length, width">length.html" ;"title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height.
Height is measure of vertical distance, either vertical ...

/depth'', and ''length
Length is a measure of distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

''.
In Euclidean geometry

Coordinate systems

In mathematics,analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

(also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes
A Cartesian coordinate system (, ) in a plane is a coordinate system
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the o ...

are given, each perpendicular to the other two at the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* , a Wolverine comic book mini-series published by Marvel Comics in 2002
* , a 1999 ''Buffy the Vampire Slayer'' comic book series
* , a major ''Judge Dred ...

, the point at which they cross. They are usually labeled , and . Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, each number giving the distance of that point from the origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* , a Wolverine comic book mini-series published by Marvel Comics in 2002
* , a 1999 ''Buffy the Vampire Slayer'' comic book series
* , a major ''Judge Dred ...

measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.
Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates
240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height .
A cylindrical coordinate system is a three-dimensional coordinate system that s ...

and spherical coordinates
File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is of ...

, though there are an infinite number of possible methods. For more, see Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

.
Below are images of the above-mentioned systems.
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Image:Cylindrical Coordinates.svg, Cylindrical coordinate system
240px, A cylindrical coordinate system with origin , polar axis , and longitudinal axis . The dot is the point with radial distance , angular coordinate , and height .
A cylindrical coordinate system is a three-dimensional coordinate system
In ...

Image:Spherical Coordinates (Colatitude, Longitude).svg, Spherical coordinate system
File:3D Spherical 2.svg, 240px, Spherical coordinates as often used in ''mathematics'': radial distance , azimuthal angle , and polar angle . The meanings of and have been swapped compared to the physics convention. As in physics, (rho) is oft ...

Lines and planes

Two distinct points always determine a (straight)line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a 2009 independent film by Nancy Schwartzman
Lite ...

. Three distinct points are either collinear
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanarIn geometry, a set of points in space are coplanar if there exists a geometric Plane (mathematics), plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear points, non-collinear, t ...

, or determine the entire space.
Two distinct lines can either intersect, be parallel
Parallel may refer to:
Computing
* Parallel algorithm
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...

or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line.
A hyperplane
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.
Varignon's theorem states that the midpoints of any quadrilateral in ℝparallelogram
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

, and hence are coplanar.
Spheres and balls

Asphere
A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a circle in two-dimensional space. A sphere is the Locus (mathematics), set of points that are ...

in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance from a central point . The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball). The volume of the ball is given by
:$V\; =\; \backslash frac\backslash pi\; r^$.
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space . If a point has coordinates, , then characterizes those points on the unit 3-sphere centered at the origin.
Polytopes

In three dimensions, there are nine regular polytopes: the five convexPlatonic solid
In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...

s and the four nonconvex Kepler-Poinsot polyhedra.
Surfaces of revolution

Asurface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

generated by revolving a plane curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

about a fixed line in its plane as an axis is called a surface of revolution
A surface of revolution is a surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermos ...

. The plane curve is called the '' generatrix'' of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.
Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone
A cone is a three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder
A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base.
This traditi ...

.
Quadric surfaces

In analogy with theconic section
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
:$Ax^2\; +\; By^2\; +\; Cz^2\; +\; Fxy\; +\; Gyz\; +\; Hxz\; +\; Jx\; +\; Ky\; +\; Lz\; +\; M\; =\; 0,$
where and are real numbers and not all of and are zero, is called a quadric surface.
There are six types of non-degenerate quadric surfaces:
# Ellipsoid
An ellipsoid is a surface that may be obtained from a sphere
A sphere (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country loca ...

# Hyperboloid of one sheet
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

# Hyperboloid of two sheets
# Elliptic cone
#
# Hyperbolic paraboloid
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...

The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surface
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

s, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus
Regulus , designated α Leonis (Latinisation of names, Latinized to Alpha Leonis, abbreviated Alpha Leo, α Leo), is the brightest object in the constellation Leo (constellation), Leo and one of the List of brightest stars, brightest ...

.
In linear algebra

Another way of viewing three-dimensional space is found inlinear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, where the idea of independence is crucial. Space has three dimensions because the length of a box
File:Box with cover MET DP241878.jpg, alt=A small, elaborate box, featuring a hinged lid, two swing doors at the front and a small pull-out drawer; the interior is entirely red and features small items that seem to be part of a toilette set, An el ...

is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

s.
Dot product, angle, and length

A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in can be represented by an ordered triple of real numbers. These numbers are called the components of the vector. The dot product of two vectors and is defined as: :$\backslash mathbf\backslash cdot\; \backslash mathbf\; =\; A\_1B\_1\; +\; A\_2B\_2\; +\; A\_3B\_3.$ The magnitude of a vector is denoted by . The dot product of a vector with itself is :$\backslash mathbf\; A\backslash cdot\backslash mathbf\; A\; =\; \backslash ,\; \backslash mathbf\; A\backslash ,\; ^2\; =\; A\_1^2\; +\; A\_2^2\; +\; A\_3^2,$ which gives : $\backslash ,\; \backslash mathbf\; A\backslash ,\; =\; \backslash sqrt\; =\; \backslash sqrt,$ the formula for theEuclidean length
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occa ...

of the vector.
Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors and is given by
:$\backslash mathbf\; A\backslash cdot\backslash mathbf\; B\; =\; \backslash ,\; \backslash mathbf\; A\backslash ,\; \backslash ,\backslash ,\; \backslash mathbf\; B\backslash ,\; \backslash cos\backslash theta,$
where is the angle
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

between and .
Cross product

Thecross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

or vector product is a binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

on two vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

s in three-dimensional space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Gre ...

and is denoted by the symbol ×. The cross product a × b of the vectors a and b is a vector that is perpendicular
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

to both and therefore normal to the plane containing them. It has many applications in mathematics, physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, and engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

.
The space and product form an algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...

, which is neither commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

nor associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, but is a Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

with the cross product being the Lie bracket.
One can in ''n'' dimensions take the product of vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.
In calculus

Gradient, divergence and curl

In a rectangular coordinate system, the gradient is given by :$\backslash nabla\; f\; =\; \backslash frac\; \backslash mathbf\; +\; \backslash frac\; \backslash mathbf\; +\; \backslash frac\; \backslash mathbf$ The divergence of acontinuously differentiable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

F = ''U'' i + ''V'' j + ''W'' k is equal to the scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...

-valued function:
:$\backslash operatorname\backslash ,\backslash mathbf\; =\; \backslash nabla\backslash cdot\backslash mathbf\; =\backslash frac\; +\backslash frac\; +\backslash frac.$
Expanded in Cartesian coordinates
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

(see Del in cylindrical and spherical coordinatesThis is a list of some vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculu ...

for spherical
of a sphere
A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values ...

and cylindrical
A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base.
This traditi ...

coordinate representations), the curl ∇ × F is, for F composed of x, ''F''unit vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s for the ''x''-, ''y''-, and ''z''-axes, respectively. This expands as follows:
:$\backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; \backslash mathbf\; +\; \backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; \backslash mathbf\; +\; \backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; \backslash mathbf$
Line integrals, surface integrals, and volume integrals

For somescalar field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

''f'' : ''U'' ⊆ Rcurve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

''C'' ⊂ ''U'' is defined as
:$\backslash int\backslash limits\_C\; f\backslash ,\; ds\; =\; \backslash int\_a^b\; f(\backslash mathbf(t))\; ,\; \backslash mathbf\text{'}(t),\; \backslash ,\; dt.$
where r: , b
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

→ ''C'' is an arbitrary bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

parametrization of the curve ''C'' such that r(''a'') and r(''b'') give the endpoints of ''C'' and $a\; <\; b$.
For a vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

F : ''U'' ⊆ Rcurve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

''C'' ⊂ ''U'', in the direction of r, is defined as
:$\backslash int\backslash limits\_C\; \backslash mathbf(\backslash mathbf)\backslash cdot\backslash ,d\backslash mathbf\; =\; \backslash int\_a^b\; \backslash mathbf(\backslash mathbf(t))\backslash cdot\backslash mathbf\text{'}(t)\backslash ,dt.$
where · is the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

and r: , b
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

→ ''C'' is a bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

parametrization of the curve ''C'' such that r(''a'') and r(''b'') give the endpoints of ''C''.
A surface integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a generalization of multiple integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s to integration over surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

s. It can be thought of as the double integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

analog of the line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. To find an explicit formula for the surface integral, we need to the surface of interest, ''S'', by considering a system of curvilinear coordinates
In , curvilinear coordinates are a for in which the s may be curved. These coordinates may be derived from a set of s by using a transformation that is (a one-to-one map) at each point. This means that one can convert a point given in a C ...

on ''S'', like the latitude and longitude
In geography
Geography (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

on a sphere
A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a circle in two-dimensional space. A sphere is the Locus (mathematics), set of points that are ...

. Let such a parameterization be x(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

. Then, the surface integral is given by
:$\backslash iint\_\; f\; \backslash ,\backslash mathrm\; dS\; =\; \backslash iint\_\; f(\backslash mathbf(s,\; t))\; \backslash left\backslash ,\; \backslash times\; \backslash right\backslash ,\; \backslash mathrm\; ds\backslash ,\; \backslash mathrm\; dt$
where the expression between bars on the right-hand side is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...

of the cross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

of the partial derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s of x(''s'', ''t''), and is known as the surface element
Element may refer to:
Science
* Chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all ...

. Given a vector field v on ''S'', that is a function that assigns to each x in ''S'' a vector v(x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
A volume integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

refers to an integral
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

over a 3-dimension
In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...

al domain.
It can also mean a triple integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

within a region ''D'' in Rfunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

$f(x,y,z),$ and is usually written as:
:$\backslash iiint\backslash limits\_D\; f(x,y,z)\backslash ,dx\backslash ,dy\backslash ,dz.$
Fundamental theorem of line integrals

The fundamental theorem of line integrals, says that aline integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

through a gradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
Let $\backslash varphi\; :\; U\; \backslash subseteq\; \backslash mathbb^n\; \backslash to\; \backslash mathbb$. Then
:$\backslash varphi\backslash left(\backslash mathbf\backslash right)-\backslash varphi\backslash left(\backslash mathbf\backslash right)\; =\; \backslash int\_\; \backslash nabla\backslash varphi(\backslash mathbf)\backslash cdot\; d\backslash mathbf.$
Stokes' theorem

Stokes' theorem
Stokes' theorem, also known as Kelvin–Stokes theorem
Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)"
:ja:培風館, Ba ...

relates the surface integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the curl
Curl or CURL may refer to:
Science and technology
* Curl (mathematics)
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the p ...

of a vector field
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Product ...

F over a surface Σ in Euclidean three-space to the line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the vector field over its boundary ∂Σ:
:$\backslash iint\_\; \backslash nabla\; \backslash times\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\backslash mathbf\; =\; \backslash oint\_\; \backslash mathbf\; \backslash cdot\; \backslash mathrm\; \backslash mathbf.$
Divergence theorem

Suppose is a subset of $\backslash mathbb^n$ (in the case of represents a volume in 3D space) which iscompact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

and has a piecewise
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

smooth boundary (also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then the divergence theorem
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

says:
:
The left side is a volume integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

over the volume , the right side is the surface integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

over the boundary of the volume . The closed manifold is quite generally the boundary of oriented by outward-pointing , and is the outward pointing unit normal field of the boundary . ( may be used as a shorthand for .)
In topology

Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie aknot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bend
Bend or bends may refer to:
Materials
* Bend, a curvature in a pipe, tube, o ...

in a piece of string.
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...

the generic three-dimensional spaces are 3-manifold
. All of the cubes in the image are the same cube, since light in the manifold wraps around into closed loops, the effect is that the cube is tiling all of space. This space has finite volume and no boundary.
In mathematics
Mathematics (from ...

s, which locally resemble $^3$.
In finite geometry

Many ideas of dimension can be tested withfinite geometry
A finite geometry is any geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθ ...

. The simplest instance is PG(3,2)
In finite geometry, PG(3,2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane.
It has 15 points, 35 lines, and 15 planes. It also has the following properties:
* Each point is contained in ...

, which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(''q''), there is a projective space PG(3,''q'') of three dimensions. For example, any three skew lines in PG(3,''q'') are contained in exactly one regulus
Regulus , designated α Leonis (Latinisation of names, Latinized to Alpha Leonis, abbreviated Alpha Leo, α Leo), is the brightest object in the constellation Leo (constellation), Leo and one of the List of brightest stars, brightest ...

.Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry'', page 72, Cambridge University Press
See also

* Dimensional analysis * Distance from a point to a plane * Four-dimensional space * * Graph of a function of two variables, Three-dimensional graph * Solid geometry * Two-dimensional spaceNotes

References

* * George B. Arfken, Arfken, George B. and Hans J. Weber. ''Mathematical Methods For Physicists'', Academic Press; 6 edition (June 21, 2005). . *External links

* *Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry

Keith Matthews from University of Queensland, 1991 {{Dimension topics Euclidean solid geometry, * Analytic geometry Multi-dimensional geometry Three-dimensional coordinate systems 3 (number)