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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 παρά, ''para'': "beside", "subsidiary"; and μέτρον, ''metron'': "measure"), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, ob ...
s) are required to determine the position of an element (i.e.,
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, ...
). This is the informal meaning of the term
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
. In
physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its motion and behavior through space and time, and the related ent ...
and
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ...
of
numbers A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be re ...
can be understood as a location in -dimensional space. When , the set of all such locations is called three-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...
(or simply Euclidean space when the context is clear). It is commonly represented by the symbol . This serves as a three-parameter model of the physical
universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the un ...
(that is, the spatial part, without considering time), in which all known
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic particl ...
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, a 3-manifold is a space ...
s. In this classical example, when the three values refer to measurements in different directions (
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is signif ...
), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space ( plane). 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. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Internati ...
'', ''
height 200px, A cuboid demonstrating the dimensions length, width">length.html" style="text-decoration: none;"class="mw-redirect" title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height. H ...
'', ''depth'', and ''
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Internati ...
''.


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 using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering ...
(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 that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the s ...
are given, each perpendicular to the other two at the
origin Origin, origins, or original may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * Origins (''Judge Dredd'' story), a major ''Judge Dredd'' ...
, 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 Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s, each number giving the distance of that point from the
origin Origin, origins, or original may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * Origins (''Judge Dredd'' story), a major ''Judge Dredd'' ...
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 spe ...

cylindrical coordinates
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 ...
, though there are an infinite number of possible methods. For more, see
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-di ...
. Below are images of the above-mentioned systems. Image:Coord XYZ.svg,
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the s ...
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 that spe ...
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 ...

Spherical coordinate system


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 Literatu ...
. Three distinct points are either
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned obj ...
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 that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, ...
, or determine the entire space. Two distinct lines can either intersect, be
parallel Parallel may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM mainframes * Parallel communication * P ...
or be
skew Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not li ...
. Two parallel lines, or
two intersecting lines 2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures. ...
, 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, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dim ...
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, a linear equation is an equation that may be put in the form :a_1x_1+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variables (or unknowns), and b, a_1, \ldots, a_n are the coefficients, which are often real numbers. The coeffi ...
, 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 300px, Area(''EFGH'') = (1/2)Area(''ABCD'') Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is name ...
states that the midpoints of any quadrilateral in ℝ3 form a
parallelogram In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal ...
, and hence are coplanar.


Spheres and balls

A
sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...
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 = \frac\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 convex
Platonic solid In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent (identical in shape and size), regular (all angles equal and all sides equal), polygonal faces with the same number of faces meeting at each ...
s and the four nonconvex Kepler-Poinsot polyhedra.


Surfaces of revolution

A
surface Water droplet lying on a damask. Surface tension">damask.html" style="text-decoration: none;"class="mw-redirect" title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating ...
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, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that app ...
about a fixed line in its plane as an axis is called a
surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on wh ...
. 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 geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecti ...
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 Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can havi ...
.


Quadric surfaces

In analogy with the
conic section In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of ...
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 In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ' ...
quadric surfaces: #
Ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the zer ...
#
Hyperboloid of one sheet In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deformi ...
#
Hyperboloid of two sheets In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deformi ...
# Elliptic cone #
Elliptic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane se ...
#
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 m ...
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, a surface ''S'' is ruled (also called a scroll) if through every point of ''S'' there is a straight line that lies on ''S''. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical direc ...
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 (Latinized to Alpha Leonis, abbreviated Alpha Leo, α Leo), is the brightest object in the constellation Leo and one of the brightest stars in the night sky, lying approximately 79 light years from the S ...
.


In linear algebra

Another way of viewing three-dimensional space is found in
linear algebra#REDIRECT Linear algebra#REDIRECT Linear algebra {{R from other capitalisation ...
{{R from other capitalisation ...
, where the idea of independence is crucial. Space has three dimensions because the length of a
box A box (plural: boxes) is a type of container or rectangular prism used for the storage or transportation of its contents. The size of a box may vary, from the very smallest (such as a matchbox) to the size of a large appliance, and can be used fo ...
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), 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 ...
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: :\mathbf\cdot \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 :\mathbf A\cdot\mathbf A = \, \mathbf A\, ^2 = A_1^2 + A_2^2 + A_3^2, which gives : \, \mathbf A\, = \sqrt = \sqrt, the formula for the Euclidean length of the vector. Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors and is given by :\mathbf A\cdot\mathbf B = \, \mathbf A\, \,\, \mathbf B\, \cos\theta, where is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are als ...
between and .


Cross product

The
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space \mathbb^3, and is denoted by the symbol \times. Given ...

cross product
or vector product is a
binary operation In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operatio ...
on two
vector 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 ...
s in three-dimensional
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be par ...
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, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
to both and therefore normal to the plane containing them. It has many applications in mathematics,
physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its motion and behavior through space and time, and the related ent ...
, 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 specializ ...
. 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 product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and ...
, which is neither
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of th ...
nor
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for ...
, but is a
Lie algebra In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, \ (x, y) \mapsto , y/math>, t ...
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 :\nabla f = \frac \mathbf + \frac \mathbf + \frac \mathbf The divergence of a
continuously differentiable In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each ...
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

vector field
F = ''U'' i + ''V'' j + ''W'' k is equal to the scalar-valued function: :\operatorname\,\mathbf = \nabla\cdot\mathbf =\frac +\frac +\frac. Expanded in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the s ...
(see
Del in cylindrical and spherical coordinatesThis is a list of some vector calculus formulae for working with common curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse t ...
for
spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...

spherical
and
cylindrical A cylinder (from Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can havi ...
coordinate representations), the curl ∇ × F is, for F composed of 'F''x, ''F''y, ''F''z :\begin \mathbf & \mathbf & \mathbf \\ \\ & & \\ \\ F_x & F_y & F_z \end where i, j, and k are the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vect ...
s for the ''x''-, ''y''-, and ''z''-axes, respectively. This expands as follows: :\left(\frac - \frac\right) \mathbf + \left(\frac - \frac\right) \mathbf + \left(\frac - \frac\right) \mathbf


Line integrals, surface integrals, and volume integrals

For some
scalar field In mathematics and physics, a scalar field or scalar-valued function associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a ph ...

scalar field
''f'' : ''U'' ⊆ R''n'' → R, the line integral along a
piecewise smooth In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piec ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that app ...
''C'' ⊂ ''U'' is defined as :\int\limits_C f\, ds = \int_a^b f(\mathbf(t)) , \mathbf'(t), \, dt. where r: , b→ ''C'' is an arbitrary
bijective In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element ...
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 and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

vector field
F : ''U'' ⊆ R''n'' → R''n'', the line integral along a
piecewise smooth In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piec ...
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that app ...
''C'' ⊂ ''U'', in the direction of r, is defined as :\int\limits_C \mathbf(\mathbf)\cdot\,d\mathbf = \int_a^b \mathbf(\mathbf(t))\cdot\mathbf'(t)\,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 ...

dot product
and r: , b→ ''C'' is a
bijective In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element ...
parametrization of the curve ''C'' such that r(''a'') and r(''b'') give the endpoints of ''C''. A
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may i ...
is a generalization of
multiple integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number pl ...
s to integration over
surface Water droplet lying on a damask. Surface tension">damask.html" style="text-decoration: none;"class="mw-redirect" title="Water droplet lying on a damask">Water droplet lying on a damask. Surface tension is high enough to prevent floating ...
s. It can be thought of as the
double integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number pl ...
analog of the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alth ...
. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, ''S'', by considering a system of
curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invert ...
on ''S'', like the
latitude and longitude A geographic coordinate system (GCS) is a coordinate system associated with positions on Earth (geographic position). A GCS can give positions: *as spherical coordinate system using latitude, longitude, and elevation; *as map coordinates p ...
on a
sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...
. Let such a parameterization be x(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the plane. Then, the surface integral is given by : \iint_ f \,\mathrm dS = \iint_ f(\mathbf(s, t)) \left\, \times \right\, \mathrm ds\, \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 mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space \mathbb^3, and is denoted by the symbol \times. Given ...

cross product
of the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Parti ...
s of x(''s'', ''t''), and is known as the surface
element Element may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of one body aroun ...
. 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 (particularly multivariable calculus), a volume integral refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applicati ...
refers to an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with diffe ...
over a 3-
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
al domain. It can also mean a
triple integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number pl ...
within a region ''D'' in R3 of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriented ...
f(x,y,z), and is usually written as: :\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.


Fundamental theorem of line integrals

The fundamental theorem of line integrals, says that a
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alth ...
through a
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the vector whose components are the partial derivatives of f a ...

gradient
field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let \varphi : U \subseteq \mathbb^n \to \mathbb. Then : \varphi\left(\mathbf\right)-\varphi\left(\mathbf\right) = \int_ \nabla\varphi(\mathbf)\cdot d\mathbf.


Stokes' theorem

Stokes' theorem An illustration of Stokes' theorem, with surface , its boundary and the normal vector . Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fuji ...
relates the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may i ...
of the
curl cURL (pronounced 'curl') is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL", which was first released in 1997. Hi ...
of a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

vector field
F over a surface Σ in Euclidean three-space to the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alth ...
of the vector field over its boundary ∂Σ: : \iint_ \nabla \times \mathbf \cdot \mathrm\mathbf = \oint_ \mathbf \cdot \mathrm \mathbf.


Divergence theorem

Suppose is a subset of \mathbb^n (in the case of represents a volume in 3D space) which is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
and has a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piec ...
smooth boundary (also indicated with ). If is a continuously differentiable vector field defined on a neighborhood of , then the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclos ...

divergence theorem
says: : The left side is a
volume integral In mathematics (particularly multivariable calculus), a volume integral refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applicati ...
over the volume , the right side is the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may i ...
over the boundary of the volume . The closed manifold is quite generally the boundary of oriented by outward-pointing normals, 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 a
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ''bend ...
in a piece of string. In
differential geometry Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in th ...
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, a 3-manifold is a space ...
s, which locally resemble ^3.


In finite geometry

Many ideas of dimension can be tested with
finite geometry A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer sc ...
. The simplest instance is PG(3,2), which has
Fano plane In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2. It is the finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 line ...

Fano plane
s as its 2-dimensional subspaces. It is an instance of
Galois geometry Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Galo ...
, a study of
projective geometry#REDIRECT projective geometry#REDIRECT projective geometry {{R from other capitalisation ...
{{R from other capitalisation ...
using
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtracti ...
s. Thus, for any Galois field GF(''q''), there is a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, ...
PG(3,''q'') of three dimensions. For example, any three
skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same p ...
in PG(3,''q'') are contained in exactly one
regulus Regulus , designated α Leonis (Latinized to Alpha Leonis, abbreviated Alpha Leo, α Leo), is the brightest object in the constellation Leo and one of the brightest stars in the night sky, lying approximately 79 light years from the S ...
.
Albrecht Beutelspacher Albrecht Beutelspacher (born 5 June 1950) is a German mathematician. Biography Beutelspacher studied 1969-1973 math, physics and philosophy at the University of Tübingen and received his PhD 1976 from the University of Mainz. His PhD advisor was ...
& Ute Rosenbaum (1998) ''Projective Geometry'', page 72,
Cambridge University Press Cambridge University Press (CUP) is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the Queen's Printer. Cambridge Universit ...


See also

*
Dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs ...
*
Distance from a point to a planeIn Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane or the nearest point on the plane. It can be found starting with a change of variables that moves the origin to ...
*
Four-dimensional space A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', t ...
* *
Three-dimensional graphA three-dimensional graph may refer to * A graph (discrete mathematics), embedded into a three-dimensional space * The graph of a function of two variables, embedded into a three-dimensional space {{mathdab ...
*
Two-dimensional space 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). The set ...


Notes


References

* * 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 , mottoeng = By means of knowledge and hard work , established = , endowment = A$224.3 million , budget = A$2.1 billion , type = Public research university , chancellor = Peter Varghese , vice_chancellor = Deborah Terry , city = Brisbane, Qu ...

University of Queensland
, 1991 {{Dimension topics * Analytic geometry Multi-dimensional geometry 3 (number)