View source for Three-dimensional space

{{short description|Geometric model of the physical space}}
{{redir|Three-dimensional||3D (disambiguation)}}
[[File:Coord planes color.svg|thumb|A representation of a three-dimensional [[Cartesian coordinate system|Cartesian coordinate system]] with the ''x''-axis pointing towards the observer]]

In [[Geometry|geometry]], a '''three-dimensional space''' ('''3D space''', '''3-space''' or, rarely, '''tri-dimensional space''') is a mathematical space in which three values (''coordinates'') are required to determine the [[Position (geometry)|position]] of a [[Point (geometry)|point]]. Most commonly, it is the '''three-dimensional Euclidean space''', that is, the [[Euclidean space]] of [[Physics:Dimension|dimension]] three, which models physical space. More general three-dimensional spaces are called ''3-manifolds''. 
The term may also refer colloquially to a subset of space, a ''three-dimensional region'' (or 3D [[Domain (mathematical analysis)|domain]]),<ref name="IEV-i241"/> a ''solid figure''.

Technically, a [[Tuple|tuple]] of {{math|''n''}} [[Real number|numbers]] can be understood as the Cartesian coordinates of a location in a {{math|''n''}}-dimensional Euclidean space. The set of these {{mvar|n}}-tuples is commonly denoted <math>\R^n,</math> and can be identified to the pair formed by a {{mvar|n}}-dimensional Euclidean space and a [[Cartesian coordinate system]].
When {{math|1=''n'' = 3}}, this space is called the {{anchor|3D EUCLIDEAN SPACE|3D Euclidean space}}three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear).<ref>{{Cite web|title=Euclidean space - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Euclidean_space|access-date=2020-08-12|website=encyclopediaofmath.org|language=en}}</ref> In [[Physics:Classical physics|classical physics]], it serves as a model of the physical [[Astronomy:Universe|universe]], in which all known [[Physics:Matter|matter]] exists. When relativity theory is considered, it can be considered a local subspace of [[Physics:Space-time|space-time]].<ref name="IEV">{{cite web | title=Details for IEV number 113-01-02: "space" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=113-01-02 | language=ja | access-date=2023-11-07}}</ref> While this space remains the most compelling and useful way to model the world as it is experienced,<ref>{{Cite web|title=Euclidean space {{!}} geometry|url=https://www.britannica.com/science/Euclidean-space|access-date=2020-08-12|website=Encyclopedia Britannica|language=en}}</ref> it is only one example of a large variety of spaces in three dimensions called [[3-manifold|3-manifold]]s. In this classical example, when the three values refer to measurements in different directions ([[Coordinate system|coordinates]]), any three directions can be chosen, provided that these directions do not lie in the same [[Plane (geometry)|plane]]. Furthermore, if these directions are pairwise [[Perpendicular|perpendicular]], the three values are often labeled by the terms ''width/breadth'', ''[[Height|height]]/depth'', and ''[[Physics:Length|length]]''.

==History==
Books XI to XIII of [[Euclid's Elements]] dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regular [[Platonic solid]]s in a sphere.

In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of [[Analytic geometry|analytic geometry]] developed by [[Biography:René Descartes|René Descartes]] in his work ''[[La Géométrie]]'' and [[Biography:Pierre de Fermat|Pierre de Fermat]] in the manuscript ''Ad locos planos et solidos isagoge'' (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.

In the 19th century, developments of the geometry of three-dimensional space came with [[Biography:William Rowan Hamilton|William Rowan Hamilton]]'s development of the quaternions. In fact, it was Hamilton who coined the terms [[Scalar (mathematics)|scalar]] and [[Euclidean vector|vector]], and they were first defined within [[Classical Hamiltonian quaternions|his geometric framework for quaternions]]. Three dimensional space could then be described by quaternions <math>q = a + ui + vj + wk</math> which had vanishing scalar component, that is, <math>a = 0</math>. While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements <math>i,j,k</math>, as well as the [[Dot product|dot product]] and [[Cross product|cross product]], which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions.

It was not until [[Biography:Josiah Willard Gibbs|Josiah Willard Gibbs]] that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook ''[[Vector Analysis]]'' written by [[Biography:Edwin Bidwell Wilson|Edwin Bidwell Wilson]] based on Gibbs' lectures.

Also during the 19th century came developments in the abstract formalism of vector spaces, with the work of [[Biography:Hermann Grassmann|Hermann Grassmann]] and [[Biography:Giuseppe Peano|Giuseppe Peano]], the latter of whom first gave the modern definition of vector spaces as an algebraic structure.

==In Euclidean geometry==

===Coordinate systems===
{{main|Coordinate system}}
{{General geometry |expanded=3d}}

In mathematics, [[Analytic geometry|analytic geometry]] (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the [[Origin (mathematics)|origin]], the point at which they cross. They are usually labeled {{math|''x'', ''y''}}, and {{math|''z''}}. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of [[Real number|real number]]s, each number giving the distance of that point from the [[Origin (mathematics)|origin]] measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.<ref name="Hughes">{{cite book|last1=Hughes-Hallett|first1=Deborah|last2=McCallum|first2=William G.|last3=Gleason|first3=Andrew M.|title=Calculus : Single and Multivariable|date=2013|publisher=John wiley|isbn=978-0470-88861-2|edition=6}}</ref>

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. For more, see [[Euclidean space]].

Below are images of the above-mentioned systems.

<gallery>
Image:Coord XYZ.svg|[[Cartesian coordinate system]]
Image:Cylindrical Coordinates.svg|[[Cylindrical coordinate system]]
Image:Spherical Coordinates (Colatitude, Longitude).svg|[[Spherical coordinate system]]
</gallery>

===Lines and planes===

Two distinct points always determine a (straight) [[Line (geometry)|line]]. Three distinct points are either collinear or determine a unique [[Plane (geometry)|plane]]. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space.

Two distinct lines can either intersect, be parallel or be [[Skew lines|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|hyperplane]] 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|linear equation]], 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 ℝ<sup>3</sup> form a [[Parallelogram|parallelogram]], and hence are coplanar.

===Spheres and balls===
{{main|Sphere}}
[[File:Sphere wireframe 10deg 6r.svg|right|thumb|A perspective projection of a sphere onto two dimensions]]
A [[Sphere|sphere]] 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 {{math|''r''}} from a central point {{mvar|P}}. The solid enclosed by the sphere is called a [[Ball (mathematics)|'''ball''']] (or, more precisely a '''3-ball''').

The volume of the ball is given by

<math display = block>V = \frac{4}{3}\pi r^{3},</math>
and the surface area of the sphere is
<math display = block>A = 4\pi r^2.</math>
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 {{math|'''R'''<sup>4</sup>}}. If a point has coordinates, {{math|''P''(''x'', ''y'', ''z'', ''w'')}}, then {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> + ''w''<sup>2</sup> = 1}} characterizes those points on the unit 3-sphere centered at the origin.

This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space.

===Polytopes===
{{main|Polyhedron}}
In three dimensions, there are nine regular polytopes: the five convex [[Platonic solid]]s and the four nonconvex Kepler-Poinsot polyhedra.

{| class=wikitable
|+ Regular polytopes in three dimensions
|- align=center
!Class
!colspan=5|[[Platonic solid]]s
!colspan=4|Kepler-Poinsot polyhedra
|-
!Symmetry
![[Tetrahedral symmetry|T<sub>d</sub>]]
!colspan=2|[[Octahedral symmetry|O<sub>h</sub>]]
!colspan=6|[[Icosahedral symmetry|I<sub>h</sub>]]
|-
![[Coxeter group]]
!A<sub>3</sub>, [3,3]
!colspan=2|B<sub>3</sub>, [4,3]
!colspan=6|H<sub>3</sub>, [5,3]
|- align=center
!Order
|24
|colspan=2|48
|colspan=6|120
|- align=center
![[Regular polyhedron|Regular<br>polyhedron]]
|[[File:Tetrahedron.svg|50px]]<br>[[Tetrahedron|{3,3}]]
|[[File:Hexahedron.svg|50px]]<br>[[Cube|{4,3}]]
|[[File:Octahedron.svg|50px]]<br>[[Octahedron|{3,4}]]
|[[File:Dodecahedron.svg|50px]]<br>[[Dodecahedron|{5,3}]]
|[[File:Icosahedron.svg|50px]]<br>[[Icosahedron|{3,5}]]
|[[File:SmallStellatedDodecahedron.jpg|50px]]<br>[[Small stellated dodecahedron|{5/2,5}]]
|[[File:GreatDodecahedron.jpg|50px]]<br>[[Great dodecahedron|{5,5/2}]]
|[[File:GreatStellatedDodecahedron.jpg|50px]]<br>[[Great stellated dodecahedron|{5/2,3}]]
|[[File:GreatIcosahedron.jpg|50px]]<br>[[Great icosahedron|{3,5/2}]]
|}

===Surfaces of revolution===
{{main|Surface of revolution}}
A [[Surface (topology)|surface]] generated by revolving a plane [[Curve|curve]] about a fixed line in its plane as an axis is called a [[Surface of revolution|surface of revolution]]. The plane curve is called the ''[[Generatrix|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 with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder.

===Quadric surfaces===
In analogy with the [[Conic section|conic section]]s, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
<math display="block">Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0,</math>
where {{math|''A'', ''B'', ''C'', ''F'', ''G'', ''H'', ''J'', ''K'', ''L''}} and {{math|''M''}} are real numbers and not all of {{math|''A'', ''B'', ''C'', ''F'', ''G''}} and {{math|''H''}} are zero, is called a '''quadric surface'''.<ref name=Brannan>{{harvnb|Brannan|Esplen|Gray|1999|pages=34–35}}</ref>

There are six types of [[Degeneracy (mathematics)|non-degenerate]] quadric surfaces:
# [[Ellipsoid]]
# Hyperboloid of one sheet
# Hyperboloid of two sheets
# Elliptic cone
# Elliptic paraboloid
# Hyperbolic paraboloid

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 {{pi}} and all the lines of {{math|'''R'''<sup>3</sup>}} through that conic that are normal to {{pi}}).<ref name=Brannan /> 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|ruled surface]]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.<ref>{{harvnb|Brannan|Esplen|Gray|1999|pages=41–42}}</ref> Each family is called a [[Regulus (geometry)|regulus]].

==In linear algebra==
Another way of viewing three-dimensional space is found in [[Linear algebra|linear algebra]], where the idea of independence is crucial. Space has three dimensions because the length of a [[Cuboid|box]] 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 [[Coordinate vector|vector]]s.

===Dot product, angle, and length===

The dot product of two vectors {{math|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>''3''</sub>]}} and {{math|1='''B''' = [''B''<sub>1</sub>, ''B''<sub>2</sub>, ''B''<sub>''3''</sub>]}} is defined as:<ref>{{harvnb|Anton|1994|page=133}}</ref>

:<math>\mathbf{A}\cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3 = \sum_{i=1}^3 A_i B_i.</math>

The magnitude of a vector {{math|'''A'''}} is denoted by {{math|{{!}}{{!}}'''A'''{{!}}{{!}}}}. The dot product of a vector {{math|1='''A''' = [''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>''3''</sub>]}} with itself is
:<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2 = A_1^2 + A_2^2 + A_3^2,</math>
which gives
: <math> \|\mathbf A\| = \sqrt{\mathbf A\cdot\mathbf A} = \sqrt{A_1^2 + A_2^2 + A_3^2},</math>
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 {{math|'''A'''}} and {{math|'''B'''}} is given by<ref>{{harvnb|Anton|1994|page=131}}</ref>
:<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,</math>
where {{math|''θ''}} is the [[Angle|angle]] between {{math|'''A'''}} and {{math|'''B'''}}.

===Cross product===
{{main|Cross product}}
The [[Cross product|cross product]] or '''vector product''' is a [[Binary operation|binary operation]] on two [[Euclidean vector|vector]]s in three-dimensional [[Euclidean space|space]] and is denoted by the symbol  ×. The cross product '''A''' × '''B''' of the vectors '''A''' and '''B''' is a vector that is [[Perpendicular|perpendicular]] to both and therefore [[Normal (geometry)|normal]] to the plane containing them. It has many applications in mathematics, [[Physics:Physics|physics]], and [[Engineering:Engineering|engineering]].

In function language, the cross product is a function <math>\times: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^3</math>.

The components of the cross product are {{nowrap|<math>\mathbf{A}\times\mathbf{B} = [A_2 B_3 - B_2 A_3, A_3 B_1 - B_3 A_1, A_1 B_2 - B_1 A_2]</math>,}} and can also be written in components, using Einstein summation convention as <math>(\mathbf{A}\times\mathbf{B})_i = \varepsilon_{ijk} A_j B_k</math> where <math>\varepsilon_{ijk}</math> is the [[Levi-Civita symbol]]. It has the property that <math>\mathbf{A}\times \mathbf{B} = -\mathbf{B}\times \mathbf{A}</math>.

Its magnitude is related to the angle <math>\theta</math> between <math>\mathbf{A}</math> and <math>\mathbf{B}</math> by the identity
<math display = block> \left\|\mathbf{A}\times \mathbf{B}\right\| = \left\|\mathbf{A}\right\| \cdot \left\|\mathbf{B}\right\| \cdot \left|\sin\theta\right|.</math>

The space and product form an [[Algebra over a field|algebra over a field]], which is not commutative nor associative, but is a [[Lie algebra]] with the cross product being the Lie bracket. Specifically, the space together with the product, <math>(\mathbb{R}^3,\times)</math> is isomorphic to the Lie algebra of three-dimensional rotations, denoted <math>\mathfrak{so}(3)</math>. In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the [[Jacobi identity]]. For any three vectors <math>\mathbf{A}, \mathbf{B}</math> and <math>\mathbf{C}</math>

<math display = block>\mathbf{A}\times(\mathbf{B}\times\mathbf{C}) + \mathbf{B}\times(\mathbf{C}\times\mathbf{A}) + \mathbf{C}\times(\mathbf{A}\times\mathbf{B}) = 0</math>

One can in ''n'' dimensions take the product of {{nowrap|''n'' − 1}} 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-dimensional cross product|seven dimensions]].<ref name=Massey2>{{cite journal |title=Cross products of vectors in higher dimensional Euclidean spaces |first=WS |last=Massey |year=1983 |jstor=2323537|quote=If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space. |pages=697–701 |journal=The American Mathematical Monthly |volume=90 |issue=10 |doi=10.2307/2323537}}</ref>
[[File:Cross product vector.svg|thumb|right|The cross-product in respect to a right-handed coordinate system]]

===Abstract description===

It can be useful to describe three-dimensional space as a three-dimensional vector space <math>V</math> over the real numbers. This differs from <math>\mathbb{R}^3</math> in a subtle way. By definition, there exists a basis <math>\mathcal{B} = \{e_1,e_2,e_3\}</math> for <math>V</math>. This corresponds to an [[Isomorphism|isomorphism]] between <math>V</math> and <math>\mathbb{R}^3</math>: the construction for the isomorphism is found [[Basis (linear algebra)#Coordinates|here]]. However, there is no 'preferred' or 'canonical basis' for <math>V</math>.

On the other hand, there is a preferred basis for <math>\mathbb{R}^3</math>, which is due to its description as a [[Cartesian product]] of copies of <math>\mathbb{R}</math>, that is, <math>\mathbb{R}^3 = \mathbb{R}\times \mathbb{R}\times \mathbb{R}</math>. This allows the definition of canonical projections, <math>\pi_i:\mathbb{R}^3 \rightarrow \mathbb{R}</math>, where <math>1 \leq i \leq 3</math>. For example, <math>\pi_1(x_1,x_2,x_3) = x</math>. This then allows the definition of the [[Standard basis|standard basis]] <math>\mathcal{B}_{\text{Standard}} = \{E_1, E_2, E_3\}</math> defined by
<math display = block>\pi_i(E_j) = \delta_{ij}</math>
where <math>\delta_{ij}</math> is the [[Kronecker delta]]. Written out in full, the standard basis is
<math display = block>E_1 = \begin{pmatrix}1 \\ 0\\ 0\end{pmatrix}, E_2 = \begin{pmatrix}0 \\ 1\\ 0\end{pmatrix}, E_3 = \begin{pmatrix}0 \\ 0\\ 1\end{pmatrix}.</math>

Therefore <math>\mathbb{R}^3</math> can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, <math>V</math> can be obtained by starting with <math>\mathbb{R}^3</math> and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis.

As opposed to a general vector space <math>V</math>, the space <math>\mathbb{R}^3</math> is sometimes referred to as a coordinate space.<ref>{{Harvard citations|last = Lang|year = 1987|loc = ch. I.1|nb = yes}}</ref>

Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space <math>\mathbb{R}^3</math> assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space.

Computationally, it is necessary to work with the more concrete description <math>\mathbb{R}^3</math> in order to do concrete computations.

====Affine description====
A more abstract description still is to model physical space as a three-dimensional affine space <math>E(3)</math> over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of <math>\mathbb{R}^3</math>, the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' for distinguishing them from Euclidean vector spaces.{{sfn|Berger|1987|loc=Chapter 9}}

This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.

====Inner product space====
The above discussion does not involve the [[Dot product|dot product]]. The dot product is an example of an inner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines notions of length and angle (and therefore in particular the notion of orthogonality). For any inner product, there exist bases under which the inner product agrees with the dot product, but again, there are many different possible bases, none of which are preferred. They differ from one another by a rotation, an element of the group of rotations [[SO(3)]].

==In calculus==
{{main|Vector calculus}}

===Gradient, divergence and curl===
In a rectangular coordinate system, the gradient of a (differentiable) function <math>f: \mathbb{R}^3 \rightarrow \mathbb{R}</math> is given by

:<math>\nabla f = \frac{\partial f}{\partial x} \mathbf{i} +
\frac{\partial f}{\partial y}  \mathbf{j} +
\frac{\partial f}{\partial z} \mathbf{k}</math>

and in [[Index notation|index notation]] is written

<math display=block>(\nabla f)_i = \partial_i f.</math>

The divergence of a (differentiable) [[Vector field|vector field]] '''F''' = ''U'' '''i''' + ''V'' '''j''' + ''W'' '''k''', that is, a function <math>\mathbf{F}:\mathbb{R}^3 \rightarrow \mathbb{R}^3</math>, is equal to the [[Scalar (mathematics)|scalar]]-valued function:

:<math>\operatorname{div}\,\mathbf{F} = \nabla\cdot\mathbf{F}
=\frac{\partial U}{\partial x}
+\frac{\partial V}{\partial y}
+\frac{\partial W}{\partial z
}. </math>

In index notation, with Einstein summation convention this is 
<math display=block>\nabla \cdot \mathbf{F} = \partial_i F_i.</math>

Expanded in [[Cartesian coordinate system|Cartesian coordinates]] (see [[Del in cylindrical and spherical coordinates]] for [[Spherical coordinate system|spherical]] and [[Cylindrical coordinate system|cylindrical]] coordinate representations), the curl ∇ × '''F''' is, for '''F''' composed of [''F''<sub>x</sub>, ''F''<sub>y</sub>, ''F''<sub>z</sub>]:

:<math>\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\  \\
{\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\
 \\  F_x & F_y & F_z \end{vmatrix}</math>

where '''i''', '''j''', and '''k''' are the [[Unit vector|unit vector]]s for the ''x''-, ''y''-, and ''z''-axes, respectively. This expands as follows:<ref>Arfken, p. 43.</ref>

:<math>\left(\frac{\partial F_z}{\partial y}  - \frac{\partial F_y}{\partial z}\right) \mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}.</math>

In index notation, with Einstein summation convention this is 
<math display=block> (\nabla \times \mathbf{F})_i = \epsilon_{ijk}\partial_j F_k,</math>
where <math>\epsilon_{ijk}</math> is the totally antisymmetric symbol, the [[Levi-Civita symbol]].

===Line, surface, and volume integrals===
For some [[Scalar field|scalar field]] ''f'' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R''', the ''[[Line integral|line integral]]'' along a piecewise smooth curve ''C'' ⊂  ''U'' is defined as
:<math>\int\limits_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.</math>
where '''r''': [a, b] → ''C'' is an arbitrary bijective parametrization of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C'' and <math>a < b</math>.

For a [[Vector field|vector field]] '''F''' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>, the line integral along a piecewise smooth [[Curve|curve]] ''C'' ⊂ ''U'', in the direction of '''r''', is defined as

:<math>\int\limits_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.</math>

where · is the [[Dot product|dot product]] and '''r''': [a, b] → ''C'' is a bijective [[Parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C''.

A ''[[Surface integral|surface integral]]'' is a generalization of [[Multiple integral|multiple integral]]s to integration over [[Surface (topology)|surface]]s. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to [[Coordinate system|parameterize]] the surface of interest, ''S'', by considering a system of [[Curvilinear coordinates|curvilinear coordinates]] on ''S'', like the [[Earth:Geographic coordinate system|latitude and longitude]] on a [[Sphere|sphere]]. Let such a parameterization be '''x'''(''s'', ''t''), where (''s'', ''t'') varies in some region ''T'' in the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|plane]]. Then, the surface integral is given by

:<math>
\iint_{S} f \,\mathrm dS
= \iint_{T} f(\mathbf{x}(s, t)) \left\|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right\| \mathrm ds\, \mathrm dt
</math>
where the expression between bars on the right-hand side is the [[Magnitude (mathematics)|magnitude]] of the [[Cross product|cross product]] of the [[Partial derivative|partial derivative]]s of '''x'''(''s'', ''t''), and is known as the surface [[Volume element|element]]. 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|volume integral]]'' is an [[Integral|integral]] over a ''three-dimensional [[Domain (mathematical analysis)|domain]]'' or region.
When the integrand is trivial (unity), the volume integral is simply the region's ''[[Physics:Volume|volume]]''.<ref name="IEV-h648">{{cite web | title=IEC 60050 — Details for IEV number 102-04-40: "volume" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-04-40 | language=ja | access-date=2023-09-19}}</ref><ref name="IEV-i241">{{cite web | title=IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain" | website=International Electrotechnical Vocabulary | url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-04-39 | language=ja | access-date=2023-09-19}}</ref>
It can also mean a [[Multiple integral|triple integral]] within a region ''D'' in '''R'''<sup>3</sup> of a [[Function (mathematics)|function]] <math>f(x,y,z),</math> and is usually written as:

:<math>\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.</math>

===Fundamental theorem of line integrals===
The fundamental theorem of line integrals, says that a [[Line integral|line integral]] through a [[Gradient|gradient]] field can be evaluated by evaluating the original scalar field at the endpoints of the curve.

Let <math> \varphi : U \subseteq \mathbb{R}^n \to \mathbb{R}</math>. Then

:<math> \varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right) = \int_{\gamma[\mathbf{p},\,\mathbf{q}]} \nabla\varphi(\mathbf{r})\cdot d\mathbf{r}. </math>

===Stokes' theorem===
{{main|Physics:Stokes' theorem}}
[[Physics:Stokes' theorem|Stokes' theorem]] relates the [[Surface integral|surface integral]] of the [[Curl (mathematics)|curl]] of a [[Vector field|vector field]] F over a surface Σ in Euclidean three-space to the [[Line integral|line integral]] of the vector field over its boundary ∂Σ:

:<math> \iint_{\Sigma} \nabla \times \mathbf{F} \cdot \mathrm{d}\mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d} \mathbf{r}. </math>

===Divergence theorem===
{{main|Divergence theorem}}
Suppose {{mvar|V}} is a subset of <math>\mathbb{R}^n</math> (in the case of {{math|''n'' {{=}} 3, ''V''}} represents a volume in 3D space) which is [[Compact space|compact]] and has a piecewise smooth boundary {{mvar|S}} (also indicated with {{math|∂''V'' {{=}} ''S''&thinsp;}}). If {{math|'''F'''}} is a continuously differentiable vector field defined on a neighborhood of {{mvar|V}}, then the [[Divergence theorem|divergence theorem]] says:<ref name=spiegel>{{cite book |author1=M. R. Spiegel |author2=S. Lipschutz |author3=D. Spellman | title = Vector Analysis | edition = 2nd | series = Schaum's Outlines | publisher = McGraw Hill | location = US | year = 2009 | isbn = 978-0-07-161545-7 }}</ref>

:{{oiint
| preintegral = <math>\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV=</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>(\mathbf{F}\cdot\mathbf{n})\,dS .</math>
}}

The left side is a [[Volume integral|volume integral]] over the volume {{mvar|V}}, the right side is the [[Surface integral|surface integral]] over the boundary of the volume {{mvar|V}}. The closed manifold {{math|∂''V''}} is quite generally the boundary of {{mvar|V}} oriented by outward-pointing normals, and {{math|'''n'''}} is the outward pointing unit normal field of the boundary {{math|∂''V''}}. ({{math|''d'''''S'''}} may be used as a shorthand for {{math|'''n'''''dS''}}.)

==In topology==
[[File:WikipediaGlobeOnePiece.stl|thumb|upright=1.2|[[Wikipedia|Wikipedia]]'s globe logo in 3-D]]
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 theory|knot]] in a piece of string.<ref>{{cite book |first1=Dale |last1=Rolfsen |title=Knots and Links |publisher=Publish or Perish |location=Berkeley, California |year=1976 |isbn=0-914098-16-0}}</ref>

In [[Differential geometry|differential geometry]] the generic three-dimensional spaces are [[3-manifold|3-manifold]]s, which locally resemble <math>{\mathbb{R}}^3</math>.

==In finite geometry==
Many ideas of dimension can be tested with [[Finite geometry|finite geometry]]. The simplest instance is [[PG(3,2)]], which has [[Fano plane]]s as its 2-dimensional subspaces. It is an instance of [[Galois geometry]], a study of [[Projective geometry|projective geometry]] using [[Finite field|finite field]]s. Thus, for any Galois field GF(''q''), there is a [[Projective space|projective space]] PG(3,''q'') of three dimensions. For example, any three [[Skew lines|skew lines]] in PG(3,''q'') are contained in exactly one [[Regulus (geometry)|regulus]].<ref>Albrecht Beutelspacher & Ute Rosenbaum (1998) ''Projective Geometry'', page 72, {{wipe|Cambridge University Press}}  {{ISBN|0-521-48277-1}}</ref>

==See also==
* 3D rotation
** [[Rotation formalisms in three dimensions]]
* [[Dimensional analysis]]
* [[Distance from a point to a plane]]
* [[Four-dimensional space]]
* {{section link|Skew lines|Distance}}
* Three-dimensional graph
* [[Solid geometry]]

==Notes==
{{reflist}}

==References==
* {{citation|first=Howard|last=Anton|title=Elementary Linear Algebra|edition=7th|year=1994|publisher=John Wiley & Sons|isbn=978-0-471-58742-2}}
* [[Biography:George B. Arfken|Arfken, George B.]] and Hans J. Weber. ''Mathematical Methods For Physicists'', Academic Press; 6 edition (June 21, 2005). {{isbn|978-0-12-059876-2}}.
* {{citation|first1=David A.|last1=Brannan|first2=Matthew F.|last2=Esplen|first3=Jeremy J.|last3=Gray|title=Geometry|year=1999|publisher=Cambridge University Press|isbn=978-0-521-59787-6}}

==External links==
* {{MathWorld |title=Four-Dimensional Geometry |id=Four-DimensionalGeometry}}
* [http://www.numbertheory.org/book/cha8.pdf Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry] Keith Matthews from University of Queensland, 1991

[[Category:Euclidean solid geometry|*]]
[[Category:Analytic geometry]]
[[Category:Multi-dimensional geometry]]
[[Category:Three-dimensional coordinate systems]]
[[Category:3 (number)]]
[[Category:Space]]

{{Sourceattribution|Three-dimensional space}}