Three-dimensional space

{{short description|Geometric model of the physical space}} {{For|a broader, less mathematical treatment related to this topic|Space}} {{redir|Three-dimensional||3D (disambiguation)}} [[File:Coord planes color.svg|thumb|A representation of a three-dimensional Cartesian coordinate system]]

In geometry, a '''three-dimensional space''' is a mathematical space in which three values (termed ''coordinates'') are required to determine the position of a point. Alternatively, it can be referred to as '''3D space''', '''3-space''' or, rarely, '''tri-dimensional space'''. Most commonly, it means the '''three-dimensional Euclidean space''', that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called ''3-manifolds''. The term may refer colloquially to a subset of space, a ''three-dimensional region'' (or 3D domain),<ref name="IEV-i241"/> a ''solid figure''.

Technically, a tuple of {{math|''n''}} 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 classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of 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 widely used 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 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms ''width/breadth'', ''height/depth'', and ''length''.

==History== The philosopher Aristotle recognised the existence of three dimensions:{{Quote|A magnitude if divisible one way is a line, if two ways a surface, and if three a body. Beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all.<ref>Aristotle (350 BC), [https://archive.org/details/decaeloleofric00arisuoft/decaeloleofric00arisuoft/page/n11/mode/2up ''De Caelo''] (Latin title), Book 1, translated by Stocks, J. L., published in 1922</ref>}}

Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of perpendicularity, parallelism, and orthogonality of lines and planes, the construction and properties of angles, and parallelepiped solids. Book XII discusses infinitesimals and the method of exhaustion for finding the area of a circle or the volume of a pyramid,<ref name=Artmann_2012/> cone, cylinder, or sphere.<ref>{{cite web | title=Euclid's Elements of Geometry | first=Richard | last=Fitzpatrick | date=August 26, 2014 | publisher=University of Texas | url=https://farside.ph.utexas.edu/books/Euclid/Euclid.html | access-date=2025-11-04 |archive-url=https://web.archive.org/web/20251122110646/https://farside.ph.utexas.edu/books/Euclid/Euclid.html |archive-date=2025-11-22}}</ref> Book XIII describes the construction of the five regular Platonic solids in a sphere, covering the cube, octahedra, icosahedra and dodecahedra.<ref name=Artmann_2012>{{cite book | title=Euclid—The Creation of Mathematics | first=Benno | last=Artmann | publisher=Springer Science & Business Media | year=2012 | isbn=978-1-4612-1412-0 | pages=9–10 | url=https://books.google.com/books?id=F8XgBwAAQBAJ&pg=PA9 }}</ref>

In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of analytic geometry developed by René Descartes in his work ''La Géométrie''.<ref>{{cite book | title=The Foundations of Geometry and the Non-Euclidean Plane | series=Mathematics and Statistics | first=G. E. | last=Martin | publisher=Springer Science & Business Media | year=2012 | page=51 | isbn=978-1-4612-5725-7 | url=https://books.google.com/books?id=_ynUBwAAQBAJ&pg=PA51 }}</ref> Pierre de Fermat independently developed similar ideas in the manuscript ''Ad locos planos et solidos isagoge'' (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime.<ref>{{cite book | title=Relay Race To Infinity, The: Developments In Mathematics From Euclid To Fermat | first1=Derek Allan | last1=Holton | first2=John | last2=Stillwell | publisher=World Scientific | year=2024 | isbn=978-981-12-9634-5 | page=158 | url=https://books.google.com/books?id=okEyEQAAQBAJ&pg=PA194 }}</ref> Fermat's work on seeking extrema of a curve would lay the groundwork for differential calculus.<ref>{{cite book | chapter=Mathematical Faits Divers | first=J.-B. | last=Hiriart-Urrety | title=Convexity and Duality in Optimization: Proceedings of the Symposium on Convexity and Duality in Optimization Held at the University of Groningen, The Netherlands June 22, 1984 | series=Lecture Notes in Economics and Mathematical Systems | editor-first=Jacob | editor-last=Ponstein | publisher=Springer Science & Business Media | year=2012 | page=3 | isbn=978-3-642-45610-7 | chapter-url=https://books.google.com/books?id=L1TsCAAAQBAJ&pg=PA3 }}</ref> Isaac Newton introduced the polar coordinate system as an alternative non-Cartesian system that is useful for certain geometries.<ref>{{cite journal | title=Newton as an Originator of Polar Coördinates | first=C. B. | last=Boyer | journal=The American Mathematical Monthly | volume=56 | issue=2 | date=February 1949 | pages=73-78 | doi=10.2307/2306162 | jstor=2306162 | publisher=Taylor & Francis, Ltd. }}</ref>

The 18th century, Alexis Clairaut studied algebraic curves in space, the concept of tangent space and curvature, and the use of calculus for this purpose.<ref>{{cite web | title=The Four Curves of Alexis Clairaut | display-authors=1 | first1=Taner | last1=Kiral | first2=Jonathan | last2=Murdock | first3=Colin B. P. | last3=McKinney | journal=Convergence | publisher=Mathematical Association of America | url=https://old.maa.org/press/periodicals/convergence/the-four-curves-of-alexis-clairaut | access-date=2025-11-05 }}</ref><ref name=Struik_1933>{{cite journal | title=Outline of a History of Differential Geometry: I | first=D. J. | last=Struik |author-link=Dirk Jan Struik | journal=Isis | volume=19 | issue=1 | date=April 1933 | pages=92–120 | jstor=225188 | publisher=The University of Chicago Press }}</ref> Leonhard Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation,<ref>{{cite book | title=Leonhard Euler: Mathematical Genius in the Enlightenment | first=Ronald | last=Calinger | publisher=Princeton University Press | year=2019 | page=76 | isbn=978-0-691-19640-4 | url=https://books.google.com/books?id=TM2bDwAAQBAJ&pg=PA76 }}</ref> and later introduced the first set of intrinsic coordinate systems on a surface,<ref name=Struik_1933/> beginning the theory of ''intrinsic geometry'' upon which modern geometric ideas are based. In 1760, Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures,<ref>{{cite book | title=The Legacy of Leonhard Euler: A Tricentennial Tribute | first=Lokenath | last=Debnath | publisher=World Scientific | year=2010 | isbn=978-1-84816-526-7 | page=137 | url=https://books.google.com/books?id=K2liU-SHl6EC&pg=PA137 }}</ref> known as Euler's theorem. Later in the century, Gaspard Monge made important contributions to the study of curves and surfaces in space.<ref name=Struik_1933/> The work of Euler and Monge laid the foundations for differential geometry.

In the 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton's development of the quaternions, a hypercomplex number system. For this purpose, Hamilton coined the terms scalar and vector, and they were first defined in the three dimensional sense within his geometric framework for quaternions.<ref>{{cite book | chapter=Introduction | first=Adrian | last=Rice | title=Mathematics in Victorian Britain | display-editors=1 | editor1-first=Raymond | editor1-last=Flood |editor-link=Raymond Flood (mathematician) | editor2-first=Adrian | editor2-last=Rice | editor3-first=Robin | editor3-last=Wilson |editor3-link=Robin Wilson (mathematician) | publisher=OUP Oxford | year=2011 | page=5 | isbn=978-0-19-960139-4 | chapter-url=https://books.google.com/books?id=l5YiddUUfl4C&pg=PA5 }}</ref> Three dimensional space could then be described by quaternions <math>q = a + ui + vj + wk</math> which had a vanishing scalar component, that is, <math>a = 0</math>.<ref name=Morais_2014>{{cite book | title=Real Quaternionic Calculus Handbook | display-authors=1 | first1=João Pedro | last1=Morais | first2=Svetlin | last2=Georgiev | first3=Wolfgang | last3=Sprößig | publisher=Springer Science & Business Media | year=2014 | isbn=978-3-0348-0622-0 | pages=1–13 | url=https://books.google.com/books?id=YnS8BAAAQBAJ&pg=PA1 }}</ref>

While not explicitly studied by Hamilton, this work indirectly introduced notions of basis, here given by the quaternion elements <math>i,j,k</math>, as well as the dot product and 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 Josiah Willard Gibbs that these two products were identified in their own right,<ref name=Morais_2014/> 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 Edwin Bidwell Wilson based on Gibbs' lectures.<ref>{{cite book | title=Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics | series=Yale bicentennial publications | first1=Josiah Willard | last1=Gibbs | first2=Edwin Bidwell | last2=Wilson | edition=2nd | publisher=Yale University Press | year=1901 | pages=ix, 55 | url=https://books.google.com/books?id=R5IKAAAAYAAJ&pg=PA55 }}</ref>

Further development came in the abstract formalism of vector spaces, with the work of Hermann Grassmann and Giuseppe Peano, the latter of whom first gave the modern definition of vector spaces as an algebraic structure.<ref>{{cite book | title=Linear Algebra: A Geometric Approach | first1=Theodore | last1=Shifrin | first2=Malcolm | last2=Adams | publisher=W. H. Freeman & Company | year=2002 | page=215 | isbn=978-0-7167-4337-8 | url=https://books.google.com/books?id=e-X8wgy9a5gC&pg=PA215 }}</ref> The development of matrix mathematics and its application to n-dimensional geometry was made by Arthur Cayley.<ref>{{cite web | title=Arthur Cayley | website=MacTutor | first1=J. J. | last1=O'Connor | first2=E. F. | last2=Robertson | date=November 2014 | publisher=School of Mathematics and Statistics, University of St Andrews, Scotland | url=https://mathshistory.st-andrews.ac.uk/Biographies/Cayley/ | access-date=2025-11-05 }}</ref>

==In Euclidean geometry==

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

In mathematics, 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, 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 numbers, each number giving the distance of that point from the 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|author-link=Deborah Hughes Hallett|last2=McCallum|first2=William G.|author2-link=William G. McCallum|last3=Gleason|first3=Andrew M.|author3-link=Andrew M. Gleason|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.<ref>{{cite book | title=A Student's Guide to Vectors and Tensors | series=Student's Guides | first=Daniel A. | last=Fleisch | publisher=Cambridge University Press | year=2011 | isbn=978-1-139-50394-5 | pages=15–18 | url=https://books.google.com/books?id=eu1wCIRDwSEC&pg=PA15 }}</ref><ref>{{cite book | title=Mathematics for Physical Science and Engineering: Symbolic Computing Applications in Maple and Mathematica | first=Frank E. | last=Harris | publisher=Academic Press | year=2014 | isbn=978-0-12-801049-5 | pages=202–205 | url=https://books.google.com/books?id=TbbrAgAAQBAJ&pg=PA202 }}</ref> 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. Three distinct points are either collinear or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space.<ref>{{cite book | title=Introduction to the Geometry of N Dimensions | series=Dover Books on Mathematics | first=D. M. Y. | last=Sommerville | author-link=Duncan Sommerville | edition=reprint | publisher=Courier Dover Publications | orig-year=1929 | year=2020 | isbn=978-0-486-84248-6 | pages=3–6 | url=https://books.google.com/books?id=4vXDDwAAQBAJ&pg=PA3 }}</ref>

Two distinct lines can either intersect, be parallel 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.<ref name=Bronštejn_Semendjaev_2013>{{cite book | title=Handbook of Mathematics | first1=Ilja N. | last1=Bronštejn | first2=Konstantin A. | last2=Semendjaev | edition=3rd | publisher=Springer | year=2013 | isbn=978-3-662-25651-0 | page=177 | url=https://books.google.com/books?id=mPXxCAAAQBAJ&pg=PA177 }}</ref>

right|thumb|Relations between up to three planes; only in example 12 do three planes meet to form a point Two distinct planes can either meet in a common line or are parallel (i.e., do not meet).<ref name=Bronštejn_Semendjaev_2013/> 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.<ref>{{cite book | title=Geometry | series=Springer Undergraduate Mathematics Series | first=Roger | last=Fenn | publisher=Springer Science & Business Media | year=2012 | page=152 | isbn=978-1-4471-0325-7 | url=https://books.google.com/books?id=b1HlBwAAQBAJ&pg=PA152 }}</ref>

A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane.<ref name=Bronštejn_Semendjaev_2013/> In the last case, lines can be formed in the plane that are parallel to the given line.

A 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, 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.<ref>{{cite book | title=Topics in Mathematical Analysis and Differential Geometry | series=Pure Mathematics | volume=24 | first=Nicolas K. | last=Laos | publisher=World Scientific | year=1998 | isbn=978-981-02-3180-4 | pages=220–221 | url=https://books.google.com/books?id=1r7dSn4ZqogC&pg=PA220 }}</ref>

Varignon's theorem states that the midpoints of any quadrilateral in <math>\mathbb{R}^{3}</math> form a parallelogram, and hence are coplanar.<ref>{{cite book | title=Geometry by Its History | series=Undergraduate Texts in Mathematics | first1=Alexander | last1=Ostermann | first2=Gerhard | last2=Wanner | publisher=Springer Science & Business Media | year=2012 | isbn=978-3-642-29163-0 | url=https://books.google.com/books?id=eOSqPHwWJX8C&pg=PA264 }}</ref>

===Spheres and balls=== {{main|Sphere}} [[File:Sphere wireframe 10deg 6r.svg|right|thumb|A perspective projection of a sphere onto two dimensions]] A sphere in 3-space (also called a ''2-sphere'' because, like all surfaces, it is intrinsically two-dimensional) 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'' (or ''3-ball'').<ref>{{cite book | title=Geometry of Lengths, Areas, and Volumes | volume=108 | first=James W. | last=Cannon | publisher=American Mathematical Society | year=2017 | isbn=978-1-4704-3714-5 | page=29 | url=https://books.google.com/books?id=sSI_DwAAQBAJ&pg=PA29 }}</ref>

The volume of the ball is given by<ref name=Johnston-Wilder_Mason_2005>{{cite book | title=Developing Thinking in Geometry | editor1-first=Sue | editor1-last=Johnston-Wilder | editor2-first=John | editor2-last=Mason | publisher=Paul Chapman Educational Publishing | year=2005 | isbn=978-1-4129-1169-6 | page=106 | url=https://books.google.com/books?id=kig6NvsG9OIC&pg=PA106 }}</ref> <math display = block>V = \frac{4}{3}\pi r^{3},</math> and the surface area of the sphere is<ref name=Johnston-Wilder_Mason_2005/> <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'''4}}. If a point has coordinates, {{math|''P''(''x'', ''y'', ''z'', ''w'')}}, then {{math|1=''x''2 + ''y''2 + ''z''2 + ''w''2 = 1}} characterizes those points on the unit 3-sphere centered at the origin.<ref>{{cite web | title=Hypersphere | last=Weisstein | first=Eric W. | work=Wolfram MathWorld | url=https://mathworld.wolfram.com/Hypersphere.html | access-date=2025-11-06 |archive-url=https://web.archive.org/web/20260202033726/https://mathworld.wolfram.com/Hypersphere.html |archive-date=2026-02-02}}</ref>

This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space.<ref>{{cite book | title=Discrete and Computational Geometry | first1=Joseph | last1=O'Rourke | first2=Satyan L. | last2=Devadoss | publisher=Princeton University Press | year=2011 | isbn=978-1-4008-3898-1 | url=https://books.google.com/books?id=InJL6iAaIQQC&pg=PA146 }}</ref> 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 solids and the four nonconvex Kepler-Poinsot polyhedra.<ref>{{cite book | title=Gems of Geometry | first=John | last=Barnes | edition=2nd | publisher=Springer Science & Business Media | year=2012 | isbn=978-3-642-30964-9 | page=46 | url=https://books.google.com/books?id=7YCUBUd-4BQC&pg=PA46 }}</ref>

{| class=wikitable |+ Regular polytopes in three dimensions |- align=center !Class !colspan=5|Platonic solids !colspan=4|Kepler-Poinsot polyhedra |- !Symmetry !Td !colspan=2|Oh !colspan=6|Ih |- !Coxeter group !A3, [3,3] !colspan=2|B3, [4,3] !colspan=6|H3, [5,3] |- align=center !Order |24 |colspan=2|48 |colspan=6|120 |- align=center !Regular polyhedron |50px {3,3} |50px {4,3} |50px {3,4} |50px {5,3} |50px {3,5} |50px {5/2,5} |50px {5,5/2} |50px {5/2,3} |50px {3,5/2} |}

===Surfaces of revolution=== {{main|Surface of revolution}} A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. 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.<ref name=Caliò_Lazzari_2020>{{cite book | title=Elements of Mathematics with numerical applications | first1=Franca | last1=Caliò | first2=Alessandro | last2=Lazzari | publisher=Società Editrice Esculapio | year=2020 | isbn=978-88-358-1755-0 | pages=149–151 | url=https://books.google.com/books?id=bX_gDwAAQBAJ&pg=PA149 }}</ref><ref name=Parker_1987>{{cite book | title=Sceno-graphic Techniques | first=Wilford Oren | last=Parker | publisher=SIU Press | year=1987 | isbn=978-0-8093-1350-1 | pages=74–76 | url=https://books.google.com/books?id=Be14QTT8SL4C&pg=PA74 }}</ref>

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.<ref name=Caliò_Lazzari_2020/><ref name=Parker_1987/>

===Quadric surfaces=== {{main|Quadric surface}} In analogy with the conic sections, 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_et_al_2011>{{cite book | title=Geometry | display-authors=1 | first1=David A. | last1=Brannan | first2=Matthew F. | last2=Esplen | first3=Jeremy J. | last3=Gray | edition=2nd, revised | publisher=Cambridge University Press | year=2011 | isbn=978-1-139-50370-9 | pages=42–43, 48–52 | url=https://books.google.com/books?id=UlrmKjIjrzQC&pg=PA42 }}</ref>

There are six types of non-degenerate quadric surfaces:<ref name=Brannan_et_al_2011/> # 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'''3}} through that conic that are normal to {{pi}}).<ref name=Brannan_et_al_2011/> Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.{{cn|date=November 2025}}

Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, 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 of one family intersects, with just one exception, every member of the other family.<ref name=Brannan_et_al_2011/> Each family is called a regulus.<ref>{{cite book | title=The Collected Mathematical Papers of Arthur Cayley | volume=11 | first=Arthur | last=Cayley | publisher=Cambridge University Press | year=1896 | page=633 | url=https://books.google.com/books?id=-O66AkmM1a4C&pg=PA633 }}</ref>

==In linear algebra== In linear algebra, the perspective of three-dimensional space is crucially dependent on the concept of independence. Space has three dimensions because the length of a 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 vectors.<ref name=Towers_1988>{{cite book | title=Guide to Linear Algebra | series=Mathematical Guides | first=David A. | last=Towers | publisher=Bloomsbury Publishing | year=1988 | isbn=978-1-349-09318-2 | pages=6–8 | url=https://books.google.com/books?id=ZZBKEAAAQBAJ&pg=PA8 }}</ref>

===Dot product, angle, and length=== {{main|Dot product}}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 <math>\mathbb{R}^{3}</math> 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 {{math|1='''A''' = [''A''1, ''A''2, ''A''''3'']}} and {{math|1='''B''' = [''B''1, ''B''2, ''B''''3'']}} is defined as:<ref name=Williams_2007>{{cite book | title=Linear Algebra with Applications | first=Gareth | last=Williams | publisher=Jones & Bartlett Publishing | year=2007 | isbn=978-0-7637-5753-3 | pages=38–40 | url=https://books.google.com/books?id=HLQ9ocWuCzMC&pg=PA38 }}</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''1, ''A''2, ''A''''3'']}} with itself is :<math>\mathbf A\cdot\mathbf A = \|\mathbf A\|^2 = A_1^2 + A_2^2 + A_3^2,</math> which gives<ref name=Williams_2007/> : <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 name=Williams_2007/> :<math>\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,</math> where {{math|''θ''}} is the angle between {{math|'''A'''}} and {{math|'''B'''}}.

For a physical example, consider a block on an inclined plane that is being pulled downward by a gravitational force. The dot product can be used to compute the work <math>W</math> performed by the constant force vector <math>\mathbf g</math> that is applied at an angle <math>\theta</math> to the downslope direction of motion <math>\mathbf d</math>. That is:<ref>{{cite book | title=Advanced Engineering Mathematics | edition=4th | first1=Dennis G. | last1=Zill | first2=Warren S. | last2=Wright | publisher=Jones & Bartlett Publishers | year=2009 | isbn=978-0-7637-8241-2 | page=311 | url=https://books.google.com/books?id=jbJDUFZ27yMC&pg=PA311 }}</ref> : <math>W = \mathbf g\cdot\mathbf d = \|\mathbf g\|\,\|\mathbf d\|\cos\theta</math>

===Cross product=== {{main|Cross product}} The cross product or '''vector product''' is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product '''A''' × '''B''' of the vectors '''A''' and '''B''' is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.<ref name=Rogawski_2007>{{cite book | title=Multivariable Calculus | first=Jon | last=Rogawski | publisher=Macmillan | year=2007 | isbn=978-1-4292-1069-0 | page=684–686 | url=https://books.google.com/books?id=S0TEA8R_TwIC&pg=PA684 }}</ref> For example, it can be used to compute the amount of torque on a bolt being turned by a wrench, or the Lorentz force on an electron travelling through a magnetic field.<ref>{{cite book | title=A Brief on Tensor Analysis | series=Undergraduate Texts in Mathematics | first=J. G. | last=Simmonds | publisher=Springer Science & Business Media | year=2012 | isbn=978-1-4684-0141-7 | page=11 | url=https://books.google.com/books?id=s0HUBwAAQBAJ&pg=PA11 }}</ref>

In function language, the cross product is a function <math>\times: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^3</math>.<ref name=Woit_2017/>

thumb|right|The cross-product in respect to a right-handed coordinate system 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.<ref>{{cite book | title=Einstein's Physics: Atoms, Quanta, and Relativity – Derived, Explained, and Appraised | first=Ta-Pei | last=Cheng | publisher=OUP Oxford | year=2013 | isbn=978-0-19-164877-9 | url=https://books.google.com/books?id=cqR8gzXKQKoC&pg=PT459 }}</ref> It has the property that <math>\mathbf{A}\times \mathbf{B} = -\mathbf{B}\times \mathbf{A}</math>.<ref name=Rogawski_2007/>

Its magnitude is related to the angle <math>\theta</math> between <math>\mathbf{A}</math> and <math>\mathbf{B}</math> by the identity<ref name=Rogawski_2007/> <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, which is not commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.<ref name=Quillen_Blower_2020/> 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>.<ref name=Woit_2017>{{cite book | title=Quantum Theory, Groups and Representations: An Introduction | first=Peter | last=Woit | publisher=Springer | year=2017 | isbn=978-3-319-64612-1 | pages=73–75 | url=https://books.google.com/books?id=G248DwAAQBAJ&pg=PA75 }}</ref> 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><ref name=Quillen_Blower_2020>{{cite book | title=Topics in Cyclic Theory | volume=97 | series=London Mathematical Society Student Texts | first1=Daniel G. | last1=Quillen | first2=Gordon | last2=Blower | publisher=Cambridge University Press | year=2020 | isbn=978-1-108-85955-4 | url=https://books.google.com/books?id=8VDuDwAAQBAJ&pg=PA18 }}</ref>

<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 dimensions.<ref name=Massey2>{{cite journal | title=Cross products of vectors in higher dimensional Euclidean spaces | first=W. S. | last=Massey | year=1983 | pages=697–701 | journal=The American Mathematical Monthly | volume=90 | issue=10 | jstor=2323537 | doi=10.2307/2323537 | 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. }}</ref>

===Abstract description=== {{See also|vector space}}

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 between <math>V</math> and <math>\mathbb{R}^3</math>:<ref name=Towers_1988/> the construction for the isomorphism is found 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>, the three-dimensional Euclidean space.<ref>{{cite book | title=Algebraic Topology | display-authors=1 | first1=Clark | last1=Bray | first2=Adrian | last2=Butscher | first3=Simon | last3=Rubinstein-Salzedo | publisher=Springer Nature | year=2021 | page=2 | isbn=978-3-030-70608-1 | url=https://books.google.com/books?id=Twc0EAAAQBAJ&pg=PA2 }}</ref> 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 <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<ref>{{cite book | title=Linear Algebra with Mathematica: An Introduction Using Mathematica | first=Fred | last=Szabo | publisher=Academic Press | year=2000 | isbn=978-0-12-680135-4 | pages=267–268 | url=https://books.google.com/books?id=aqruCh5Hb2cC&pg=PA267 }}</ref>

<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>{{cite book | title=Computational Homology | volume=157 | series=Applied Mathematical Sciences | display-authors=1 | first1=Tomasz | last1=Kaczynski | first2=Konstantin | last2=Mischaikow | first3=Marian | last3=Mrozek | publisher=Springer Science & Business Media | year=2006 | isbn=978-0-387-21597-6 | page=429 | url=https://books.google.com/books?id=hbcuBAAAQBAJ&pg=PA429 }}</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==== {{See also | affine space | Euclidean space}} 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.<ref name=Moretti_2023/> 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.<ref>{{cite book | title=A Course in Algebra | volume=56 | series=Graduate studies in mathematics | first=Ėrnest Borisovich | last=Vinberg | publisher=American Mathematical Society | year=2003 | isbn=978-0-8218-3413-8 | pages=239–247 | url=https://books.google.com/books?id=kd24d3mwaecC&pg=PA247 }}</ref>

This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.<ref name=Moretti_2023>{{cite book | title=Analytical Mechanics: Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity | first=Valter | last=Moretti | translator-first=Simon G. | translator-last=Chiossi | publisher=Springer Nature | year=2023 | isbn=978-3-031-27612-5 | pages=2–7 | url=https://books.google.com/books?id=3SrCEAAAQBAJ&pg=PA2 }}</ref>

====Inner product space==== {{See also | inner product space}} The above discussion does not involve the 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).<ref>{{cite book | title=Intermediate Dynamics: A Linear Algebraic Approach | series=Mechanical Engineering Series | first=R. A. | last=Howland | publisher=Springer Science & Business Media | year=2006 | isbn=978-0-387-28316-6 | pages=49–51 | url=https://books.google.com/books?id=tHOOeOs0jUAC&pg=PA49 }}</ref> For any inner product, there exist bases under which the inner product agrees with the dot product,{{cn|date=November 2025}} 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}} Vector calculus is concerned with infinitesimal and cumulative changes to vector fields, primarily in three-dimensional Euclidean space, <math>\mathbb{R}^3</math>. For differentiation, the del (<math>\nabla</math>), or nabla, operator is used.

===Gradient, divergence and curl=== The gradient indicates the direction of greatest increase of a function, and its magnitude. An example is a flow of particles, with the gradient being the magnitude and direction of the flow at a location.<ref>{{cite book | title=An Invitation to Mathematical Physics and Its History | first=Jont | last=Allen | publisher=Springer Nature | year=2020 | isbn=978-3-030-53759-3 | pages=239–240 | url=https://books.google.com/books?id=cpH-DwAAQBAJ&pg=PA239 }}</ref> In a rectangular coordinate system, the gradient of a differentiable function <math>f: \mathbb{R}^3 \rightarrow \mathbb{R}</math> is given by<ref name=Sussman_Wisdom_2025>{{cite book | title=Functional Differential Geometry | first1=Gerald Jay | last1=Sussman | first2=Jack | last2=Wisdom | publisher=MIT Press | year=2025 | page=154 | isbn=978-0-262-05289-4 | url=https://books.google.com/books?id=ZulZEQAAQBAJ&pg=PA154 }}</ref>

:<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>

where '''i''', '''j''', and '''k''' are the unit vectors for the ''x''-, ''y''-, and ''z''-axes, respectively. In index notation it is written<ref name=Bedford_Drumheller_2023>{{cite book | title=Introduction to Elastic Wave Propagation | first=Anthony | last=Bedford | first2=Douglas S. | last2=Drumheller | edition=2nd | publisher=Springer Nature | year=2023 | isbn=978-3-031-32875-6 | pages=1–4 | url=https://books.google.com/books?id=BSTbEAAAQBAJ&pg=PA4 }}</ref>

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

The divergence indicates the net flux of a vector field around a point, such as an increase or decrease of particle density. That is, whether the location is a source or sink.<ref>{{cite book | title=Classical Mechanics: A Computational Approach with Examples Using Mathematica and Python | first1=Christopher W. | last1=Kulp | first2=Vasilis | last2=Pagonis | publisher=CRC Press | year=2020 | page=92 | isbn=978-1-351-02437-2 | url=https://books.google.com/books?id=PRn9DwAAQBAJ&pg=PA92 }}</ref> The divergence of a (differentiable) 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-valued function:<ref name=Sussman_Wisdom_2025/>

:<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<ref name=Bedford_Drumheller_2023/> <math display=block>\nabla \cdot \mathbf{F} = \partial_i F_i.</math>

The curl (or rotor) is a vector indicating the rotational circulation of a vector field. Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × '''F''' is, for '''F''' composed of [''F''x, ''F''y, ''F''z]:<ref>{{cite book | title=Vector Calculus | series=Springer Undergraduate Mathematics Series | first=Paul C. | last=Matthews | publisher=Springer Science & Business Media | year=2000 | page=60 | isbn=978-3-540-76180-8 | url=https://books.google.com/books?id=9wmR7kDdO8EC&pg=PA60 }}</ref>

:<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>

This expands as follows:<ref name=Sussman_Wisdom_2025/>

:<math>\operatorname{curl}\,\mathbf{F} = \nabla\times\mathbf{F} = \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<ref name=Bedford_Drumheller_2023/> <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=== right|thumb|Illustration of a line integral along curve C in a vector field F A line integral of a function along a curve can be thought of as a continuous summation of the function value along every infinitesimal increment of that curve. For some scalar field ''f'' : ''U'' ⊆ '''R'''''n'' → '''R''', the ''line integral'' along a piecewise smooth curve ''C'' ⊂ ''U'' is defined as<ref name=Karpfinger_2022/> :<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 (one-to-one correspondence) 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 '''F''' : ''U'' ⊆ '''R'''''n'' → '''R'''''n'', the line integral along a piecewise smooth curve ''C'' ⊂ ''U'', in the direction of '''r''', is defined as<ref name=Karpfinger_2022>{{cite book | title=Calculus and Linear Algebra in Recipes: Terms, phrases and numerous examples in short learning units | first=Christian | last=Karpfinger | publisher=Springer Nature | year=2022 | isbn=978-3-662-65458-3 | page=640 | url=https://books.google.com/books?id=7xWbEAAAQBAJ&pg=PA640 }}</ref>

:<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 <math>\cdot</math> is the dot product and '''r''': [a, b] → ''C'' is a bijective parametrization of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C''. A subtype of line integral found in physics is the plane closed loop, which determines the circulation of the function around the loop<ref>{{cite book | title=Vectors in Physics and Engineering | first=Alan | last=Durrant | publisher=Routledge | year=2019 | page=225 | isbn=978-1-351-40556-0 | url=https://books.google.com/books?id=rFEPEAAAQBAJ&pg=PA225 }}</ref>

:<math>\oint_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}.</math>

A ''surface integral'' is a generalization of multiple integrals to integration over surfaces. 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 parameterize the surface of interest, ''S'', by considering a system of curvilinear coordinates on ''S'', like the latitude and longitude on a sphere. 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{{cn|date=November 2025}}

:<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 of the cross product of the partial derivatives of '''x'''(''s'', ''t''), and is known as the surface 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'' is an integral over a ''three-dimensional domain'' or region. When the integrand is trivial (unity), the volume integral is simply the region's ''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 triple integral within a region ''D'' in '''R'''3 of a 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=== {{main|Fundamental theorem of line integrals}} The fundamental theorem of line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.<ref>{{cite web | title=Lecture 25: Fundamental Theorem of Line Integrals | first=Oliver | last=Knill | work=Multivariable Calculus | publisher=Department of Mathematics, Harvard University | url=https://people.math.harvard.edu/~knill/teaching/math21a2022/handouts/lecture25.pdf | access-date=2025-11-08 |archive-url=https://web.archive.org/web/20231109214140/https://people.math.harvard.edu/~knill/teaching/math21a2022/handouts/lecture25.pdf |archive-date=2023-11-09}}</ref>

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|Stokes' theorem}} Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:<ref>{{cite web | title=Lecture 22: Stokes’ Theorem and Applications | first=Martin | last=Evans | date=April 23, 2002 | publisher=The University of Edinburgh, Department of Physics & Astronomy | url=https://www2.ph.ed.ac.uk/~mevans/mp2h/VTF/lecture22.pdf | access-date=2025-11-08 }}</ref>

:<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 and has a piecewise smooth boundary {{mvar|S}} (also indicated with {{math|∂''V'' {{=}} ''S''}}). If {{math|'''F'''}} is a continuously differentiable vector field defined on a neighborhood of {{mvar|V}}, then the 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 over the volume {{mvar|V}}, the right side is the 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'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 in a piece of string.<ref>{{cite book | first=Dale | last=Rolfsen | title= Knots and Links | issue=346 | series=AMS Chelsea Publishing | publisher=American Mathematical Society | location=Providence, Rhode Island | year=1976 | isbn=978-0-8218-3436-7 | url=https://books.google.com/books?id=naYJBAAAQBAJ&pg=PA9 }}</ref>{{Page needed|date=November 2025|reason=The paper covers two-dimensional knots, so a page number is needed for this reference}}

In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble <math>{\mathbb{R}}^3</math>. Globally, the same 3-manifold can curve in a variety of manners, as long as it remains continuous.<ref>{{cite book | title=Tensor Calculus and Analytical Dynamics | series=Engineering Mathematics | first=John G. | last=Papastavridis | publisher=Routledge | year=2018 | isbn=978-1-351-41162-2 | page=22 | url=https://books.google.com/books?id=KlIPEAAAQBAJ&pg=PA22 }}</ref> An example of this is the curved spacetime found in General Relativity.

==In finite geometry== Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces.<ref>{{cite book | title=A Course in Combinatorics and Graphs | series=Compact Textbooks in Mathematics | first1=Simeon | last1=Ball | first2=Oriol | last2=Serra | publisher=Springer Nature | year=2024 | isbn=978-3-031-55384-4 | page=77 | url=https://books.google.com/books?id=SI4CEQAAQBAJ&pg=PA77 }}</ref> 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.<ref>{{cite conference | title=Introduction | conference=Finite Geometries and Designs: Proceedings of the Second Isle of Thorns Conference 1980 | issue=3 | series=Lecture note series, London Mathematical Society | volume=49 | display-editors=1 | editor1-first=P. J. | editor1-last=Cameron | editor2-first=J. W. P. | editor2-last=Hirschfeld | editor3-first=D. R. | editor3-last=Hughes | publisher=Cambridge University Press | year=1981 | isbn=978-0-521-28378-6 | page=1 | url=https://books.google.com/books?id=qMawcdK0s-8C&pg=PA1 }}</ref> For example, any three skew lines in PG(3,''q'') are contained in exactly one regulus.<ref>{{cite book | first1=Albrecht | last1=Beutelspacher | author-link1=Albrecht Beutelspacher | first2=Ute | last2=Rosenbaum | year=1998 | title=Projective Geometry | page=72 | publisher=Cambridge University Press | isbn=978-0-521-48364-3 | url=https://books.google.com/books?id=I4OqBcaKAJ0C&pg=PA72 }}</ref>

==References== {{reflist}}

==External links== {{Wikiquote}} {{commons category|3D}} * {{wiktionary-inline|three-dimensional}} * {{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 {{Dimension topics}}

* Category:Analytic geometry 3 Category:Three-dimensional coordinate systems Space Category:Space