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n 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 60 1 it can also mean a triple integral within a region d in r 3 of a function f x y z displaystyle f x y z and is usually written as d f x y z d x d y d z displaystyle iiint limits _ d f x y z dx dy dz fundamental theorem of line integrals edit main article 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 61 let φ u r n r displaystyle varphi u subseteq mathbb r n to mathbb r then φ q φ p γ p q φ r d r displaystyle varphi left mathbf q right varphi left mathbf p right int _ gamma mathbf p mathbf q nabla varphi mathbf r cdot d mathbf r stokes theorem edit main article 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 62 σ f d σ σ f d r displaystyle iint _ sigma nabla times mathbf f cdot mathrm d mathbf sigma oint _ partial sigma mathbf f cdot mathrm d mathbf r divergence theorem edit main article divergence theorem suppose v is a subset of r n displaystyle mathbb r n in the case of n 3 v represents a volume in 3d space which is compact and has a piecewise smooth boundary s also indicated with v s if f is a continuously differentiable vector field defined on a neighborhood of v then the divergence theorem says 63 v f d v displaystyle iiint _ v left mathbf nabla cdot mathbf f right dv s displaystyle scriptstyle s f n d s displaystyle mathbf f cdot mathbf n ds the left side is a volume integral over the volume v the right side is the surface integral over the boundary of the volume v the closed manifold v is quite generally the boundary of v oriented by outward pointing normals and n is the outward pointing unit normal field of the boundary v d s may be used as a shorthand for n ds in topology edit 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 64 page needed in differential geometry the generic three dimensional spaces are 3 manifolds which locally resemble r 3 displaystyle mathbb r 3 globally the same 3 manifold can curve in a variety of manners as long as it remains continuous 65 an example of this is the curved spacetime found in general relativity in finite geometry edit 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 66 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 67 for example any three skew lines in pg 3 q are contained in exactly one regulus 68 see also edit 3d rotation rotation formalisms in three dimensions dimensional analysis distance from a point to a plane four dimensional space skew lines distance three dimensional graph solid geometry terms of orientation references edit 1 2 iec 60050 details for iev number 102 04 39 three dimensional domain international electrotechnical vocabulary in japanese retrieved 2023 09 19 euclidean space encyclopedia of mathematics encyclopediaofmath org retrieved 2020 08 12 details for iev number 113 01 02 space international electrotechnical vocabulary in japanese retrieved 2023 11 07 euclidean space geometry encyclopedia britannica retrieved 2020 08 12 aristotle 350 bc de caelo latin title book 1 translated by stocks j l published in 1922 1 2 artmann benno 2012 euclid the creation of mathematics springer science business media pp 9 10 isbn 978 1 4612 1412 0 fitzpatrick richard august 26 2014 euclid s elements of geometry university of texas archived from the original on 2025 11 22 retrieved 2025 11 04 martin g e 2012 the foundations of geometry and the non euclidean plane mathematics and statistics springer science business media p 51 isbn 978 1 4612 5725 7 holton derek allan stillwell john 2024 relay race to infinity the developments in mathematics from euclid to fermat world scientific p 158 isbn 978 981 12 9634 5 hiriart urrety j b 2012 mathematical faits divers in ponstein jacob ed 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 lecture notes in economics and mathematical systems springer science business media p 3 isbn 978 3 642 45610 7 boyer c b february 1949 newton as an originator of polar coördinates the american mathematical monthly 56 2 taylor francis ltd 73 78 doi 10 2307 2306162 jstor 2306162 kiral taner et al the four curves of alexis clairaut convergence mathematical association of america retrieved 2025 11 05 1 2 3 struik d j april 1933 outline of a history of differential geometry i isis 19 1 the university of chicago press 92 120 jstor 225188 calinger ronald 2019 leonhard euler mathematical genius in the enlightenment princeton university press p 76 isbn 978 0 691 19640 4 debnath lokenath 2010 the legacy of leonhard euler a tricentennial tribute world scientific p 137 isbn 978 1 84816 526 7 rice adrian 2011 introduction in flood raymond et al eds mathematics in victorian britain oup oxford p 5 isbn 978 0 19 960139 4 1 2 morais joão pedro et al 2014 real quaternionic calculus handbook springer science business media pp 1 13 isbn 978 3 0348 0622 0 gibbs josiah willard wilson edwin bidwell 1901 vector analysis a text book for the use of students of mathematics and physics yale bicentennial publications 2nd ed yale university press pp ix 55 shifrin theodore adams malcolm 2002 linear algebra a geometric approach w h freeman company p 215 isbn 978 0 7167 4337 8 o connor j j robertson e f november 2014 arthur cayley mactutor school of mathematics and statistics university of st andrews scotland retrieved 2025 11 05 hughes hallett deborah mccallum william g gleason andrew m 2013 calculus single and multivariable 6 ed john wiley isbn 978 0470 88861 2 fleisch daniel a 2011 a student s guide to vectors and tensors student s guides cambridge university press pp 15 18 isbn 978 1 139 50394 5 harris frank e 2014 mathematics for physical science and engineering symbolic computing applications in maple and mathematica academic press pp 202 205 isbn 978 0 12 801049 5 sommerville d m y 2020 1929 introduction to the geometry of n dimensions dover books on mathematics reprint ed courier dover publications pp 3 6 isbn 978 0 486 84248 6 1 2 3 bronštejn ilja n semendjaev konstantin a 2013 handbook of mathematics 3rd ed springer p 177 isbn 978 3 662 25651 0 fenn roger 2012 geometry springer undergraduate mathematics series springer science business media p 152 isbn 978 1 4471 0325 7 laos nicolas k 1998 topics in mathematical analysis and differential geometry pure mathematics vol 24 world scientific pp 220 221 isbn 978 981 02 3180 4 ostermann alexander wanner gerhard 2012 geometry by its history undergraduate texts in mathematics springer science business media isbn 978 3 642 29163 0 cannon james w 2017 geometry of lengths areas and volumes vol 108 american mathematical society p 29 isbn 978 1 4704 3714 5 1 2 johnston wilder sue mason john eds 2005 developing thinking in geometry paul chapman educational publishing p 106 isbn 978 1 4129 1169 6 weisstein eric w hypersphere wolfram mathworld archived from the original on 2026 02 02 retrieved 2025 11 06 o rourke joseph devadoss satyan l 2011 discrete and computational geometry princeton university press isbn 978 1 4008 3898 1 barnes john 2012 gems of geometry 2nd ed springer science business media p 46 isbn 978 3 642 30964 9 1 2 caliò franca lazzari alessandro 2020 elements of mathematics with numerical applications società editrice esculapio pp 149 151 isbn 978 88 358 1755 0 1 2 parker wilford oren 1987 sceno graphic techniques siu press pp 74 76 isbn 978 0 8093 1350 1 1 2 3 4 brannan david a et al 2011 geometry 2nd revised ed cambridge university press pp 42 43 48 52 isbn 978 1 139 50370 9 cayley arthur 1896 the collected mathematical papers of arthur cayley vol 11 cambridge university press p 633 1 2 towers david a 1988 guide to linear algebra mathematical guides bloomsbury publishing pp 6 8 isbn 978 1 349 09318 2 1 2 3 williams gareth 2007 linear algebra with applications jones bartlett publishing pp 38 40 isbn 978 0 7637 5753 3 zill dennis g wright warren s 2009 advanced engineering mathematics 4th ed jones bartlett publishers p 311 isbn 978 0 7637 8241 2 1 2 3 rogawski jon 2007 multivariable calculus macmillan p 684 686 isbn 978 1 4292 1069 0 simmonds j g 2012 a 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cambridge university press p 72 isbn 978 0 521 48364 3 external links edit wikiquote has quotations related to three dimensional space wikimedia commons has media related to 3d the dictionary definition of three dimensional at wiktionary weisstein eric w four dimensional geometry mathworld elementary linear algebra chapter 8 three dimensional geometry keith matthews from university of queensland 1991 v t e dimension dimensional spaces vector space euclidean space affine space projective space free module manifold algebraic variety spacetime other dimensions krull lebesgue covering inductive hausdorff minkowski fractal degrees of freedom polytopes and shapes hyperplane hypersurface hypercube hyperrectangle demihypercube hypersphere cross polytope simplex hyperpyramid number systems hypercomplex numbers cayley dickson construction dimensions by number zero one two three four five six seven eight n dimensions see also hyperspace codimension category retrieved from https en wikipedia org w index php title three dimensional_space oldid 1365678357 categories euclidean solid geometry analytic geometry multi dimensional geometry three dimensional coordinate systems 3 number space hidden categories articles with short description short description is different from wikidata cs1 japanese language sources ja pages using sidebar with the child parameter all articles with unsourced statements articles with unsourced statements from november 2025 wikipedia articles needing page number citations from november 2025 commons category link is on wikidata this page was last edited on 23 july 2026 at 20 40 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts 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