1806:
576:
50:
5366:. 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
1792:
6320:
1927:
2325:
2303:
7968:
8626:
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107:
1778:
2314:
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6083:
7821:
2281:
6044:
8758:
2292:
2248:
2270:
2259:
7117:
5805:
4202:
6315:{\displaystyle \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} .}
7660:
5540:
5227:
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.
5197:
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
1857:
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
341:
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
6843:
5900:
3936:
7536:
5060:
2686:, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a
6967:
4097:
3172:
1868:
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
2922:
5698:
6608:
3844:
3057:
3265:
4901:
3424:
7577:
5438:
6039:{\displaystyle {\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}}}
6406:
4567:
3686:
7817:
2117:
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.
3772:
6718:
7425:
5686:
5307:
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
4803:
7417:
4617:
2664:
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
2544:
5866:
7112:{\displaystyle \iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {x} (s,t))\left\|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right\|\mathrm {d} s\,\mathrm {d} t}
2706:
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
5430:
7339:
1873:, 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.
7890:
4359:
524:
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
3602:
1861:
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.
5604:
4724:
4862:
3995:
2018:
7979:
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
4069:
3068:
293:. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called
4035:
8366:
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.
8023:
3719:
5800:{\displaystyle \operatorname {div} \,\mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}+{\frac {\partial W}{\partial z}}.}
4197:{\displaystyle \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}
2781:
256:
7707:
5335:
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5192:
5140:
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4290:
2752:
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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
2954:
3795:
6673:
3202:
507:
7655:{\displaystyle \iint _{\Sigma }\nabla \times \mathbf {F} \cdot \mathrm {d} \mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {r} .}
6496:
5305:
475:
3364:
8996:
5163:
5111:
4456:
4403:
4379:
4261:
5535:{\displaystyle \nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} }
5875:
3931:{\displaystyle \left\|\mathbf {A} \times \mathbf {B} \right\|=\left\|\mathbf {A} \right\|\cdot \left\|\mathbf {B} \right\|\cdot \left|\sin \theta \right|.}
1805:
8689:
4519:
1704:
5344:
This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.
3609:
375:(Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.
7762:
6838:{\displaystyle \int \limits _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt.}
3728:
7174:), 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.
2377:
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.
7531:{\displaystyle \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)=\int _{\gamma }\nabla \varphi (\mathbf {r} )\cdot d\mathbf {r} .}
5055:{\displaystyle 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}}.}
2409:
2380:
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
5817:
8095:
5639:
8609:
8583:
8546:
8526:
8446:
7356:
7350:
2725:
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
2384:
with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular
804:
4732:
8273:
8122:
7373:
71:
8991:
4572:
8792:
8981:
8742:
8565:
8501:
8471:
5554:
770:
395:
93:
4808:
8107:
5392:
1974:
4218:
477:. While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements
7269:
1791:
8682:
7190:
5337:, the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called
1753:
measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.
1697:
1651:
1257:
716:
187:
7849:
5883:
5811:
4295:
1797:
273:, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). In
6926:
2329:
2307:
1721:(also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three
8777:
6958:
5879:
5871:
3433:
1811:
1783:
517:, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions.
263:
111:
5093:
can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely,
64:
58:
8986:
8494:
4654:
2023:
1672:
1282:
3167:{\displaystyle \|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }}={\sqrt {A_{1}^{2}+A_{2}^{2}+A_{3}^{2}}},}
8675:
8412:
8204:
8155:
2643:
1690:
3959:
1745:. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of
75:
8712:
8661:
4044:
2638:
2626:
2157:
2137:
659:
8920:
8915:
7123:
6918:
4037:. In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the
1757:
765:
622:
537:
Also during the 19th century came developments in the abstract formalism of vector spaces, with the work of
2406:, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
8905:
8900:
8880:
8486:
8253:
8112:
7217:
4435:
2917:{\displaystyle \mathbf {A} \cdot \mathbf {B} =A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}=\sum _{i=1}^{3}A_{i}B_{i}.}
1851:
1777:
1161:
872:
750:
635:
379:
4004:
8910:
8890:
8885:
7997:
7987:
3691:
2658:
2368:
2354:
1761:
933:
894:
853:
848:
701:
350:
213:
1946:
because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance
1876:
229:
7683:
5311:
5234:
5201:
5168:
5116:
5067:
4464:
4408:
4266:
3184:
Without reference to the components of the vectors, the dot product of two non-zero
Euclidean vectors
2728:
1882:
1550:
1297:
8132:
8102:
6411:
5689:
3941:
2185:
2169:
1750:
1726:
1601:
1524:
1372:
1277:
799:
694:
608:
531:
521:
387:
338:
8630:
7827:
1854:, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
9001:
8787:
8782:
8257:
8050:
6861:
5353:
4622:
2653:
2385:
2177:
2164:
1606:
1463:
1317:
1222:
1112:
983:
973:
836:
711:
706:
689:
664:
652:
604:
599:
580:
153:
38:
4867:
4497:
4074:
3822:
3800:
3052:{\displaystyle \mathbf {A} \cdot \mathbf {A} =\|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2},}
575:
545:, the latter of whom first gave the modern definition of vector spaces as an algebraic structure.
401:
297:. In this classical example, when the three values refer to measurements in different directions (
8961:
8802:
8757:
8357:
8078:
7752:
7671:
7560:
7131:
6902:
6442:
5278:
A more abstract description still is to model physical space as a three-dimensional affine space
3956:
with the cross product being the Lie bracket. Specifically, the space together with the product,
3722:
2687:
2381:
2360:
2318:
2239:
1957:
1565:
1292:
1132:
760:
684:
674:
645:
630:
143:
7552:
7547:
5142:
and 'forgetting' the
Cartesian product structure, or equivalently the standard choice of basis.
2702:, where the idea of independence is crucial. Space has three dimensions because the length of a
363:
7222:
1756:
Other popular methods of describing the location of a point in three-dimensional space include
8797:
8643:
8640:
8605:
8579:
8561:
8542:
8522:
8497:
8467:
8442:
8269:
8261:
7367:
field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
7205:
6910:
6909:
analog of the line integral. To find an explicit formula for the surface integral, we need to
6898:
4491:
3347:
3260:{\displaystyle \mathbf {A} \cdot \mathbf {B} =\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,}
2708:
2340:
1718:
1636:
1626:
1555:
1424:
1402:
1352:
1327:
1262:
1186:
841:
733:
679:
640:
559:
538:
358:
354:
298:
286:
274:
6603:{\displaystyle \int \limits _{C}f\,ds=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.}
3780:
301:), any three directions can be chosen, provided that these directions do not lie in the same
8727:
8597:
8534:
8349:
8062:
7905:
7556:
6893:
6652:
6479:
3319:
3315:
3178:
2324:
2302:
1832:
1616:
1357:
1067:
945:
880:
738:
723:
588:
391:
368:
302:
157:
3419:{\displaystyle \times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}
1926:
480:
8772:
8717:
8046:
8034:
7897:
7179:
6906:
5379:
5281:
5273:
4895:
4038:
3323:
2397:
1870:
1824:
1765:
1722:
1039:
912:
902:
745:
728:
669:
526:
165:
4243:
It can be useful to describe three-dimensional space as a three-dimensional vector space
1611:
1580:
1514:
1484:
1362:
1307:
1302:
1242:
454:
9006:
8854:
8839:
8340:
Massey, WS (1983). "Cross products of vectors in higher dimensional
Euclidean spaces".
8127:
7967:
7924:
7724:
7147:
5546:
5148:
5096:
4727:
4441:
4388:
4364:
4246:
2699:
2223:
2152:
2133:
1667:
1641:
1575:
1519:
1392:
1272:
1252:
1232:
1137:
542:
378:
In the 19th century, developments of the geometry of three-dimensional space came with
343:
173:
8625:
2754:
can be represented by an ordered triple of real numbers. These numbers are called the
2679:). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
8975:
8844:
8553:
8229:
7720:
7568:
7360:
7127:
6474:
5363:
3343:
3307:
3301:
2683:
2648:
2403:
2195:
1931:
1909:
1843:
1646:
1631:
1560:
1377:
1337:
1287:
1062:
1025:
992:
830:
826:
514:
306:
4224:
2062:
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the
8864:
8829:
8722:
8054:
7564:
6679:
6454:
6401:{\displaystyle (\nabla \times \mathbf {F} )_{i}=\epsilon _{ijk}\partial _{j}F_{k},}
5609:
5269:
4562:{\displaystyle \mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} }
4238:
2285:
1585:
1534:
1347:
1202:
1117:
907:
192:
7972:
3681:{\displaystyle (\mathbf {A} \times \mathbf {B} )_{i}=\varepsilon _{ijk}A_{j}B_{k}}
106:
8180:
7812:{\displaystyle \iiint _{V}\left(\mathbf {\nabla } \cdot \mathbf {F} \right)\,dV=}
17:
8949:
8732:
8090:
7980:
6849:
6062:
5389:
In a rectangular coordinate system, the gradient of a (differentiable) function
5359:
4382:
3953:
3949:
3945:
3767:{\displaystyle \mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} }
3355:
2720:
2313:
2296:
2252:
1746:
1621:
1494:
1312:
1247:
1175:
1147:
1122:
510:
383:
209:
178:
148:
2335:
8944:
8824:
8601:
8117:
8070:
8042:
7991:
3998:
3606:
and can also be written in components, using
Einstein summation convention as
2373:
2274:
2127:
1865:
1847:
1479:
1458:
1448:
1438:
1397:
1342:
1237:
1227:
1127:
978:
294:
290:
8925:
8834:
8747:
8698:
8657:
8648:
7195:
6622:
5231:
Computationally, it is necessary to work with the more concrete description
2633:
1828:
1489:
1207:
1170:
1034:
1006:
169:
7747:
is a continuously differentiable vector field defined on a neighborhood of
8156:"IEC 60050 — Details for IEV number 102-04-39: "three-dimensional domain""
2280:
8849:
8812:
8737:
7364:
7184:
5681:{\displaystyle \mathbf {F} :\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}}
1836:
1764:, though there are an infinite number of possible methods. For more, see
1570:
1529:
1499:
1387:
1382:
1332:
1057:
1016:
964:
858:
821:
567:
278:
123:
2291:
2247:
2114:
characterizes those points on the unit 3-sphere centered at the origin.
8859:
8361:
8038:
3351:
3350:
to the plane containing them. It has many applications in mathematics,
2269:
2258:
2132:
In three dimensions, there are nine regular polytopes: the five convex
1504:
1217:
1011:
955:
755:
4798:{\displaystyle {\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}}
30:
For a broader, less mathematical treatment related to this topic, see
7200:
6930:
2703:
1939:
1921:
1453:
1443:
1322:
1267:
1142:
1105:
1093:
1048:
1001:
919:
584:
325:
318:
282:
8353:
7412:{\displaystyle \varphi :U\subseteq \mathbb {R} ^{n}\to \mathbb {R} }
2682:
Both the hyperboloid of one sheet and the hyperbolic paraboloid are
1835:. On the other hand, four distinct points can either be collinear,
8816:
7965:
6698:
5367:
4223:
3277:
2364:
1925:
1509:
1433:
1367:
1212:
816:
811:
311:
200:
105:
31:
8658:
Elementary Linear
Algebra - Chapter 8: Three-dimensional Geometry
4612:{\displaystyle \pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} }
398:. Three dimensional space could then be described by quaternions
7122:
where the expression between bars on the right-hand side is the
4228:
The cross-product in respect to a right-handed coordinate system
2263:
1100:
950:
349:
In the 17th century, three-dimensional space was described with
182:. The term may also refer colloquially to a subset of space, a
8671:
8574:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999),
7198:
is trivial (unity), the volume integral is simply the region's
6325:
In index notation, with
Einstein summation convention this is
2539:{\displaystyle Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,}
43:
5861:{\displaystyle \nabla \cdot \mathbf {F} =\partial _{i}F_{i}.}
8667:
4739:
4438:. However, there is no 'preferred' or 'canonical basis' for
4301:
4001:
to the Lie algebra of three-dimensional rotations, denoted
2698:
Another way of viewing three-dimensional space is found in
7933:
is the outward pointing unit normal field of the boundary
5425:{\displaystyle f:\mathbb {R} ^{3}\rightarrow \mathbb {R} }
2066:: points equidistant to the origin of the euclidean space
1729:, the point at which they cross. They are usually labeled
1749:, each number giving the distance of that point from the
8413:"IEC 60050 — Details for IEV number 102-04-40: "volume""
7334:{\displaystyle \iiint \limits _{D}f(x,y,z)\,dx\,dy\,dz.}
289:
is considered, it can be considered a local subspace of
37:"Three-dimensional" redirects here. For other uses, see
4569:. This allows the definition of canonical projections,
2367:
about a fixed line in its plane as an axis is called a
7831:
5909:
5341:
for distinguishing them from
Euclidean vector spaces.
5021:
4972:
4923:
3361:
In function language, the cross product is a function
1725:
are given, each perpendicular to the other two at the
8441:. Schaum's Outlines (2nd ed.). US: McGraw Hill.
8045:
as its 2-dimensional subspaces. It is an instance of
8000:
7885:{\displaystyle (\mathbf {F} \cdot \mathbf {n} )\,dS.}
7852:
7830:
7765:
7686:
7580:
7428:
7376:
7272:
7225:
6970:
6721:
6655:
6499:
6414:
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6086:
5903:
5820:
5701:
5642:
5557:
5441:
5395:
5314:
5284:
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5171:
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5119:
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4657:
4625:
4575:
4522:
4500:
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4444:
4411:
4391:
4367:
4298:
4292:
in a subtle way. By definition, there exists a basis
4269:
4249:
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4047:
4007:
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3731:
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3205:
3071:
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2784:
2731:
2412:
2026:
1977:
1885:
483:
457:
404:
232:
4354:{\displaystyle {\mathcal {B}}=\{e_{1},e_{2},e_{3}\}}
8937:
8873:
8811:
8765:
8705:
7567:F over a surface Σ in Euclidean three-space to the
3326:and is denoted by the symbol ×. The cross product
176:. More general three-dimensional spaces are called
8017:
7884:
7838:
7811:
7701:
7654:
7530:
7411:
7333:
7255:
7111:
6837:
6667:
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6314:
6038:
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5799:
5680:
5598:
5534:
5424:
5329:
5299:
5252:
5219:
5186:
5157:
5134:
5105:
5085:
5054:
4886:
4856:
4797:
4718:
4643:
4611:
4561:
4508:
4482:
4461:On the other hand, there is a preferred basis for
4450:
4426:
4397:
4373:
4353:
4284:
4255:
4196:
4085:
4063:
4029:
3989:
3930:
3833:
3811:
3789:
3766:
3713:
3680:
3596:
3418:
3259:
3166:
3051:
2916:
2746:
2538:
2054:
2012:
1900:
1879:states that the midpoints of any quadrilateral in
1823:Two distinct points always determine a (straight)
501:
469:
443:
309:, the three values are often labeled by the terms
250:
8437:M. R. Spiegel; S. Lipschutz; D. Spellman (2009).
8303:
8291:
1771:Below are images of the above-mentioned systems.
5194:is sometimes referred to as a coordinate space.
4434:: the construction for the isomorphism is found
3597:{\displaystyle \mathbf {A} \times \mathbf {B} =}
386:. In fact, it was Hamilton who coined the terms
305:. Furthermore, if these directions are pairwise
8181:"Euclidean space - Encyclopedia of Mathematics"
1956:. The solid enclosed by the sphere is called a
451:which had vanishing scalar component, that is,
222:-dimensional Euclidean space. The set of these
8069:) of three dimensions. For example, any three
6077:-axes, respectively. This expands as follows:
5599:{\displaystyle (\nabla f)_{i}=\partial _{i}f.}
258:and can be identified to the pair formed by a
8683:
8541:, Academic Press; 6 edition (June 21, 2005).
6697:, the line integral along a piecewise smooth
4898:. Written out in full, the standard basis is
4719:{\displaystyle \pi _{1}(x_{1},x_{2},x_{3})=x}
1698:
8:
5876:Del in cylindrical and spherical coordinates
4857:{\displaystyle \pi _{i}(E_{j})=\delta _{ij}}
4792:
4753:
4348:
4309:
3242:
3234:
3230:
3222:
3080:
3072:
2983:
2974:
1842:Two distinct lines can either intersect, be
8466:. Berkeley, California: Publish or Perish.
8205:"Details for IEV number 113-01-02: "space""
8033:Many ideas of dimension can be tested with
8690:
8676:
8668:
8150:
8148:
7719:represents a volume in 3D space) which is
7571:of the vector field over its boundary ∂Σ:
7158:, that is a function that assigns to each
5886:coordinate representations), the curl ∇ ×
5358:The above discussion does not involve the
3990:{\displaystyle (\mathbb {R} ^{3},\times )}
2013:{\displaystyle V={\frac {4}{3}}\pi r^{3},}
1705:
1691:
1420:
939:
574:
563:
8417:International Electrotechnical Vocabulary
8287:
8285:
8209:International Electrotechnical Vocabulary
8160:International Electrotechnical Vocabulary
8009:
8004:
8003:
8002:
7999:
7990:the generic three-dimensional spaces are
7872:
7864:
7856:
7851:
7829:
7799:
7789:
7781:
7770:
7764:
7693:
7689:
7688:
7685:
7644:
7639:
7631:
7622:
7610:
7605:
7597:
7585:
7579:
7520:
7506:
7487:
7486:
7478:
7471:
7455:
7436:
7427:
7405:
7404:
7395:
7391:
7390:
7375:
7321:
7314:
7307:
7277:
7271:
7224:
7101:
7100:
7092:
7071:
7065:
7046:
7040:
7012:
7000:
6985:
6984:
6975:
6969:
6961:. Then, the surface integral is given by
6825:
6807:
6786:
6778:
6772:
6767:
6755:
6751:
6740:
6732:
6726:
6720:
6654:
6590:
6585:
6567:
6561:
6544:
6532:
6527:
6513:
6504:
6498:
6441:is the totally antisymmetric symbol, the
6419:
6413:
6389:
6379:
6363:
6350:
6341:
6330:
6304:
6282:
6272:
6252:
6242:
6229:
6207:
6197:
6177:
6167:
6154:
6132:
6122:
6102:
6092:
6085:
6022:
6010:
5998:
5972:
5955:
5938:
5926:
5919:
5912:
5904:
5902:
5849:
5839:
5827:
5819:
5774:
5751:
5728:
5720:
5706:
5705:
5700:
5672:
5668:
5667:
5657:
5653:
5652:
5643:
5641:
5584:
5571:
5556:
5527:
5507:
5499:
5479:
5471:
5451:
5440:
5418:
5417:
5408:
5404:
5403:
5394:
5321:
5317:
5316:
5313:
5283:
5244:
5240:
5239:
5236:
5211:
5207:
5206:
5203:
5178:
5174:
5173:
5170:
5150:
5126:
5122:
5121:
5118:
5098:
5077:
5073:
5072:
5069:
5016:
5007:
4967:
4958:
4918:
4909:
4903:
4875:
4869:
4845:
4829:
4816:
4810:
4786:
4773:
4760:
4744:
4738:
4737:
4734:
4726:. This then allows the definition of the
4701:
4688:
4675:
4662:
4656:
4624:
4605:
4604:
4595:
4591:
4590:
4580:
4574:
4555:
4554:
4547:
4546:
4539:
4538:
4529:
4525:
4524:
4521:
4502:
4501:
4499:
4474:
4470:
4469:
4466:
4443:
4418:
4414:
4413:
4410:
4390:
4366:
4342:
4329:
4316:
4300:
4299:
4297:
4276:
4272:
4271:
4268:
4263:over the real numbers. This differs from
4248:
4180:
4172:
4161:
4150:
4142:
4131:
4120:
4112:
4101:
4099:
4078:
4076:
4064:{\displaystyle \mathbf {A} ,\mathbf {B} }
4056:
4048:
4046:
4009:
4008:
4006:
3972:
3968:
3967:
3961:
3894:
3878:
3861:
3853:
3846:
3826:
3824:
3804:
3802:
3782:
3759:
3751:
3740:
3732:
3730:
3699:
3693:
3672:
3662:
3646:
3633:
3624:
3616:
3611:
3585:
3575:
3562:
3552:
3539:
3529:
3516:
3506:
3493:
3483:
3470:
3460:
3445:
3437:
3435:
3410:
3406:
3405:
3395:
3391:
3390:
3380:
3376:
3375:
3366:
3237:
3233:
3225:
3214:
3206:
3204:
3153:
3148:
3135:
3130:
3117:
3112:
3106:
3096:
3088:
3086:
3075:
3070:
3040:
3035:
3022:
3017:
3004:
2999:
2986:
2977:
2966:
2958:
2956:
2905:
2895:
2885:
2874:
2861:
2851:
2838:
2828:
2815:
2805:
2793:
2785:
2783:
2738:
2734:
2733:
2730:
2452:
2436:
2420:
2411:
2043:
2025:
2001:
1984:
1976:
1892:
1888:
1887:
1884:
482:
456:
403:
239:
235:
234:
231:
94:Learn how and when to remove this message
8266:Calculus : Single and Multivariable
3429:The components of the cross product are
2142:
110:A representation of a three-dimensional
57:This article includes a list of general
8521:(7th ed.), John Wiley & Sons,
8144:
8096:Rotation formalisms in three dimensions
4490:, which is due to its description as a
1773:
1659:
1593:
1542:
1471:
1423:
1185:
1047:
1024:
991:
963:
566:
396:his geometric framework for quaternions
277:, it serves as a model of the physical
8390:
5362:. The dot product is an example of an
5260:in order to do concrete computations.
3777:Its magnitude is related to the angle
2675:through that conic that are normal to
2144:Regular polytopes in three dimensions
2020:and the surface area of the sphere is
805:Straightedge and compass constructions
8327:
8315:
7357:fundamental theorem of line integrals
7351:Fundamental theorem of line integrals
7345:Fundamental theorem of line integrals
5608:The divergence of a (differentiable)
5145:As opposed to a general vector space
394:, and they were first defined within
342:the construction of the five regular
27:Geometric model of the physical space
7:
8997:Three-dimensional coordinate systems
8378:
8539:Mathematical Methods For Physicists
7919:is quite generally the boundary of
6449:Line, surface, and volume integrals
4030:{\displaystyle {\mathfrak {so}}(3)}
4013:
4010:
1969:The volume of the ball is given by
1827:. Three distinct points are either
262:-dimensional Euclidean space and a
118:-axis pointing towards the observer
8018:{\displaystyle {\mathbb {R} }^{3}}
7782:
7640:
7626:
7623:
7606:
7591:
7586:
7497:
7102:
7093:
7078:
7068:
7053:
7043:
6986:
6376:
6335:
6290:
6275:
6260:
6245:
6215:
6200:
6185:
6170:
6140:
6125:
6110:
6095:
5978:
5974:
5961:
5957:
5944:
5940:
5836:
5821:
5785:
5777:
5762:
5754:
5739:
5731:
5714:
5581:
5561:
5518:
5510:
5490:
5482:
5462:
5454:
5442:
3714:{\displaystyle \varepsilon _{ijk}}
373:Ad locos planos et solidos isagoge
63:it lacks sufficient corresponding
25:
8342:The American Mathematical Monthly
6933:. Let such a parameterization be
5113:can be obtained by starting with
1858:of planes are mutually parallel.
1839:, or determine the entire space.
771:Noncommutative algebraic geometry
251:{\displaystyle \mathbb {R} ^{n},}
162:three-dimensional Euclidean space
8756:
8624:
8108:Distance from a point to a plane
8057:. Thus, for any Galois field GF(
7908:over the boundary of the volume
7865:
7857:
7819:
7790:
7702:{\displaystyle \mathbb {R} ^{n}}
7645:
7632:
7611:
7598:
7521:
7507:
7488:
7479:
7456:
7437:
7072:
7047:
7013:
6808:
6787:
6779:
6756:
6741:
6733:
6568:
6545:
6342:
6305:
6230:
6155:
5927:
5920:
5913:
5828:
5721:
5707:
5644:
5528:
5500:
5472:
5330:{\displaystyle \mathbb {R} ^{3}}
5253:{\displaystyle \mathbb {R} ^{3}}
5220:{\displaystyle \mathbb {R} ^{3}}
5187:{\displaystyle \mathbb {R} ^{3}}
5135:{\displaystyle \mathbb {R} ^{3}}
5086:{\displaystyle \mathbb {R} ^{3}}
4483:{\displaystyle \mathbb {R} ^{3}}
4427:{\displaystyle \mathbb {R} ^{3}}
4285:{\displaystyle \mathbb {R} ^{3}}
4181:
4173:
4162:
4151:
4143:
4132:
4121:
4113:
4102:
4079:
4057:
4049:
3895:
3879:
3862:
3854:
3827:
3805:
3760:
3752:
3741:
3733:
3625:
3617:
3446:
3438:
3238:
3226:
3215:
3207:
3097:
3089:
3076:
2978:
2967:
2959:
2794:
2786:
2747:{\displaystyle \mathbb {R} ^{3}}
2590:are real numbers and not all of
2371:. The plane curve is called the
2334:
2323:
2312:
2301:
2290:
2279:
2268:
2257:
2246:
1901:{\displaystyle \mathbb {R} ^{3}}
1804:
1790:
1776:
152:) are required to determine the
48:
8304:Brannan, Esplen & Gray 1999
8292:Brannan, Esplen & Gray 1999
8077:) are contained in exactly one
7949:may be used as a shorthand for
7146:), and is known as the surface
6434:{\displaystyle \epsilon _{ijk}}
4210:dimensions take the product of
2761:The dot product of two vectors
2363:generated by revolving a plane
1934:of a sphere onto two dimensions
8578:, Cambridge University Press,
7869:
7853:
7839:{\displaystyle \scriptstyle S}
7511:
7503:
7492:
7475:
7401:
7304:
7286:
7247:
7229:
7088:
7036:
7032:
7029:
7017:
7009:
6905:. It can be thought of as the
6822:
6816:
6800:
6797:
6791:
6783:
6745:
6737:
6586:
6582:
6576:
6562:
6558:
6555:
6549:
6541:
6347:
6332:
5663:
5568:
5558:
5414:
5294:
5288:
4835:
4822:
4707:
4668:
4601:
4185:
4169:
4155:
4139:
4125:
4109:
4024:
4018:
3984:
3963:
3940:The space and product form an
3899:
3891:
3883:
3875:
3867:
3849:
3630:
3613:
3591:
3453:
3401:
2941:. The dot product of a vector
2715:Dot product, angle, and length
2072:. If a point has coordinates,
1164:- / other-dimensional
1:
8629:The dictionary definition of
7923:oriented by outward-pointing
6917:, by considering a system of
6625:parametrization of the curve
5812:Einstein summation convention
5385:Gradient, divergence and curl
4644:{\displaystyle 1\leq i\leq 3}
2055:{\displaystyle A=4\pi r^{2}.}
1798:Cylindrical coordinate system
8230:"Euclidean space | geometry"
4887:{\displaystyle \delta _{ij}}
4509:{\displaystyle \mathbb {R} }
4086:{\displaystyle \mathbf {C} }
3834:{\displaystyle \mathbf {B} }
3812:{\displaystyle \mathbf {A} }
2237:
2221:
2147:
444:{\displaystyle q=a+ui+vj+wk}
226:-tuples is commonly denoted
8644:"Four-Dimensional Geometry"
8489:& Ute Rosenbaum (1998)
8037:. The simplest instance is
7263:and is usually written as:
3725:. It has the property that
1812:Spherical coordinate system
1784:Cartesian coordinate system
264:Cartesian coordinate system
160:. Most commonly, it is the
112:Cartesian coordinate system
9023:
8992:Multi-dimensional geometry
8596:(3rd ed.), Springer,
8495:Cambridge University Press
8268:(6 ed.). John wiley.
8118:Skew lines § Distance
7669:
7545:
7348:
5377:
5351:
5267:
4236:
3299:
2927:The magnitude of a vector
2718:
2395:
2352:
2125:
1942:in 3-space (also called a
1919:
1912:, and hence are coplanar.
557:
534:based on Gibbs' lectures.
36:
29:
8958:
8754:
8602:10.1007/978-1-4757-1949-9
8519:Elementary Linear Algebra
7994:, which locally resemble
7256:{\displaystyle f(x,y,z),}
6913:the surface of interest,
4381:. This corresponds to an
2644:Hyperboloid of two sheets
2233:
2230:
2213:
2206:
2184:
2176:
2156:
2151:
1850:. Two parallel lines, or
212:can be understood as the
8982:Euclidean solid geometry
8662:University of Queensland
7904:, the right side is the
6953:) varies in some region
6884:) give the endpoints of
6645:) give the endpoints of
5810:In index notation, with
4041:. For any three vectors
2639:Hyperboloid of one sheet
2158:Kepler-Poinsot polyhedra
2138:Kepler-Poinsot polyhedra
660:Non-Archimedean geometry
184:three-dimensional region
8254:Hughes-Hallett, Deborah
8234:Encyclopedia Britannica
8123:Three-dimensional graph
7150:. Given a vector field
6919:curvilinear coordinates
6897:is a generalization of
5339:Euclidean affine spaces
3790:{\displaystyle \theta }
2625:There are six types of
2136:and the four nonconvex
1758:cylindrical coordinates
766:Noncommutative geometry
146:in which three values (
128:three-dimensional space
78:more precise citations.
8517:Anton, Howard (1994),
8487:Albrecht Beutelspacher
8462:Rolfsen, Dale (1976).
8185:encyclopediaofmath.org
8113:Four-dimensional space
8019:
7983:in a piece of string.
7976:
7912:. The closed manifold
7886:
7840:
7813:
7703:
7656:
7541:
7532:
7413:
7335:
7257:
7113:
6927:latitude and longitude
6839:
6708:, in the direction of
6669:
6668:{\displaystyle a<b}
6604:
6435:
6402:
6316:
6040:
5862:
5801:
5682:
5636:, that is, a function
5600:
5536:
5426:
5331:
5301:
5254:
5221:
5188:
5159:
5136:
5107:
5087:
5056:
4888:
4858:
4799:
4720:
4645:
4613:
4563:
4510:
4484:
4452:
4428:
4399:
4375:
4355:
4286:
4257:
4229:
4198:
4087:
4065:
4031:
3991:
3932:
3835:
3813:
3791:
3768:
3715:
3682:
3598:
3420:
3346:to both and therefore
3261:
3168:
3053:
2918:
2890:
2748:
2618:are zero, is called a
2540:
2349:Surfaces of revolution
2056:
2014:
1962:(or, more precisely a
1935:
1932:perspective projection
1902:
1852:two intersecting lines
1831:or determine a unique
734:Discrete/Combinatorial
503:
471:
445:
382:'s development of the
380:William Rowan Hamilton
252:
119:
8020:
7988:differential geometry
7971:
7887:
7841:
7814:
7730:(also indicated with
7704:
7657:
7533:
7414:
7336:
7258:
7204:. It can also mean a
7114:
6840:
6670:
6605:
6436:
6403:
6317:
6041:
5872:Cartesian coordinates
5863:
5802:
5683:
5601:
5537:
5427:
5332:
5302:
5255:
5222:
5189:
5160:
5137:
5108:
5088:
5057:
4889:
4859:
4800:
4721:
4646:
4614:
4564:
4511:
4485:
4453:
4429:
4400:
4376:
4356:
4287:
4258:
4227:
4199:
4088:
4066:
4032:
3992:
3933:
3836:
3814:
3792:
3769:
3716:
3683:
3599:
3421:
3322:in three-dimensional
3262:
3169:
3054:
2919:
2870:
2749:
2669:and all the lines of
2659:Hyperbolic paraboloid
2541:
2369:surface of revolution
2355:Surface of revolution
2057:
2015:
1952:from a central point
1929:
1903:
1762:spherical coordinates
717:Discrete differential
549:In Euclidean geometry
504:
502:{\displaystyle i,j,k}
472:
446:
353:, with the advent of
351:Cartesian coordinates
281:, in which all known
253:
214:Cartesian coordinates
140:tri-dimensional space
109:
8874:Dimensions by number
8660:Keith Matthews from
8592:Lang, Serge (1987),
8560:, Berlin: Springer,
8258:McCallum, William G.
8133:Terms of orientation
8103:Dimensional analysis
7998:
7975:'s globe logo in 3-D
7850:
7828:
7763:
7723:and has a piecewise
7684:
7578:
7426:
7374:
7270:
7223:
7194:or region. When the
6968:
6901:to integration over
6719:
6653:
6497:
6412:
6329:
6084:
5901:
5818:
5699:
5640:
5555:
5439:
5393:
5312:
5300:{\displaystyle E(3)}
5282:
5235:
5202:
5169:
5149:
5117:
5097:
5068:
4902:
4868:
4809:
4733:
4655:
4623:
4573:
4520:
4498:
4465:
4442:
4409:
4389:
4365:
4296:
4267:
4247:
4233:Abstract description
4098:
4075:
4045:
4005:
3960:
3942:algebra over a field
3845:
3823:
3801:
3781:
3729:
3692:
3610:
3434:
3365:
3342:is a vector that is
3203:
3177:the formula for the
3069:
2955:
2782:
2729:
2410:
2402:In analogy with the
2024:
1975:
1883:
532:Edwin Bidwell Wilson
522:Josiah Willard Gibbs
481:
455:
402:
337:Books XI to XIII of
230:
172:three, which models
8537:and Hans J. Weber.
8491:Projective Geometry
8051:projective geometry
7896:The left side is a
7132:partial derivatives
6777:
6537:
5354:inner product space
5348:Inner product space
3158:
3140:
3122:
3045:
3027:
3009:
2654:Elliptic paraboloid
2145:
984:Pythagorean theorem
470:{\displaystyle a=0}
216:of a location in a
39:3D (disambiguation)
8803:Degrees of freedom
8706:Dimensional spaces
8641:Weisstein, Eric W.
8262:Gleason, Andrew M.
8029:In finite geometry
8015:
7977:
7882:
7836:
7835:
7809:
7753:divergence theorem
7699:
7672:Divergence theorem
7666:Divergence theorem
7652:
7528:
7409:
7331:
7282:
7253:
7189:three-dimensional
7109:
6899:multiple integrals
6835:
6763:
6731:
6665:
6600:
6523:
6509:
6443:Levi-Civita symbol
6431:
6398:
6312:
6036:
6030:
5858:
5797:
5692:-valued function:
5688:, is equal to the
5678:
5596:
5532:
5422:
5327:
5297:
5264:Affine description
5250:
5217:
5184:
5155:
5132:
5103:
5083:
5052:
5043:
4994:
4945:
4884:
4854:
4795:
4716:
4641:
4609:
4559:
4506:
4480:
4448:
4424:
4395:
4371:
4351:
4282:
4253:
4230:
4194:
4083:
4061:
4027:
3987:
3928:
3831:
3809:
3787:
3764:
3723:Levi-Civita symbol
3711:
3678:
3594:
3416:
3257:
3164:
3144:
3126:
3108:
3049:
3031:
3013:
2995:
2914:
2744:
2629:quadric surfaces:
2536:
2143:
2052:
2010:
1936:
1898:
1877:Varignon's theorem
554:Coordinate systems
499:
467:
441:
371:in the manuscript
248:
144:mathematical space
120:
8987:Analytic geometry
8969:
8968:
8778:Lebesgue covering
8743:Algebraic variety
8632:three-dimensional
8611:978-1-4757-1949-9
8585:978-0-521-59787-6
8547:978-0-12-059876-2
8535:Arfken, George B.
8528:978-0-471-58742-2
8448:978-0-07-161545-7
7273:
7085:
7060:
6722:
6500:
6297:
6267:
6222:
6192:
6147:
6117:
5985:
5968:
5951:
5792:
5769:
5746:
5525:
5497:
5469:
5158:{\displaystyle V}
5106:{\displaystyle V}
4747:
4492:Cartesian product
4451:{\displaystyle V}
4398:{\displaystyle V}
4374:{\displaystyle V}
4256:{\displaystyle V}
3159:
3101:
2694:In linear algebra
2346:
2345:
1992:
1916:Spheres and balls
1719:analytic geometry
1715:
1714:
1680:
1679:
1403:List of geometers
1086:Three-dimensional
1075:
1074:
560:Coordinate system
539:Hermann Grassmann
520:It was not until
509:, as well as the
355:analytic geometry
339:Euclid's Elements
287:relativity theory
275:classical physics
104:
103:
96:
18:Three-dimensional
16:(Redirected from
9014:
8766:Other dimensions
8760:
8728:Projective space
8692:
8685:
8678:
8669:
8654:
8653:
8628:
8614:
8588:
8570:
8531:
8504:
8484:
8478:
8477:
8459:
8453:
8452:
8434:
8428:
8427:
8425:
8424:
8409:
8403:
8400:
8394:
8388:
8382:
8375:
8369:
8368:
8337:
8331:
8325:
8319:
8313:
8307:
8306:, pp. 41–42
8301:
8295:
8294:, pp. 34–35
8289:
8280:
8279:
8275:978-0470-88861-2
8250:
8244:
8243:
8241:
8240:
8226:
8220:
8219:
8217:
8216:
8201:
8195:
8194:
8192:
8191:
8177:
8171:
8170:
8168:
8167:
8152:
8063:projective space
8024:
8022:
8021:
8016:
8014:
8013:
8008:
8007:
7970:
7957:
7948:
7939:
7932:
7922:
7918:
7911:
7906:surface integral
7903:
7900:over the volume
7892:
7891:
7889:
7888:
7883:
7868:
7860:
7846:
7845:
7843:
7842:
7837:
7823:
7822:
7818:
7816:
7815:
7810:
7798:
7794:
7793:
7785:
7775:
7774:
7750:
7746:
7740:
7729:
7718:
7709:(in the case of
7708:
7706:
7705:
7700:
7698:
7697:
7692:
7679:
7661:
7659:
7658:
7653:
7648:
7643:
7635:
7630:
7629:
7614:
7609:
7601:
7590:
7589:
7557:surface integral
7537:
7535:
7534:
7529:
7524:
7510:
7496:
7495:
7491:
7482:
7463:
7459:
7444:
7440:
7418:
7416:
7415:
7410:
7408:
7400:
7399:
7394:
7340:
7338:
7337:
7332:
7281:
7262:
7260:
7259:
7254:
7208:within a region
7118:
7116:
7115:
7110:
7105:
7096:
7091:
7087:
7086:
7084:
7076:
7075:
7066:
7061:
7059:
7051:
7050:
7041:
7016:
7005:
7004:
6989:
6980:
6979:
6894:surface integral
6844:
6842:
6841:
6836:
6815:
6811:
6790:
6782:
6776:
6771:
6759:
6744:
6736:
6730:
6712:, is defined as
6674:
6672:
6671:
6666:
6621:is an arbitrary
6609:
6607:
6606:
6601:
6589:
6575:
6571:
6565:
6548:
6536:
6531:
6508:
6480:piecewise smooth
6440:
6438:
6437:
6432:
6430:
6429:
6407:
6405:
6404:
6399:
6394:
6393:
6384:
6383:
6374:
6373:
6355:
6354:
6345:
6321:
6319:
6318:
6313:
6308:
6303:
6299:
6298:
6296:
6288:
6287:
6286:
6273:
6268:
6266:
6258:
6257:
6256:
6243:
6233:
6228:
6224:
6223:
6221:
6213:
6212:
6211:
6198:
6193:
6191:
6183:
6182:
6181:
6168:
6158:
6153:
6149:
6148:
6146:
6138:
6137:
6136:
6123:
6118:
6116:
6108:
6107:
6106:
6093:
6045:
6043:
6042:
6037:
6035:
6034:
6027:
6026:
6015:
6014:
6003:
6002:
5990:
5986:
5984:
5973:
5969:
5967:
5956:
5952:
5950:
5939:
5934:
5930:
5923:
5916:
5867:
5865:
5864:
5859:
5854:
5853:
5844:
5843:
5831:
5806:
5804:
5803:
5798:
5793:
5791:
5783:
5775:
5770:
5768:
5760:
5752:
5747:
5745:
5737:
5729:
5724:
5710:
5687:
5685:
5684:
5679:
5677:
5676:
5671:
5662:
5661:
5656:
5647:
5605:
5603:
5602:
5597:
5589:
5588:
5576:
5575:
5541:
5539:
5538:
5533:
5531:
5526:
5524:
5516:
5508:
5503:
5498:
5496:
5488:
5480:
5475:
5470:
5468:
5460:
5452:
5431:
5429:
5428:
5423:
5421:
5413:
5412:
5407:
5336:
5334:
5333:
5328:
5326:
5325:
5320:
5306:
5304:
5303:
5298:
5259:
5257:
5256:
5251:
5249:
5248:
5243:
5226:
5224:
5223:
5218:
5216:
5215:
5210:
5193:
5191:
5190:
5185:
5183:
5182:
5177:
5164:
5162:
5161:
5156:
5141:
5139:
5138:
5133:
5131:
5130:
5125:
5112:
5110:
5109:
5104:
5092:
5090:
5089:
5084:
5082:
5081:
5076:
5061:
5059:
5058:
5053:
5048:
5047:
5012:
5011:
4999:
4998:
4963:
4962:
4950:
4949:
4914:
4913:
4893:
4891:
4890:
4885:
4883:
4882:
4863:
4861:
4860:
4855:
4853:
4852:
4834:
4833:
4821:
4820:
4804:
4802:
4801:
4796:
4791:
4790:
4778:
4777:
4765:
4764:
4749:
4748:
4745:
4743:
4742:
4725:
4723:
4722:
4717:
4706:
4705:
4693:
4692:
4680:
4679:
4667:
4666:
4650:
4648:
4647:
4642:
4618:
4616:
4615:
4610:
4608:
4600:
4599:
4594:
4585:
4584:
4568:
4566:
4565:
4560:
4558:
4550:
4542:
4534:
4533:
4528:
4515:
4513:
4512:
4507:
4505:
4489:
4487:
4486:
4481:
4479:
4478:
4473:
4457:
4455:
4454:
4449:
4433:
4431:
4430:
4425:
4423:
4422:
4417:
4404:
4402:
4401:
4396:
4380:
4378:
4377:
4372:
4360:
4358:
4357:
4352:
4347:
4346:
4334:
4333:
4321:
4320:
4305:
4304:
4291:
4289:
4288:
4283:
4281:
4280:
4275:
4262:
4260:
4259:
4254:
4219:seven dimensions
4216:
4203:
4201:
4200:
4195:
4184:
4176:
4165:
4154:
4146:
4135:
4124:
4116:
4105:
4092:
4090:
4089:
4084:
4082:
4070:
4068:
4067:
4062:
4060:
4052:
4036:
4034:
4033:
4028:
4017:
4016:
3996:
3994:
3993:
3988:
3977:
3976:
3971:
3937:
3935:
3934:
3929:
3924:
3920:
3902:
3898:
3886:
3882:
3870:
3866:
3865:
3857:
3841:by the identity
3840:
3838:
3837:
3832:
3830:
3818:
3816:
3815:
3810:
3808:
3796:
3794:
3793:
3788:
3773:
3771:
3770:
3765:
3763:
3755:
3744:
3736:
3720:
3718:
3717:
3712:
3710:
3709:
3687:
3685:
3684:
3679:
3677:
3676:
3667:
3666:
3657:
3656:
3638:
3637:
3628:
3620:
3605:
3603:
3601:
3600:
3595:
3590:
3589:
3580:
3579:
3567:
3566:
3557:
3556:
3544:
3543:
3534:
3533:
3521:
3520:
3511:
3510:
3498:
3497:
3488:
3487:
3475:
3474:
3465:
3464:
3449:
3441:
3425:
3423:
3422:
3417:
3415:
3414:
3409:
3400:
3399:
3394:
3385:
3384:
3379:
3316:binary operation
3291:
3285:
3275:
3266:
3264:
3263:
3258:
3241:
3229:
3218:
3210:
3195:
3189:
3179:Euclidean length
3173:
3171:
3170:
3165:
3160:
3157:
3152:
3139:
3134:
3121:
3116:
3107:
3102:
3100:
3092:
3087:
3079:
3058:
3056:
3055:
3050:
3044:
3039:
3026:
3021:
3008:
3003:
2991:
2990:
2981:
2970:
2962:
2947:
2940:
2932:
2923:
2921:
2920:
2915:
2910:
2909:
2900:
2899:
2889:
2884:
2866:
2865:
2856:
2855:
2843:
2842:
2833:
2832:
2820:
2819:
2810:
2809:
2797:
2789:
2774:
2767:
2753:
2751:
2750:
2745:
2743:
2742:
2737:
2678:
2674:
2668:
2617:
2611:
2589:
2583:
2545:
2543:
2542:
2537:
2457:
2456:
2441:
2440:
2425:
2424:
2392:Quadric surfaces
2338:
2327:
2316:
2305:
2294:
2283:
2272:
2261:
2250:
2146:
2113:
2094:
2071:
2061:
2059:
2058:
2053:
2048:
2047:
2019:
2017:
2016:
2011:
2006:
2005:
1993:
1985:
1955:
1951:
1907:
1905:
1904:
1899:
1897:
1896:
1891:
1819:Lines and planes
1808:
1794:
1780:
1744:
1738:
1717:In mathematics,
1707:
1700:
1693:
1421:
940:
873:Zero-dimensional
578:
564:
508:
506:
505:
500:
476:
474:
473:
468:
450:
448:
447:
442:
369:Pierre de Fermat
272:
261:
257:
255:
254:
249:
244:
243:
238:
225:
221:
208:
99:
92:
88:
85:
79:
74:this article by
65:inline citations
52:
51:
44:
21:
9022:
9021:
9017:
9016:
9015:
9013:
9012:
9011:
8972:
8971:
8970:
8965:
8954:
8933:
8869:
8807:
8761:
8752:
8718:Euclidean space
8701:
8696:
8639:
8638:
8621:
8612:
8591:
8586:
8573:
8568:
8552:
8529:
8516:
8513:
8508:
8507:
8485:
8481:
8474:
8464:Knots and Links
8461:
8460:
8456:
8449:
8439:Vector Analysis
8436:
8435:
8431:
8422:
8420:
8411:
8410:
8406:
8401:
8397:
8389:
8385:
8381:, ch. I.1
8376:
8372:
8354:10.2307/2323537
8348:(10): 697–701.
8339:
8338:
8334:
8326:
8322:
8314:
8310:
8302:
8298:
8290:
8283:
8276:
8252:
8251:
8247:
8238:
8236:
8228:
8227:
8223:
8214:
8212:
8203:
8202:
8198:
8189:
8187:
8179:
8178:
8174:
8165:
8163:
8154:
8153:
8146:
8141:
8087:
8047:Galois geometry
8035:finite geometry
8031:
8001:
7996:
7995:
7973:Knowledge (XXG)
7966:
7964:
7950:
7941:
7934:
7928:
7920:
7913:
7909:
7901:
7898:volume integral
7848:
7847:
7826:
7825:
7824:
7820:
7780:
7776:
7766:
7761:
7760:
7759:
7748:
7742:
7731:
7727:
7725:smooth boundary
7710:
7687:
7682:
7681:
7680:is a subset of
7677:
7674:
7668:
7618:
7581:
7576:
7575:
7553:Stokes' theorem
7550:
7548:Stokes' theorem
7544:
7542:Stokes' theorem
7467:
7451:
7432:
7424:
7423:
7389:
7372:
7371:
7353:
7347:
7268:
7267:
7221:
7220:
7206:triple integral
7180:volume integral
7077:
7067:
7052:
7042:
7039:
7035:
6996:
6971:
6966:
6965:
6907:double integral
6862:parametrization
6860:is a bijective
6848:where · is the
6806:
6717:
6716:
6651:
6650:
6566:
6495:
6494:
6451:
6415:
6410:
6409:
6385:
6375:
6359:
6346:
6327:
6326:
6289:
6278:
6274:
6259:
6248:
6244:
6241:
6237:
6214:
6203:
6199:
6184:
6173:
6169:
6166:
6162:
6139:
6128:
6124:
6109:
6098:
6094:
6091:
6087:
6082:
6081:
6029:
6028:
6018:
6016:
6006:
6004:
5994:
5991:
5988:
5987:
5977:
5970:
5960:
5953:
5943:
5935:
5932:
5931:
5924:
5917:
5905:
5899:
5898:
5845:
5835:
5816:
5815:
5784:
5776:
5761:
5753:
5738:
5730:
5697:
5696:
5666:
5651:
5638:
5637:
5580:
5567:
5553:
5552:
5517:
5509:
5489:
5481:
5461:
5453:
5437:
5436:
5402:
5391:
5390:
5387:
5382:
5380:vector calculus
5376:
5356:
5350:
5315:
5310:
5309:
5280:
5279:
5276:
5274:Euclidean space
5266:
5238:
5233:
5232:
5205:
5200:
5199:
5172:
5167:
5166:
5147:
5146:
5120:
5115:
5114:
5095:
5094:
5071:
5066:
5065:
5042:
5041:
5035:
5034:
5028:
5027:
5017:
5003:
4993:
4992:
4986:
4985:
4979:
4978:
4968:
4954:
4944:
4943:
4937:
4936:
4930:
4929:
4919:
4905:
4900:
4899:
4896:Kronecker delta
4871:
4866:
4865:
4841:
4825:
4812:
4807:
4806:
4782:
4769:
4756:
4736:
4731:
4730:
4697:
4684:
4671:
4658:
4653:
4652:
4651:. For example,
4621:
4620:
4589:
4576:
4571:
4570:
4523:
4518:
4517:
4496:
4495:
4468:
4463:
4462:
4440:
4439:
4412:
4407:
4406:
4387:
4386:
4363:
4362:
4338:
4325:
4312:
4294:
4293:
4270:
4265:
4264:
4245:
4244:
4241:
4235:
4211:
4096:
4095:
4073:
4072:
4043:
4042:
4039:Jacobi identity
4003:
4002:
3966:
3958:
3957:
3944:, which is not
3910:
3906:
3890:
3874:
3852:
3848:
3843:
3842:
3821:
3820:
3799:
3798:
3779:
3778:
3727:
3726:
3695:
3690:
3689:
3668:
3658:
3642:
3629:
3608:
3607:
3581:
3571:
3558:
3548:
3535:
3525:
3512:
3502:
3489:
3479:
3466:
3456:
3432:
3431:
3430:
3404:
3389:
3374:
3363:
3362:
3334:of the vectors
3304:
3298:
3287:
3281:
3271:
3201:
3200:
3191:
3185:
3181:of the vector.
3067:
3066:
2982:
2953:
2952:
2948:with itself is
2942:
2934:
2928:
2901:
2891:
2857:
2847:
2834:
2824:
2811:
2801:
2780:
2779:
2775:is defined as:
2769:
2762:
2758:of the vector.
2732:
2727:
2726:
2723:
2717:
2696:
2676:
2670:
2666:
2620:quadric surface
2613:
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2398:Quadric surface
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2134:Platonic solids
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1871:linear equation
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1766:Euclidean space
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1723:coordinate axes
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344:Platonic solids
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199:Technically, a
166:Euclidean space
164:, that is, the
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8617:
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8594:Linear algebra
8589:
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8554:Berger, Marcel
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8402:Arfken, p. 43.
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8128:Solid geometry
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2700:linear algebra
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2684:ruled surfaces
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558:Main article:
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543:Giuseppe Peano
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359:René Descartes
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8784:
8781:
8779:
8776:
8774:
8771:
8770:
8768:
8764:
8759:
8749:
8746:
8744:
8741:
8739:
8736:
8734:
8731:
8729:
8726:
8724:
8721:
8719:
8716:
8714:
8711:
8710:
8708:
8704:
8700:
8693:
8688:
8686:
8681:
8679:
8674:
8673:
8670:
8663:
8659:
8656:
8651:
8650:
8645:
8642:
8637:
8635:at Wiktionary
8634:
8633:
8627:
8623:
8622:
8618:
8613:
8607:
8603:
8599:
8595:
8590:
8587:
8581:
8577:
8572:
8569:
8567:3-540-11658-3
8563:
8559:
8555:
8551:
8548:
8544:
8540:
8536:
8533:
8530:
8524:
8520:
8515:
8514:
8510:
8503:
8502:0-521-48277-1
8499:
8496:
8492:
8488:
8483:
8480:
8475:
8473:0-914098-16-0
8469:
8465:
8458:
8455:
8450:
8444:
8440:
8433:
8430:
8419:(in Japanese)
8418:
8414:
8408:
8405:
8399:
8396:
8392:
8387:
8384:
8380:
8374:
8371:
8367:
8363:
8359:
8355:
8351:
8347:
8343:
8336:
8333:
8330:, p. 131
8329:
8324:
8321:
8318:, p. 133
8317:
8312:
8309:
8305:
8300:
8297:
8293:
8288:
8286:
8282:
8277:
8271:
8267:
8263:
8259:
8255:
8249:
8246:
8235:
8231:
8225:
8222:
8211:(in Japanese)
8210:
8206:
8200:
8197:
8186:
8182:
8176:
8173:
8162:(in Japanese)
8161:
8157:
8151:
8149:
8145:
8138:
8134:
8131:
8129:
8126:
8124:
8121:
8119:
8116:
8114:
8111:
8109:
8106:
8104:
8101:
8097:
8094:
8093:
8092:
8089:
8088:
8084:
8082:
8080:
8076:
8072:
8068:
8064:
8060:
8056:
8055:finite fields
8052:
8049:, a study of
8048:
8044:
8040:
8036:
8028:
8026:
8010:
7993:
7989:
7984:
7982:
7974:
7969:
7961:
7959:
7956:
7953:
7947:
7944:
7938:
7931:
7926:
7917:
7907:
7899:
7879:
7876:
7873:
7861:
7832:
7806:
7803:
7800:
7795:
7786:
7777:
7771:
7767:
7758:
7757:
7756:
7754:
7745:
7739:
7735:
7726:
7722:
7717:
7713:
7694:
7673:
7665:
7649:
7636:
7619:
7615:
7602:
7594:
7582:
7574:
7573:
7572:
7570:
7569:line integral
7566:
7562:
7558:
7554:
7549:
7525:
7517:
7514:
7500:
7483:
7472:
7468:
7464:
7460:
7452:
7448:
7445:
7441:
7433:
7429:
7422:
7421:
7420:
7396:
7386:
7383:
7380:
7377:
7368:
7366:
7362:
7361:line integral
7358:
7352:
7344:
7328:
7325:
7322:
7318:
7315:
7311:
7308:
7301:
7298:
7295:
7292:
7289:
7283:
7278:
7274:
7266:
7265:
7264:
7250:
7244:
7241:
7238:
7235:
7232:
7226:
7219:
7215:
7211:
7207:
7203:
7202:
7197:
7193:
7192:
7186:
7182:
7181:
7175:
7173:
7169:
7165:
7161:
7157:
7153:
7149:
7145:
7141:
7137:
7133:
7129:
7128:cross product
7125:
7106:
7097:
7081:
7062:
7056:
7026:
7023:
7020:
7006:
7001:
6997:
6993:
6990:
6981:
6976:
6972:
6964:
6963:
6962:
6960:
6956:
6952:
6948:
6944:
6940:
6936:
6932:
6928:
6924:
6920:
6916:
6912:
6908:
6904:
6900:
6896:
6895:
6889:
6887:
6883:
6879:
6875:
6871:
6867:
6864:of the curve
6863:
6859:
6855:
6851:
6832:
6829:
6826:
6819:
6812:
6803:
6794:
6773:
6768:
6764:
6760:
6752:
6748:
6727:
6723:
6715:
6714:
6713:
6711:
6707:
6703:
6700:
6696:
6692:
6688:
6684:
6681:
6676:
6662:
6659:
6656:
6648:
6644:
6640:
6636:
6632:
6628:
6624:
6620:
6616:
6597:
6594:
6591:
6579:
6572:
6552:
6538:
6533:
6528:
6524:
6520:
6517:
6514:
6510:
6505:
6501:
6493:
6492:
6491:
6489:
6485:
6481:
6477:
6476:
6475:line integral
6471:
6467:
6463:
6459:
6456:
6448:
6446:
6444:
6426:
6423:
6420:
6416:
6395:
6390:
6386:
6380:
6370:
6367:
6364:
6360:
6356:
6351:
6338:
6309:
6300:
6293:
6283:
6279:
6269:
6263:
6253:
6249:
6238:
6234:
6225:
6218:
6208:
6204:
6194:
6188:
6178:
6174:
6163:
6159:
6150:
6143:
6133:
6129:
6119:
6113:
6103:
6099:
6088:
6080:
6079:
6078:
6076:
6072:
6068:
6064:
6060:
6056:
6052:
6031:
6023:
6019:
6011:
6007:
5999:
5995:
5981:
5964:
5947:
5906:
5897:
5896:
5895:
5893:
5889:
5885:
5881:
5877:
5873:
5868:
5855:
5850:
5846:
5840:
5832:
5824:
5813:
5794:
5788:
5780:
5771:
5765:
5757:
5748:
5742:
5734:
5725:
5717:
5711:
5702:
5695:
5694:
5693:
5691:
5673:
5658:
5648:
5635:
5632:
5628:
5625:
5621:
5618:
5614:
5611:
5606:
5593:
5590:
5585:
5577:
5572:
5564:
5550:
5548:
5521:
5513:
5504:
5493:
5485:
5476:
5465:
5457:
5448:
5445:
5435:
5434:
5433:
5409:
5399:
5396:
5384:
5381:
5373:
5371:
5369:
5365:
5364:inner product
5361:
5355:
5347:
5345:
5342:
5340:
5322:
5291:
5285:
5275:
5271:
5263:
5261:
5245:
5229:
5212:
5195:
5179:
5152:
5143:
5127:
5100:
5078:
5062:
5049:
5044:
5038:
5031:
5024:
5018:
5013:
5008:
5004:
5000:
4995:
4989:
4982:
4975:
4969:
4964:
4959:
4955:
4951:
4946:
4940:
4933:
4926:
4920:
4915:
4910:
4906:
4897:
4879:
4876:
4872:
4849:
4846:
4842:
4838:
4830:
4826:
4817:
4813:
4787:
4783:
4779:
4774:
4770:
4766:
4761:
4757:
4750:
4729:
4713:
4710:
4702:
4698:
4694:
4689:
4685:
4681:
4676:
4672:
4663:
4659:
4638:
4635:
4632:
4629:
4626:
4596:
4586:
4581:
4577:
4551:
4543:
4535:
4530:
4494:of copies of
4493:
4475:
4459:
4445:
4437:
4419:
4392:
4384:
4368:
4343:
4339:
4335:
4330:
4326:
4322:
4317:
4313:
4306:
4277:
4250:
4240:
4232:
4226:
4222:
4220:
4214:
4209:
4204:
4191:
4188:
4177:
4166:
4158:
4147:
4136:
4128:
4117:
4106:
4093:
4053:
4040:
4021:
4000:
3981:
3978:
3973:
3955:
3951:
3947:
3943:
3938:
3925:
3921:
3917:
3914:
3911:
3907:
3903:
3887:
3871:
3858:
3784:
3775:
3756:
3748:
3745:
3737:
3724:
3706:
3703:
3700:
3696:
3673:
3669:
3663:
3659:
3653:
3650:
3647:
3643:
3639:
3634:
3621:
3586:
3582:
3576:
3572:
3568:
3563:
3559:
3553:
3549:
3545:
3540:
3536:
3530:
3526:
3522:
3517:
3513:
3507:
3503:
3499:
3494:
3490:
3484:
3480:
3476:
3471:
3467:
3461:
3457:
3450:
3442:
3427:
3411:
3396:
3386:
3381:
3371:
3368:
3359:
3357:
3353:
3349:
3345:
3344:perpendicular
3341:
3337:
3333:
3329:
3325:
3321:
3317:
3313:
3309:
3308:cross product
3303:
3302:Cross product
3296:Cross product
3295:
3293:
3290:
3284:
3279:
3274:
3254:
3251:
3248:
3245:
3219:
3211:
3199:
3198:
3197:
3194:
3188:
3182:
3180:
3161:
3154:
3149:
3145:
3141:
3136:
3131:
3127:
3123:
3118:
3113:
3109:
3103:
3093:
3083:
3065:
3064:
3063:
3046:
3041:
3036:
3032:
3028:
3023:
3018:
3014:
3010:
3005:
3000:
2996:
2992:
2987:
2971:
2963:
2951:
2950:
2949:
2945:
2938:
2931:
2911:
2906:
2902:
2896:
2892:
2886:
2881:
2878:
2875:
2871:
2867:
2862:
2858:
2852:
2848:
2844:
2839:
2835:
2829:
2825:
2821:
2816:
2812:
2806:
2802:
2798:
2790:
2778:
2777:
2776:
2772:
2765:
2759:
2757:
2739:
2722:
2714:
2712:
2710:
2705:
2701:
2693:
2691:
2689:
2685:
2680:
2673:
2660:
2657:
2655:
2652:
2650:
2649:Elliptic cone
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2631:
2630:
2628:
2623:
2621:
2616:
2610:
2606:
2602:
2598:
2594:
2588:
2582:
2578:
2574:
2570:
2566:
2562:
2558:
2554:
2550:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2453:
2449:
2445:
2442:
2437:
2433:
2429:
2426:
2421:
2417:
2413:
2405:
2399:
2391:
2389:
2387:
2383:
2378:
2376:
2375:
2370:
2366:
2362:
2356:
2348:
2342:
2337:
2333:
2331:
2326:
2322:
2320:
2315:
2311:
2309:
2304:
2300:
2298:
2293:
2289:
2287:
2282:
2278:
2276:
2271:
2267:
2265:
2260:
2256:
2254:
2249:
2245:
2243:
2238:
2227:
2225:
2222:
2199:
2197:
2196:Coxeter group
2194:
2193:
2190:
2182:
2174:
2168:
2166:
2163:
2162:
2159:
2154:
2148:
2141:
2139:
2135:
2129:
2121:
2119:
2115:
2111:
2107:
2103:
2099:
2092:
2088:
2084:
2080:
2076:
2070:
2065:
2049:
2044:
2040:
2036:
2033:
2030:
2027:
2007:
2002:
1998:
1994:
1989:
1986:
1981:
1978:
1970:
1967:
1965:
1961:
1960:
1950:
1945:
1941:
1933:
1928:
1923:
1915:
1913:
1911:
1910:parallelogram
1893:
1878:
1874:
1872:
1867:
1862:
1859:
1855:
1853:
1849:
1845:
1840:
1838:
1834:
1830:
1826:
1818:
1813:
1807:
1802:
1799:
1793:
1788:
1785:
1779:
1774:
1772:
1769:
1767:
1763:
1759:
1754:
1752:
1748:
1743:
1737:
1733:
1728:
1724:
1720:
1708:
1703:
1701:
1696:
1694:
1689:
1688:
1686:
1685:
1674:
1671:
1669:
1666:
1665:
1664:
1663:
1658:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1599:
1598:
1597:
1592:
1587:
1584:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1548:
1547:
1546:
1541:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1477:
1476:
1475:
1470:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1431:
1430:
1429:
1426:
1422:
1412:
1411:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1316:
1314:
1311:
1309:
1306:
1304:
1301:
1299:
1296:
1294:
1291:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1200:
1192:
1191:
1188:
1184:
1177:
1174:
1172:
1169:
1168:
1163:
1157:
1156:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
1107:
1104:
1103:
1102:
1099:
1098:
1095:
1092:
1091:
1087:
1081:
1080:
1069:
1066:
1064:
1063:Circumference
1061:
1059:
1056:
1055:
1054:
1053:
1050:
1046:
1041:
1038:
1036:
1033:
1032:
1031:
1030:
1027:
1026:Quadrilateral
1023:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
999:
998:
997:
994:
993:Parallelogram
990:
985:
982:
980:
977:
975:
972:
971:
970:
969:
966:
962:
957:
954:
952:
949:
947:
944:
943:
942:
941:
935:
929:
928:
921:
918:
914:
911:
909:
906:
905:
904:
901:
900:
896:
890:
889:
882:
879:
878:
874:
868:
867:
860:
857:
855:
852:
850:
847:
846:
843:
840:
838:
835:
832:
831:Perpendicular
828:
827:Orthogonality
825:
823:
820:
818:
815:
813:
810:
809:
806:
803:
802:
801:
791:
788:
787:
782:
781:
772:
769:
768:
767:
764:
762:
759:
757:
754:
752:
751:Computational
749:
747:
744:
740:
737:
736:
735:
732:
730:
727:
725:
722:
718:
715:
713:
710:
708:
705:
704:
703:
700:
696:
693:
691:
688:
687:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
654:
651:
647:
644:
643:
642:
639:
638:
637:
636:Non-Euclidean
634:
632:
629:
628:
624:
618:
617:
610:
606:
603:
601:
598:
597:
595:
594:
590:
586:
582:
577:
573:
572:
569:
565:
561:
553:
548:
546:
544:
540:
535:
533:
529:
528:
523:
518:
516:
515:cross product
512:
496:
493:
490:
487:
484:
464:
461:
458:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
397:
393:
389:
385:
381:
376:
374:
370:
366:
365:
360:
357:developed by
356:
352:
347:
346:in a sphere.
345:
340:
332:
330:
328:
327:
322:
320:
315:
313:
308:
307:perpendicular
304:
300:
296:
292:
288:
285:exists. When
284:
280:
276:
270:
265:
245:
240:
220:
215:
211:
207:
202:
197:
195:
194:
189:
185:
181:
180:
175:
171:
167:
163:
159:
155:
151:
150:
145:
141:
137:
133:
129:
125:
117:
113:
108:
98:
95:
87:
77:
73:
67:
66:
60:
55:
46:
45:
40:
33:
19:
8960:
8926:
8895:
8865:Hyperpyramid
8830:Hypersurface
8723:Affine space
8713:Vector space
8647:
8631:
8593:
8575:
8557:
8538:
8518:
8490:
8482:
8463:
8457:
8438:
8432:
8421:. Retrieved
8416:
8407:
8398:
8393:, Chapter 9.
8386:
8373:
8365:
8345:
8341:
8335:
8323:
8311:
8299:
8265:
8248:
8237:. Retrieved
8233:
8224:
8213:. Retrieved
8208:
8199:
8188:. Retrieved
8184:
8175:
8164:. Retrieved
8159:
8074:
8066:
8058:
8041:, which has
8032:
7985:
7978:
7954:
7951:
7945:
7942:
7936:
7929:
7915:
7895:
7743:
7737:
7733:
7715:
7711:
7675:
7565:vector field
7555:relates the
7551:
7369:
7354:
7213:
7209:
7199:
7188:
7178:
7176:
7171:
7167:
7163:
7159:
7155:
7151:
7143:
7139:
7135:
7121:
6954:
6950:
6946:
6942:
6938:
6934:
6922:
6914:
6911:parameterize
6892:
6890:
6885:
6881:
6877:
6873:
6869:
6865:
6857:
6853:
6847:
6709:
6705:
6701:
6694:
6690:
6686:
6682:
6680:vector field
6677:
6646:
6642:
6638:
6634:
6630:
6626:
6618:
6614:
6612:
6487:
6483:
6473:
6469:
6465:
6461:
6457:
6455:scalar field
6452:
6324:
6074:
6070:
6066:
6063:unit vectors
6058:
6054:
6050:
6048:
5891:
5887:
5870:Expanded in
5869:
5809:
5633:
5630:
5626:
5623:
5619:
5616:
5612:
5610:vector field
5607:
5551:
5544:
5432:is given by
5388:
5357:
5343:
5338:
5277:
5270:affine space
5230:
5196:
5165:, the space
5144:
5063:
4460:
4242:
4239:vector space
4212:
4207:
4205:
4094:
3939:
3776:
3428:
3360:
3339:
3335:
3331:
3327:
3311:
3305:
3288:
3282:
3272:
3269:
3196:is given by
3192:
3186:
3183:
3176:
3062:which gives
3061:
2943:
2936:
2929:
2926:
2770:
2763:
2760:
2755:
2724:
2697:
2681:
2671:
2663:
2624:
2619:
2614:
2608:
2604:
2600:
2596:
2592:
2586:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2548:
2401:
2379:
2372:
2358:
2131:
2116:
2109:
2105:
2101:
2097:
2090:
2086:
2082:
2078:
2074:
2068:
2063:
1971:
1968:
1963:
1958:
1948:
1943:
1937:
1875:
1863:
1860:
1856:
1841:
1822:
1770:
1755:
1747:real numbers
1741:
1735:
1731:
1716:
1535:Parameshvara
1348:Parameshvara
1118:Dodecahedron
1085:
702:Differential
536:
525:
519:
377:
372:
364:La Géométrie
362:
361:in his work
348:
336:
324:
317:
310:
268:
218:
205:
198:
193:solid figure
191:
183:
177:
161:
147:
139:
138:or, rarely,
135:
131:
127:
121:
115:
90:
81:
62:
8950:Codimension
8929:-dimensions
8850:Hypersphere
8733:Free module
8493:, page 72,
8391:Berger 1987
8091:3D rotation
8043:Fano planes
7992:3-manifolds
7962:In topology
7751:, then the
6925:, like the
6850:dot product
5884:cylindrical
5549:is written
5374:In calculus
5360:dot product
4805:defined by
4516:, that is,
4383:isomorphism
4206:One can in
3954:Lie algebra
3952:, but is a
3950:associative
3946:commutative
3356:engineering
2721:Dot product
1660:Present day
1607:Lobachevsky
1594:1700s–1900s
1551:Jyeṣṭhadeva
1543:1400s–1700s
1495:Brahmagupta
1318:Lobachevsky
1298:Jyeṣṭhadeva
1248:Brahmagupta
1176:Hypersphere
1148:Tetrahedron
1123:Icosahedron
695:Diophantine
530:written by
511:dot product
384:quaternions
299:coordinates
295:3-manifolds
179:3-manifolds
149:coordinates
76:introducing
9002:3 (number)
8976:Categories
8945:Hyperspace
8825:Hyperplane
8558:Geometry I
8511:References
8423:2023-09-19
8377:Lang
8328:Anton 1994
8316:Anton 1994
8239:2020-08-12
8215:2023-11-07
8190:2020-08-12
8166:2023-09-19
8071:skew lines
7363:through a
6945:), where (
6868:such that
6629:such that
5352:See also:
5268:See also:
5064:Therefore
4237:See also:
3999:isomorphic
2756:components
2374:generatrix
2242:polyhedron
2128:Polyhedron
1866:hyperplane
1520:al-Yasamin
1464:Apollonius
1459:Archimedes
1449:Pythagoras
1439:Baudhayana
1393:al-Yasamin
1343:Pythagoras
1238:Baudhayana
1228:Archimedes
1223:Apollonius
1128:Octahedron
979:Hypotenuse
854:Similarity
849:Congruence
761:Incidence
712:Symplectic
707:Riemannian
690:Arithmetic
665:Projective
653:Hyperbolic
581:Projecting
291:space-time
84:April 2016
59:references
8835:Hypercube
8813:Polytopes
8793:Minkowski
8788:Hausdorff
8783:Inductive
8748:Spacetime
8699:Dimension
8649:MathWorld
7862:⋅
7787:⋅
7783:∇
7768:∭
7637:⋅
7627:Σ
7624:∂
7620:∮
7612:Σ
7603:⋅
7595:×
7592:∇
7587:Σ
7583:∬
7515:⋅
7501:φ
7498:∇
7473:γ
7469:∫
7449:φ
7446:−
7430:φ
7402:→
7387:⊆
7378:φ
7275:∭
7196:integrand
7166:a vector
7124:magnitude
7079:∂
7069:∂
7063:×
7054:∂
7044:∂
6998:∬
6973:∬
6804:⋅
6765:∫
6749:⋅
6724:∫
6623:bijective
6525:∫
6502:∫
6453:For some
6417:ϵ
6377:∂
6361:ϵ
6339:×
6336:∇
6291:∂
6276:∂
6270:−
6261:∂
6246:∂
6216:∂
6201:∂
6195:−
6186:∂
6171:∂
6141:∂
6126:∂
6120:−
6111:∂
6096:∂
5979:∂
5975:∂
5962:∂
5958:∂
5945:∂
5941:∂
5880:spherical
5837:∂
5825:⋅
5822:∇
5814:this is
5786:∂
5778:∂
5763:∂
5755:∂
5740:∂
5732:∂
5718:⋅
5715:∇
5664:→
5582:∂
5562:∇
5519:∂
5511:∂
5491:∂
5483:∂
5463:∂
5455:∂
5443:∇
5415:→
4873:δ
4843:δ
4814:π
4660:π
4636:≤
4630:≤
4602:→
4578:π
4552:×
4544:×
4178:×
4167:×
4148:×
4137:×
4118:×
4107:×
3982:×
3918:θ
3915:
3904:⋅
3888:⋅
3859:×
3785:θ
3757:×
3749:−
3738:×
3697:ε
3644:ε
3622:×
3569:−
3523:−
3477:−
3443:×
3402:→
3387:×
3369:×
3252:θ
3249:
3243:‖
3235:‖
3231:‖
3223:‖
3212:⋅
3094:⋅
3081:‖
3073:‖
2984:‖
2975:‖
2964:⋅
2872:∑
2791:⋅
2634:Ellipsoid
2122:Polytopes
2037:π
1995:π
1829:collinear
1637:Minkowski
1556:Descartes
1490:Aryabhata
1485:Kātyāyana
1416:by period
1328:Minkowski
1303:Kātyāyana
1263:Descartes
1208:Aryabhata
1187:Geometers
1171:Tesseract
1035:Trapezoid
1007:Rectangle
800:Dimension
685:Algebraic
675:Synthetic
646:Spherical
631:Euclidean
170:dimension
114:with the
8962:Category
8938:See also
8738:Manifold
8576:Geometry
8556:(1987),
8264:(2013).
8085:See also
8073:in PG(3,
7676:Suppose
7365:gradient
7218:function
7185:integral
7089:‖
7037:‖
6903:surfaces
6813:′
6685: :
6573:′
6478:along a
6460: :
6065:for the
6061:are the
5890:is, for
4746:Standard
4619:, where
4385:between
3900:‖
3892:‖
3884:‖
3876:‖
3868:‖
3850:‖
3797:between
3280:between
2386:cylinder
2165:Symmetry
2064:3-sphere
1944:2-sphere
1844:parallel
1837:coplanar
1627:Poincaré
1571:Minggatu
1530:Yang Hui
1500:Virasena
1388:Yang Hui
1383:Virasena
1353:Poincaré
1333:Minggatu
1113:Cylinder
1058:Diameter
1017:Rhomboid
974:Altitude
965:Triangle
859:Symmetry
837:Parallel
822:Diagonal
792:Features
789:Concepts
680:Analytic
641:Elliptic
623:Branches
609:Timeline
568:Geometry
314:/breadth
279:universe
154:position
132:3D space
124:geometry
8860:Simplex
8798:Fractal
8362:2323537
8079:regulus
8039:PG(3,2)
7925:normals
7721:compact
7559:of the
7419:. Then
7187:over a
7148:element
7130:of the
7126:of the
6957:in the
6073:-, and
5545:and in
4894:is the
3721:is the
3352:physics
3320:vectors
3318:on two
3276:is the
2709:vectors
2688:regulus
2361:surface
2341:{3,5/2}
2330:{5/2,3}
2319:{5,5/2}
2308:{5/2,5}
2240:Regular
2095:, then
1908:form a
1652:Coxeter
1632:Hilbert
1617:Riemann
1566:Huygens
1525:al-Tusi
1515:Khayyám
1505:Alhazen
1472:1–1400s
1373:al-Tusi
1358:Riemann
1308:Khayyám
1293:Huygens
1288:Hilbert
1258:Coxeter
1218:Alhazen
1196:by name
1133:Pyramid
1012:Rhombus
956:Polygon
908:segment
756:Fractal
739:Digital
724:Complex
605:History
600:Outline
333:History
266:. When
210:numbers
186:(or 3D
142:) is a
136:3-space
72:improve
8817:shapes
8664:, 1991
8608:
8582:
8564:
8545:
8525:
8500:
8470:
8445:
8360:
8272:
8053:using
7927:, and
7755:says:
7741:). If
7201:volume
7191:domain
7183:is an
6931:sphere
6876:) and
6678:For a
6637:) and
6613:where
6482:curve
6472:, the
6408:where
6057:, and
6049:where
5690:scalar
5198:space
4864:where
3688:where
3354:, and
3348:normal
3270:where
2546:where
2149:Class
1964:3-ball
1940:sphere
1922:Sphere
1846:or be
1751:origin
1739:, and
1727:origin
1673:Gromov
1668:Atiyah
1647:Veblen
1642:Cartan
1612:Bolyai
1581:Sakabe
1561:Pascal
1454:Euclid
1444:Manava
1378:Veblen
1363:Sakabe
1338:Pascal
1323:Manava
1283:Gromov
1268:Euclid
1253:Cartan
1243:Bolyai
1233:Atiyah
1143:Sphere
1106:cuboid
1094:Volume
1049:Circle
1002:Square
920:Length
842:Vertex
746:Convex
729:Finite
670:Affine
585:sphere
392:vector
388:scalar
326:length
323:, and
321:/depth
319:height
283:matter
188:domain
61:, but
9007:Space
8921:Eight
8916:Seven
8896:Three
8773:Krull
8358:JSTOR
8139:Notes
8065:PG(3,
7714:= 3,
7563:of a
7216:of a
6959:plane
6929:on a
6856:: →
6699:curve
6617:: →
5874:(see
5368:SO(3)
3324:space
3314:is a
3278:angle
2365:curve
2297:{3,5}
2286:{5,3}
2275:{3,4}
2264:{4,3}
2253:{3,3}
2224:Order
1833:plane
1622:Klein
1602:Gauss
1576:Euler
1510:Sijzi
1480:Zhang
1434:Ahmes
1398:Zhang
1368:Sijzi
1313:Klein
1278:Gauss
1273:Euler
1213:Ahmes
946:Plane
881:Point
817:Curve
812:Angle
589:plane
587:to a
312:width
303:plane
201:tuple
190:), a
158:point
156:of a
32:Space
8906:Five
8901:Four
8881:Zero
8815:and
8606:ISBN
8580:ISBN
8562:ISBN
8543:ISBN
8523:ISBN
8498:ISBN
8468:ISBN
8443:ISBN
8379:1987
8270:ISBN
7981:knot
7561:curl
7370:Let
7355:The
6852:and
6660:<
6649:and
5882:and
5878:for
5272:and
4436:here
4405:and
4361:for
4071:and
3948:nor
3819:and
3338:and
3306:The
3286:and
3190:and
2768:and
2612:and
2584:and
2382:cone
2234:120
1959:ball
1848:skew
1825:line
1760:and
1586:Aida
1203:Aida
1162:Four
1101:Cube
1068:Area
1040:Kite
951:Area
903:Line
541:and
513:and
390:and
367:and
126:, a
8911:Six
8891:Two
8886:One
8598:doi
8350:doi
7986:In
7958:.)
7940:. (
7212:in
7162:in
7154:on
7134:of
6921:on
6486:⊂
6069:-,
5703:div
4215:− 1
3997:is
3912:sin
3310:or
3246:cos
2704:box
2231:48
2228:24
2218:,
2211:,
2204:,
2112:= 1
1966:).
1425:BCE
913:ray
271:= 3
203:of
168:of
122:In
8978::
8646:.
8604:,
8415:.
8364:.
8356:.
8346:90
8344:.
8284:^
8260:;
8256:;
8232:.
8207:.
8183:.
8158:.
8147:^
8081:.
8025:.
7955:dS
7736:=
7177:A
7142:,
6949:,
6941:,
6891:A
6888:.
6704:⊂
6693:→
6689:⊆
6675:.
6468:→
6464:⊆
6445:.
6053:,
5629:+
5622:+
5615:=
5370:.
4458:.
4221:.
3774:.
3426:.
3358:.
3330:×
3292:.
2946:=
2939:||
2935:||
2773:=
2766:=
2711:.
2690:.
2622:.
2607:,
2603:,
2599:,
2595:,
2579:,
2575:,
2571:,
2567:,
2563:,
2559:,
2555:,
2551:,
2388:.
2359:A
2140:.
2108:+
2104:+
2100:+
2089:,
2085:,
2081:,
1938:A
1930:A
1864:A
1768:.
1734:,
583:a
329:.
316:,
196:.
134:,
8927:n
8691:e
8684:t
8677:v
8652:.
8600::
8549:.
8476:.
8451:.
8426:.
8352::
8278:.
8242:.
8218:.
8193:.
8169:.
8075:q
8067:q
8059:q
8011:3
8005:R
7952:n
7946:S
7943:d
7937:V
7935:∂
7930:n
7921:V
7916:V
7914:∂
7910:V
7902:V
7880:.
7877:S
7874:d
7870:)
7866:n
7858:F
7854:(
7833:S
7807:=
7804:V
7801:d
7796:)
7791:F
7778:(
7772:V
7749:V
7744:F
7738:S
7734:V
7732:∂
7728:S
7716:V
7712:n
7695:n
7690:R
7678:V
7650:.
7646:r
7641:d
7633:F
7616:=
7607:d
7599:F
7526:.
7522:r
7518:d
7512:)
7508:r
7504:(
7493:]
7489:q
7484:,
7480:p
7476:[
7465:=
7461:)
7457:p
7453:(
7442:)
7438:q
7434:(
7406:R
7397:n
7392:R
7384:U
7381::
7329:.
7326:z
7323:d
7319:y
7316:d
7312:x
7309:d
7305:)
7302:z
7299:,
7296:y
7293:,
7290:x
7287:(
7284:f
7279:D
7251:,
7248:)
7245:z
7242:,
7239:y
7236:,
7233:x
7230:(
7227:f
7214:R
7210:D
7172:x
7170:(
7168:v
7164:S
7160:x
7156:S
7152:v
7144:t
7140:s
7138:(
7136:x
7107:t
7103:d
7098:s
7094:d
7082:t
7073:x
7057:s
7048:x
7033:)
7030:)
7027:t
7024:,
7021:s
7018:(
7014:x
7010:(
7007:f
7002:T
6994:=
6991:S
6987:d
6982:f
6977:S
6955:T
6951:t
6947:s
6943:t
6939:s
6937:(
6935:x
6923:S
6915:S
6886:C
6882:b
6880:(
6878:r
6874:a
6872:(
6870:r
6866:C
6858:C
6854:r
6833:.
6830:t
6827:d
6823:)
6820:t
6817:(
6809:r
6801:)
6798:)
6795:t
6792:(
6788:r
6784:(
6780:F
6774:b
6769:a
6761:=
6757:r
6753:d
6746:)
6742:r
6738:(
6734:F
6728:C
6710:r
6706:U
6702:C
6695:R
6691:R
6687:U
6683:F
6663:b
6657:a
6647:C
6643:b
6641:(
6639:r
6635:a
6633:(
6631:r
6627:C
6619:C
6615:r
6598:.
6595:t
6592:d
6587:|
6583:)
6580:t
6577:(
6569:r
6563:|
6559:)
6556:)
6553:t
6550:(
6546:r
6542:(
6539:f
6534:b
6529:a
6521:=
6518:s
6515:d
6511:f
6506:C
6488:U
6484:C
6470:R
6466:R
6462:U
6458:f
6427:k
6424:j
6421:i
6396:,
6391:k
6387:F
6381:j
6371:k
6368:j
6365:i
6357:=
6352:i
6348:)
6343:F
6333:(
6310:.
6306:k
6301:)
6294:y
6284:x
6280:F
6264:x
6254:y
6250:F
6239:(
6235:+
6231:j
6226:)
6219:x
6209:z
6205:F
6189:z
6179:x
6175:F
6164:(
6160:+
6156:i
6151:)
6144:z
6134:y
6130:F
6114:y
6104:z
6100:F
6089:(
6075:z
6071:y
6067:x
6059:k
6055:j
6051:i
6032:|
6024:z
6020:F
6012:y
6008:F
6000:x
5996:F
5982:z
5965:y
5948:x
5928:k
5921:j
5914:i
5907:|
5892:F
5888:F
5856:.
5851:i
5847:F
5841:i
5833:=
5829:F
5795:.
5789:z
5781:W
5772:+
5766:y
5758:V
5749:+
5743:x
5735:U
5726:=
5722:F
5712:=
5708:F
5674:3
5669:R
5659:3
5654:R
5649::
5645:F
5634:k
5631:W
5627:j
5624:V
5620:i
5617:U
5613:F
5594:.
5591:f
5586:i
5578:=
5573:i
5569:)
5565:f
5559:(
5529:k
5522:z
5514:f
5505:+
5501:j
5494:y
5486:f
5477:+
5473:i
5466:x
5458:f
5449:=
5446:f
5419:R
5410:3
5405:R
5400::
5397:f
5323:3
5318:R
5295:)
5292:3
5289:(
5286:E
5246:3
5241:R
5213:3
5208:R
5180:3
5175:R
5153:V
5128:3
5123:R
5101:V
5079:3
5074:R
5050:.
5045:)
5039:1
5032:0
5025:0
5019:(
5014:=
5009:3
5005:E
5001:,
4996:)
4990:0
4983:1
4976:0
4970:(
4965:=
4960:2
4956:E
4952:,
4947:)
4941:0
4934:0
4927:1
4921:(
4916:=
4911:1
4907:E
4880:j
4877:i
4850:j
4847:i
4839:=
4836:)
4831:j
4827:E
4823:(
4818:i
4793:}
4788:3
4784:E
4780:,
4775:2
4771:E
4767:,
4762:1
4758:E
4754:{
4751:=
4740:B
4714:x
4711:=
4708:)
4703:3
4699:x
4695:,
4690:2
4686:x
4682:,
4677:1
4673:x
4669:(
4664:1
4639:3
4633:i
4627:1
4606:R
4597:3
4592:R
4587::
4582:i
4556:R
4548:R
4540:R
4536:=
4531:3
4526:R
4503:R
4476:3
4471:R
4446:V
4420:3
4415:R
4393:V
4369:V
4349:}
4344:3
4340:e
4336:,
4331:2
4327:e
4323:,
4318:1
4314:e
4310:{
4307:=
4302:B
4278:3
4273:R
4251:V
4213:n
4208:n
4192:0
4189:=
4186:)
4182:B
4174:A
4170:(
4163:C
4159:+
4156:)
4152:A
4144:C
4140:(
4133:B
4129:+
4126:)
4122:C
4114:B
4110:(
4103:A
4080:C
4058:B
4054:,
4050:A
4025:)
4022:3
4019:(
4014:o
4011:s
3985:)
3979:,
3974:3
3969:R
3964:(
3926:.
3922:|
3908:|
3896:B
3880:A
3872:=
3863:B
3855:A
3828:B
3806:A
3761:A
3753:B
3746:=
3742:B
3734:A
3707:k
3704:j
3701:i
3674:k
3670:B
3664:j
3660:A
3654:k
3651:j
3648:i
3640:=
3635:i
3631:)
3626:B
3618:A
3614:(
3604:,
3592:]
3587:2
3583:A
3577:1
3573:B
3564:2
3560:B
3554:1
3550:A
3546:,
3541:1
3537:A
3531:3
3527:B
3518:1
3514:B
3508:3
3504:A
3500:,
3495:3
3491:A
3485:2
3481:B
3472:3
3468:B
3462:2
3458:A
3454:[
3451:=
3447:B
3439:A
3412:3
3407:R
3397:3
3392:R
3382:3
3377:R
3372::
3340:B
3336:A
3332:B
3328:A
3289:B
3283:A
3273:θ
3255:,
3239:B
3227:A
3220:=
3216:B
3208:A
3193:B
3187:A
3162:,
3155:2
3150:3
3146:A
3142:+
3137:2
3132:2
3128:A
3124:+
3119:2
3114:1
3110:A
3104:=
3098:A
3090:A
3084:=
3077:A
3047:,
3042:2
3037:3
3033:A
3029:+
3024:2
3019:2
3015:A
3011:+
3006:2
3001:1
2997:A
2993:=
2988:2
2979:A
2972:=
2968:A
2960:A
2944:A
2937:A
2930:A
2912:.
2907:i
2903:B
2897:i
2893:A
2887:3
2882:1
2879:=
2876:i
2868:=
2863:3
2859:B
2853:3
2849:A
2845:+
2840:2
2836:B
2830:2
2826:A
2822:+
2817:1
2813:B
2807:1
2803:A
2799:=
2795:B
2787:A
2771:B
2764:A
2740:3
2735:R
2677:π
2672:R
2667:π
2615:H
2609:G
2605:F
2601:C
2597:B
2593:A
2587:M
2581:L
2577:K
2573:J
2569:H
2565:G
2561:F
2557:C
2553:B
2549:A
2534:,
2531:0
2528:=
2525:M
2522:+
2519:z
2516:L
2513:+
2510:y
2507:K
2504:+
2501:x
2498:J
2495:+
2492:z
2489:x
2486:H
2483:+
2480:z
2477:y
2474:G
2471:+
2468:y
2465:x
2462:F
2459:+
2454:2
2450:z
2446:C
2443:+
2438:2
2434:y
2430:B
2427:+
2422:2
2418:x
2414:A
2216:3
2214:H
2209:3
2207:B
2202:3
2200:A
2188:h
2186:I
2180:h
2178:O
2172:d
2170:T
2110:w
2106:z
2102:y
2098:x
2093:)
2091:w
2087:z
2083:y
2079:x
2077:(
2075:P
2069:R
2050:.
2045:2
2041:r
2034:4
2031:=
2028:A
2008:,
2003:3
1999:r
1990:3
1987:4
1982:=
1979:V
1954:P
1949:r
1894:3
1889:R
1742:z
1736:y
1732:x
1706:e
1699:t
1692:v
833:)
829:(
611:)
607:(
497:k
494:,
491:j
488:,
485:i
465:0
462:=
459:a
439:k
436:w
433:+
430:j
427:v
424:+
421:i
418:u
415:+
412:a
409:=
406:q
269:n
260:n
246:,
241:n
236:R
224:n
219:n
206:n
130:(
116:x
97:)
91:(
86:)
82:(
68:.
41:.
34:.
20:)
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