234:
250:
406:
870:
210:
42:
989:. In the latter case, the orientation of a second (typically referred to as "local") coordinate system, fixed to the node, is defined based on the first (typically referred to as "global" or "world" coordinate system). For instance, the orientation of a rigid body can be represented by an orientation
683:
It may occur that systems of coordinates for two different sets of geometric figures are equivalent in terms of their analysis. An example of this is the systems of homogeneous coordinates for points and lines in the projective plane. The two systems in a case like this are said to be
855:. However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero.
785:, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to â3, so that the coordinate of each point becomes 3 more.
940:
is central to the theory of manifolds. A coordinate map is essentially a coordinate system for a subset of a given space with the property that each point has exactly one set of coordinates. More precisely, a coordinate map is a
688:. Dualistic systems have the property that results from one system can be carried over to the other since these results are only different interpretations of the same analytical result; this is known as the
515:
are the
Cartesian coordinates of the point. This introduces an "extra" coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the
774:
Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
777:
Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)
889:
are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the
Cartesian coordinate system we may speak of
674:
are used to determine the position of a line in space. When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term
1015:
The Earth as a whole is one of the most common geometric spaces requiring the precise measurement of location, and thus coordinate systems. Starting with the Greeks of the
662:
Coordinates systems are often used to specify the position of a point, but they may also be used to specify the position of more complex figures such as lines, planes,
953:. It is often not possible to provide one consistent coordinate system for an entire space. In this case, a collection of coordinate maps are put together to form an
719:, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (
809:
Given a coordinate system, if one of the coordinates of a point varies while the other coordinates are held constant, then the resulting curve is called a
715:
There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by
710:
568:
represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
2096:
844:, all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise
1170:
1160:
1573:
1514:
1407:
1342:
1317:
1286:
877:
In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a
331:(measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is
524:. In general, a homogeneous coordinate system is one where only the ratios of the coordinates are significant and not the actual values.
278:
planes are chosen and the three coordinates of a point are the signed distances to each of the planes. This can be generalized to create
961:
and additional structure can be defined on a manifold if the structure is consistent where the coordinate maps overlap. For example, a
274:
lines are chosen and the coordinates of a point are taken to be the signed distances to the lines. In three dimensions, three mutually
2280:
1959:
1739:
1695:
1630:
1597:
1548:
1455:
1371:
704:
1180:
2161:
997:
of three points. These points are used to define the orientation of the axes of the local system; they are the tips of three
2012:
1944:
1662:
1270:
823:
601:
396:
2037:
1054:
1023:
2383:
2275:
1657:
1100:
1058:
1044:
1027:
965:
is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.
886:
841:
400:
263:
240:
223:
46:
35:
2388:
2086:
1906:
1506:
1165:
1758:
1185:
1084:
1066:
121:. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered
2260:
1712:
2342:
2214:
1921:
1010:
31:
202:, where the signed distance is the distance taken as positive or negative depending on which side of the line
145:. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and
2312:
1999:
1916:
1886:
1120:
962:
581:
575:
536:
478:
306:
2270:
2126:
2081:
1110:
982:
829:
671:
571:
542:
134:
539:
are a generalization of coordinate systems generally; the system is based on the intersection of curves.
2352:
2307:
1787:
1732:
1175:
1040:
994:
859:
691:
615:
595:
591:
565:
206:
lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point.
2327:
2255:
2141:
2007:
1969:
1901:
1309:
1070:
990:
585:
441:). Spherical coordinates take this a step further by converting the pair of cylindrical coordinates (
53:. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance
347:) there is a single point, but any point is represented by many pairs of coordinates. For example, (
2204:
2027:
2017:
1866:
1851:
1807:
1146:
413:
There are two common methods for extending the polar coordinate system to three dimensions. In the
106:
2337:
2194:
2047:
1861:
1797:
1652:
1115:
1016:
623:
556:
546:
1399:
1335:
Field Theory
Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions
650:
209:
2332:
2240:
2101:
2076:
1891:
1802:
1691:
1626:
1593:
1569:
1544:
1510:
1461:
1451:
1421:
1413:
1403:
1367:
1338:
1313:
1303:
1282:
1246:
1221:
1089:
803:
233:
150:
1685:
1618:
2347:
2245:
2022:
1989:
1974:
1856:
1725:
1670:
1610:
1337:(corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9â11 (Table 1.01).
1141:
1136:
1094:
954:
796:
676:
643:
639:
552:
517:
267:
142:
110:
1019:, a variety of coordinate systems have been developed based on the types above, including:
2317:
2265:
2209:
2189:
2091:
1979:
1846:
1817:
1443:
1439:
1105:
1061:
that models the earth as an object, and are most commonly used for modeling the orbits of
869:
294:
173:
118:
739:
axis, then the coordinate transformation from polar to
Cartesian coordinates is given by
405:
249:
2357:
2322:
2302:
2219:
2052:
2042:
2032:
1954:
1926:
1911:
1896:
1812:
1681:
1494:
1392:
1387:
1125:
1048:
921:
851:
A polar coordinate system is a curvilinear system where coordinate curves are lines or
138:
1224:
574:
are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as
2377:
2294:
2199:
2111:
1984:
1360:
942:
814:
275:
271:
2362:
2166:
2151:
2116:
1964:
1949:
1614:
605:
17:
981:, coordinate systems are used to describe the (linear) position of points and the
858:
Many curves can occur as coordinate curves. For example, the coordinate curves of
1249:
821:. A coordinate system for which some coordinate curves are not lines is called a
172:
The simplest example of a coordinate system is the identification of points on a
2250:
2224:
2146:
1835:
1774:
1278:
998:
836:
A coordinate line with all other constant coordinates equal to zero is called a
178:
167:
130:
379:) are all polar coordinates for the same point. The pole is represented by (0,
2131:
986:
978:
845:
631:
421:-coordinate with the same meaning as in Cartesian coordinates is added to the
327:, there is a single line through the pole whose angle with the polar axis is
2106:
2057:
1254:
1229:
1062:
1035:
782:
767:
627:
1425:
770:
from the space to itself two coordinate transformations can be associated:
41:
2136:
2121:
1498:
1130:
1031:
974:
927:
863:
680:
is used for any coordinate system that specifies the position of a line.
609:
521:
114:
90:
1830:
1792:
1358:
Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994).
873:
Coordinate surfaces of the three-dimensional paraboloidal coordinates.
2156:
1748:
852:
667:
663:
98:
957:
covering the space. A space equipped with such an atlas is called a
908:-dimensional spaces resulting from fixing a single coordinate of an
833:
are a special but extremely common case of curvilinear coordinates.
1625:. American Institute of Aeronautics and Astronautics. p. 71.
1366:(Single Variable Version ed.). Addison-Wesley Publishing Co.
868:
122:
62:
40:
1566:
A Computational
Differential Geometry Approach to Grid Generation
1465:
1417:
1333:
Moon P, Spencer DE (1988). "Rectangular
Coordinates (x, y, z)".
1721:
622:
There are ways of describing curves without coordinates, using
30:"Coordinate" redirects here. For coordinates on the Earth, see
78:
70:
208:
1503:
Methods of
Algebraic Geometry, Volume I (Book II)
1133:, graphical representations of different coordinate systems
881:. For example, the coordinate surfaces obtained by holding
802:"Coordinate plane" redirects here. Not to be confused with
735:) have the same origin, and the polar axis is the positive
1717:
795:"Coordinate line" redirects here. Not to be confused with
255:
The
Cartesian coordinate system in three-dimensional space
532:
Some other common coordinate systems are the following:
262:
The prototypical example of a coordinate system is the
190:) is chosen on a given line. The coordinate of a point
1302:
Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021).
311:
Another common coordinate system for the plane is the
1051:
to create a planar surface of the world or a region.
2293:
2233:
2182:
2175:
2067:
1998:
1935:
1879:
1826:
1773:
1766:
1391:
1359:
1541:Mathematical Methods for Engineers and Scientists
141:or elements of a more abstract system such as a
129:-coordinate". The coordinates are taken to be
1733:
8:
1651:Voitsekhovskii, M.I.; Ivanov, A.B. (2001) ,
993:, which includes, in its three columns, the
391:Cylindrical and spherical coordinate systems
289:Depending on the direction and order of the
789:Coordinate lines/curves and planes/surfaces
483:A point in the plane may be represented in
2179:
1770:
1740:
1726:
1718:
1398:. New York City: D. van Nostrand. p.
319:and a ray from this point is taken as the
1362:Calculus: Graphical, Numerical, Algebraic
1273:; Redlin, Lothar; Watson, Saleem (2008).
711:List of common coordinate transformations
2097:Covariance and contravariance of vectors
1394:The Mathematics of Physics and Chemistry
404:
293:, the three-dimensional system may be a
1481:An Introduction to Algebraical Geometry
1203:
194:is defined as the signed distance from
1450:. New York: McGraw-Hill. p. 658.
1448:Methods of Theoretical Physics, Part I
626:that use invariant quantities such as
608:and more generally in the analysis of
125:and sometimes by a letter, as in "the
1623:Analytical Mechanics of Space Systems
1543:. Vol. 2. Springer. p. 13.
618:are used in the context of triangles.
182:. In this system, an arbitrary point
27:Method for specifying point positions
7:
290:
429:polar coordinates giving a triple (
339:. For a given pair of coordinates (
1960:Tensors in curvilinear coordinates
97:is a system that uses one or more
25:
1680:Shigeyuki Morita; Teruko Nagase;
1161:EddingtonâFinkelstein coordinates
705:Active and passive transformation
653:relates arc length and curvature.
113:or other geometric elements on a
912:-dimensional coordinate system.
658:Coordinates of geometric objects
248:
232:
1171:GullstrandâPainlevĂŠ coordinates
1154:Relativistic coordinate systems
945:from an open subset of a space
1687:Geometry of Differential Forms
1564:Liseikin, Vladimir D. (2007).
1479:Jones, Alfred Clement (1912).
286:-dimensional Euclidean space.
1:
2013:Exterior covariant derivative
1945:Tensor (intrinsic definition)
1690:. AMS Bookstore. p. 12.
969:Orientation-based coordinates
824:curvilinear coordinate system
813:. If a coordinate curve is a
602:Barycentric coordinate system
473:Homogeneous coordinate system
415:cylindrical coordinate system
409:Cylindrical coordinate system
397:Cylindrical coordinate system
282:coordinates for any point in
2038:Raising and lowering indices
1713:Hexagonal Coordinate Systems
1675:. Ginn and Co. pp. 1ff.
1669:Woods, Frederick S. (1922).
1390:; Murphy, George M. (1956).
1181:KruskalâSzekeres coordinates
1055:Geocentric coordinate system
1045:cartesian coordinate systems
1041:Projected coordinate systems
1024:Geographic coordinate system
176:with real numbers using the
105:, to uniquely determine the
2276:Gluon field strength tensor
1658:Encyclopedia of Mathematics
1101:Celestial coordinate system
1059:cartesian coordinate system
887:spherical coordinate system
842:Cartesian coordinate system
642:relates arc length and the
566:log-polar coordinate system
528:Other commonly used systems
401:Spherical coordinate system
315:. A point is chosen as the
264:Cartesian coordinate system
241:Cartesian coordinate system
224:Cartesian coordinate system
218:Cartesian coordinate system
81:) is often used instead of
47:spherical coordinate system
36:Coordinate (disambiguation)
2405:
2087:Cartan formalism (physics)
1907:Penrose graphical notation
1507:Cambridge University Press
1166:Gaussian polar coordinates
1008:
925:
919:
801:
794:
717:coordinate transformations
708:
702:
476:
394:
304:
221:
165:
29:
1759:Glossary of tensor theory
1755:
1588:Munkres, James R. (2000)
1186:Schwarzschild coordinates
1085:Absolute angular momentum
1067:Global Positioning System
1043:, including thousands of
1001:aligned with those axes.
727:) and polar coordinates (
297:or a left-handed system.
157:Common coordinate systems
2343:Gregorio Ricci-Curbastro
2215:Riemann curvature tensor
1922:Van der Waerden notation
1568:. Springer. p. 38.
1011:Spatial reference system
898:coordinate hypersurfaces
449:) to polar coordinates (
32:Spatial reference system
2313:Elwin Bruno Christoffel
2246:Angular momentum tensor
1917:Tetrad (index notation)
1887:Abstract index notation
1619:"Rigid body kinematics"
1305:Calculus: Multivariable
1121:Galilean transformation
963:differentiable manifold
598:treatment of mechanics.
588:treatment of mechanics.
582:Generalized coordinates
576:homogeneous coordinates
537:Curvilinear coordinates
485:homogeneous coordinates
479:Homogeneous coordinates
313:polar coordinate system
307:Polar coordinate system
301:Polar coordinate system
149:; this is the basis of
65:), and azimuthal angle
2127:Levi-Civita connection
1111:Fractional coordinates
1057:, a three-dimensional
874:
830:Orthogonal coordinates
543:Orthogonal coordinates
410:
214:
135:elementary mathematics
86:
34:. For other uses, see
2353:Jan Arnoldus Schouten
2308:Augustin-Louis Cauchy
1788:Differential geometry
1310:John Wiley & Sons
1176:Isotropic coordinates
1028:spherical coordinates
995:Cartesian coordinates
985:of axes, planes, and
949:to an open subset of
926:Further information:
872:
860:parabolic coordinates
616:Trilinear coordinates
592:Canonical coordinates
408:
212:
44:
2328:Carl Friedrich Gauss
2261:stressâenergy tensor
2256:Cauchy stress tensor
2008:Covariant derivative
1970:Antisymmetric tensor
1902:Multi-index notation
1539:Tang, K. T. (2006).
1071:satellite navigation
549:meet at right angles
323:. For a given angle
49:is commonly used in
2205:Nonmetricity tensor
2060:(2nd-order tensors)
2028:Hodge star operator
2018:Exterior derivative
1867:Transport phenomena
1852:Continuum mechanics
1808:Multilinear algebra
1225:"Coordinate System"
1147:Translation of axes
672:PlĂźcker coordinates
624:intrinsic equations
572:PlĂźcker coordinates
557:coordinate surfaces
547:coordinate surfaces
520:without the use of
457:) giving a triple (
383:) for any value of
18:Position coordinate
2384:Coordinate systems
2338:Tullio Levi-Civita
2281:Metric tensor (GR)
2195:Levi-Civita symbol
2048:Tensor contraction
1862:General relativity
1798:Euclidean geometry
1281:. pp. 13â19.
1247:Weisstein, Eric W.
1222:Weisstein, Eric W.
1116:Frame of reference
1047:, each based on a
1017:Hellenistic period
1005:Geographic systems
879:coordinate surface
875:
559:are not orthogonal
411:
215:
87:
2389:Analytic geometry
2371:
2370:
2333:Hermann Grassmann
2289:
2288:
2241:Moment of inertia
2102:Differential form
2077:Affine connection
1892:Einstein notation
1875:
1874:
1803:Exterior calculus
1783:Coordinate system
1592:. Prentice Hall.
1575:978-3-540-34235-9
1516:978-0-521-46900-5
1409:978-0-88275-423-9
1344:978-0-387-18430-2
1319:978-1-119-77798-4
1288:978-0-495-56521-5
1271:Stewart, James B.
1090:Alphanumeric grid
932:The concept of a
891:coordinate planes
817:, it is called a
804:Plane coordinates
634:. These include:
335:for given number
151:analytic geometry
95:coordinate system
16:(Redirected from
2396:
2348:Bernhard Riemann
2180:
2023:Exterior product
1990:Two-point tensor
1975:Symmetric tensor
1857:Electromagnetism
1771:
1742:
1735:
1728:
1719:
1701:
1676:
1665:
1637:
1636:
1611:Hanspeter Schaub
1607:
1601:
1586:
1580:
1579:
1561:
1555:
1554:
1536:
1530:
1527:
1521:
1520:
1491:
1485:
1484:
1476:
1470:
1469:
1436:
1430:
1429:
1397:
1384:
1378:
1377:
1365:
1355:
1349:
1348:
1330:
1324:
1323:
1299:
1293:
1292:
1277:(5th ed.).
1267:
1261:
1260:
1259:
1242:
1236:
1235:
1234:
1217:
1211:
1208:
1142:Rotation of axes
1137:Reference system
1095:Axes conventions
1065:, including the
983:angular position
938:coordinate chart
907:
885:constant in the
811:coordinate curve
797:Line coordinates
781:For example, in
677:line coordinates
644:tangential angle
640:Whewell equation
594:are used in the
584:are used in the
553:Skew coordinates
518:projective plane
252:
236:
143:commutative ring
21:
2404:
2403:
2399:
2398:
2397:
2395:
2394:
2393:
2374:
2373:
2372:
2367:
2318:Albert Einstein
2285:
2266:Einstein tensor
2229:
2210:Ricci curvature
2190:Kronecker delta
2176:Notable tensors
2171:
2092:Connection form
2069:
2063:
1994:
1980:Tensor operator
1937:
1931:
1871:
1847:Computer vision
1840:
1822:
1818:Tensor calculus
1762:
1751:
1746:
1709:
1704:
1698:
1679:
1672:Higher Geometry
1668:
1650:
1646:
1641:
1640:
1633:
1615:John L. Junkins
1609:
1608:
1604:
1587:
1583:
1576:
1563:
1562:
1558:
1551:
1538:
1537:
1533:
1528:
1524:
1517:
1493:
1492:
1488:
1478:
1477:
1473:
1458:
1438:
1437:
1433:
1410:
1388:Margenau, Henry
1386:
1385:
1381:
1374:
1357:
1356:
1352:
1345:
1332:
1331:
1327:
1320:
1312:. p. 657.
1301:
1300:
1296:
1289:
1275:College Algebra
1269:
1268:
1264:
1245:
1244:
1243:
1239:
1220:
1219:
1218:
1214:
1209:
1205:
1200:
1195:
1190:
1156:
1151:
1106:Coordinate-free
1080:
1013:
1007:
971:
930:
924:
918:
916:Coordinate maps
901:
838:coordinate axis
819:coordinate line
807:
800:
791:
713:
707:
701:
699:Transformations
670:. For example,
660:
651:CesĂ ro equation
530:
481:
475:
403:
395:Main articles:
393:
309:
303:
291:coordinate axes
260:
259:
258:
257:
256:
253:
245:
244:
237:
226:
220:
213:The number line
170:
164:
159:
139:complex numbers
119:Euclidean space
39:
28:
23:
22:
15:
12:
11:
5:
2402:
2400:
2392:
2391:
2386:
2376:
2375:
2369:
2368:
2366:
2365:
2360:
2358:Woldemar Voigt
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2323:Leonhard Euler
2320:
2315:
2310:
2305:
2299:
2297:
2295:Mathematicians
2291:
2290:
2287:
2286:
2284:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2237:
2235:
2231:
2230:
2228:
2227:
2222:
2220:Torsion tensor
2217:
2212:
2207:
2202:
2197:
2192:
2186:
2184:
2177:
2173:
2172:
2170:
2169:
2164:
2159:
2154:
2149:
2144:
2139:
2134:
2129:
2124:
2119:
2114:
2109:
2104:
2099:
2094:
2089:
2084:
2079:
2073:
2071:
2065:
2064:
2062:
2061:
2055:
2053:Tensor product
2050:
2045:
2043:Symmetrization
2040:
2035:
2033:Lie derivative
2030:
2025:
2020:
2015:
2010:
2004:
2002:
1996:
1995:
1993:
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1957:
1955:Tensor density
1952:
1947:
1941:
1939:
1933:
1932:
1930:
1929:
1927:Voigt notation
1924:
1919:
1914:
1912:Ricci calculus
1909:
1904:
1899:
1897:Index notation
1894:
1889:
1883:
1881:
1877:
1876:
1873:
1872:
1870:
1869:
1864:
1859:
1854:
1849:
1843:
1841:
1839:
1838:
1833:
1827:
1824:
1823:
1821:
1820:
1815:
1813:Tensor algebra
1810:
1805:
1800:
1795:
1793:Dyadic algebra
1790:
1785:
1779:
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1737:
1730:
1722:
1716:
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1708:
1707:External links
1705:
1703:
1702:
1696:
1682:Katsumi Nomizu
1677:
1666:
1647:
1645:
1642:
1639:
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1602:
1581:
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1134:
1128:
1126:Grid reference
1123:
1118:
1113:
1108:
1103:
1098:
1097:in engineering
1092:
1087:
1081:
1079:
1076:
1075:
1074:
1052:
1049:map projection
1038:
1009:Main article:
1006:
1003:
970:
967:
934:coordinate map
922:Coordinate map
920:Main article:
917:
914:
790:
787:
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709:Main article:
700:
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579:
569:
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477:Main article:
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230:
229:
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222:Main article:
219:
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166:Main article:
163:
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155:
73:). The symbol
57:, polar angle
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2401:
2390:
2387:
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2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2200:Metric tensor
2198:
2196:
2193:
2191:
2188:
2187:
2185:
2181:
2178:
2174:
2168:
2165:
2163:
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2112:Exterior form
2110:
2108:
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2100:
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2021:
2019:
2016:
2014:
2011:
2009:
2006:
2005:
2003:
2001:
1997:
1991:
1988:
1986:
1985:Tensor bundle
1983:
1981:
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1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1956:
1953:
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1948:
1946:
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1699:
1697:0-8218-1045-6
1693:
1689:
1688:
1683:
1678:
1674:
1673:
1667:
1664:
1660:
1659:
1654:
1653:"Coordinates"
1649:
1648:
1643:
1634:
1632:1-56347-563-4
1628:
1624:
1620:
1616:
1612:
1606:
1603:
1599:
1598:0-13-181629-2
1595:
1591:
1585:
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1577:
1571:
1567:
1560:
1557:
1552:
1550:3-540-30268-9
1546:
1542:
1535:
1532:
1526:
1523:
1518:
1512:
1508:
1504:
1500:
1496:
1495:Hodge, W.V.D.
1490:
1487:
1482:
1475:
1472:
1467:
1463:
1459:
1457:0-07-043316-X
1453:
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1427:
1423:
1419:
1415:
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1373:0-201-55478-X
1369:
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1250:"Coordinates"
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956:
952:
948:
944:
943:homeomorphism
939:
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929:
923:
915:
913:
911:
905:
899:
894:
892:
888:
884:
880:
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856:
854:
849:
847:
843:
839:
834:
832:
831:
826:
825:
820:
816:
815:straight line
812:
805:
798:
793:
788:
786:
784:
776:
773:
772:
771:
769:
764:
762:
758:
755: =
754:
750:
746:
743: =
742:
738:
734:
730:
726:
722:
718:
712:
706:
698:
696:
694:
693:
690:principle of
687:
681:
679:
678:
673:
669:
665:
657:
652:
648:
645:
641:
637:
636:
635:
633:
629:
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617:
614:
611:
607:
606:ternary plots
603:
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583:
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519:
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510:
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498:
494:
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487:by a triple (
486:
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472:
470:
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322:
318:
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308:
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292:
287:
285:
281:
277:
273:
272:perpendicular
269:
265:
251:
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225:
217:
211:
207:
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201:
197:
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189:
185:
181:
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169:
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156:
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144:
140:
137:, but may be
136:
132:
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100:
96:
92:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
43:
37:
33:
19:
2363:Hermann Weyl
2167:Vector space
2152:Pseudotensor
2117:Fiber bundle
2070:abstractions
1965:Mixed tensor
1950:Tensor field
1782:
1757:
1686:
1671:
1656:
1622:
1605:
1589:
1584:
1565:
1559:
1540:
1534:
1525:
1502:
1489:
1483:. Clarendon.
1480:
1474:
1447:
1434:
1393:
1382:
1361:
1353:
1334:
1328:
1304:
1297:
1274:
1265:
1253:
1240:
1228:
1215:
1206:
1014:
999:unit vectors
987:rigid bodies
972:
958:
950:
946:
937:
933:
931:
909:
903:
897:
895:
890:
882:
878:
876:
857:
850:
837:
835:
828:
822:
818:
810:
808:
792:
780:
765:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
720:
716:
714:
689:
685:
682:
675:
661:
621:
604:as used for
531:
512:
508:
504:
500:
496:
492:
488:
484:
482:
466:
462:
458:
454:
450:
446:
442:
438:
434:
430:
426:
422:
418:
414:
412:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
328:
324:
320:
316:
312:
310:
295:right-handed
288:
283:
279:
261:
243:in the plane
203:
199:
195:
191:
187:
183:
177:
171:
146:
131:real numbers
126:
102:
94:
88:
82:
74:
66:
58:
54:
50:
2303:Ălie Cartan
2251:Spin tensor
2225:Weyl tensor
2183:Mathematics
2147:Multivector
1938:definitions
1836:Engineering
1775:Mathematics
1444:Feshbach, H
1279:Brooks Cole
896:Similarly,
766:With every
596:Hamiltonian
179:number line
168:Number line
162:Number line
103:coordinates
2378:Categories
2132:Linear map
2000:Operations
1529:Woods p. 2
1210:Woods p. 1
1193:References
1069:and other
1063:satellites
979:kinematics
846:orthogonal
703:See also:
632:arc length
586:Lagrangian
321:polar axis
276:orthogonal
147:vice versa
2271:EM tensor
2107:Dimension
2058:Transpose
1663:EMS Press
1501:(1994) .
1440:Morse, PM
1255:MathWorld
1230:MathWorld
1198:Citations
1036:longitude
864:parabolas
768:bijection
759: sin
747: cos
686:dualistic
628:curvature
610:triangles
266:. In the
2137:Manifold
2122:Geodesic
1880:Notation
1684:(2001).
1617:(2003).
1590:Topology
1499:D. Pedoe
1466:52011515
1446:(1953).
1418:55010911
1131:Nomogram
1078:See also
1073:systems.
1032:latitude
975:geometry
959:manifold
928:Manifold
900:are the
840:. In a
522:infinity
499:) where
367:) and (â
117:such as
115:manifold
107:position
91:geometry
2234:Physics
2068:Related
1831:Physics
1749:Tensors
1644:Sources
1426:3017486
853:circles
731:,
723:,
692:duality
668:spheres
664:circles
495:,
491:,
465:,
461:,
453:,
445:,
437:,
433:,
371:,
359:,
351:,
343:,
109:of the
99:numbers
51:physics
2162:Vector
2157:Spinor
2142:Matrix
1936:Tensor
1694:
1629:
1596:
1572:
1547:
1513:
1464:
1454:
1424:
1416:
1406:
1370:
1341:
1316:
1285:
1026:, the
991:matrix
270:, two
188:origin
111:points
2082:Basis
1767:Scope
955:atlas
936:, or
268:plane
186:(the
123:tuple
101:, or
63:theta
1692:ISBN
1627:ISBN
1594:ISBN
1570:ISBN
1545:ISBN
1511:ISBN
1462:LCCN
1452:ISBN
1422:OCLC
1414:LCCN
1404:ISBN
1368:ISBN
1339:ISBN
1314:ISBN
1283:ISBN
1034:and
977:and
906:â 1)
862:are
751:and
649:The
638:The
630:and
564:The
507:and
425:and
417:, a
399:and
355:), (
317:pole
239:The
174:line
93:, a
45:The
1400:178
1030:of
973:In
848:.
666:or
469:).
198:to
133:in
89:In
79:rho
71:phi
2380::
1661:,
1655:,
1621:.
1613:;
1509:.
1505:.
1497:;
1460:.
1442:;
1420:.
1412:.
1402:.
1308:.
1252:.
1227:.
893:.
866:.
827:.
783:1D
763:.
695:.
555::
545::
387:.
363:+2
153:.
1741:e
1734:t
1727:v
1700:.
1635:.
1600:.
1578:.
1553:.
1519:.
1468:.
1428:.
1376:.
1347:.
1322:.
1291:.
1258:.
1233:.
951:R
947:X
910:n
904:n
902:(
883:Ď
806:.
799:.
761:θ
757:r
753:y
749:θ
745:r
741:x
737:x
733:θ
729:r
725:y
721:x
646:.
612:.
578:.
513:z
511:/
509:y
505:z
503:/
501:x
497:z
493:y
489:x
467:Ď
463:θ
459:Ď
455:Ď
451:Ď
447:z
443:r
439:z
435:θ
431:r
427:θ
423:r
419:z
385:θ
381:θ
377:Ď
375:+
373:θ
369:r
365:Ď
361:θ
357:r
353:θ
349:r
345:θ
341:r
337:r
333:r
329:θ
325:θ
284:n
280:n
204:P
200:P
196:O
192:P
184:O
127:x
85:.
83:r
77:(
75:Ď
69:(
67:Ď
61:(
59:θ
55:r
38:.
20:)
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