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1113:. Differential geometry via moving frames and exterior differential systems. Second edition. Graduate Studies in Mathematics, 175. American Mathematical Society, Providence, RI, 2016.
155:
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1084:(DVI file), in Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, edited by S.-T. Yau, vol. IV (1995), pp. 1–76, Internat. Press, Cambridge, MA
333:
697:{\displaystyle (u^{a},p_{i}^{a},\dots ,p_{I}^{a})=(u^{a}(x),{\frac {\partial u^{a}}{\partial x^{i}}},\dots ,{\frac {\partial ^{|I|}u}{\partial x^{I}}})_{1\leq |I|\leq k}}
224:
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This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply the
1183:
1224:
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280:
917:
to a system of partial differential equations by writing down the associated exterior differential system. We can frequently apply
918:
1219:
1214:
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270:
914:
1209:
44:
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451:{\displaystyle F^{r}\left(x,u,{\frac {\partial ^{|I|}u}{\partial x^{I}}}\right)=0,\quad 1\leq |I|\leq k}
1095:
66:
921:
to exterior differential systems to study their symmetries and their diffeomorphism invariants.
203:
883:{\displaystyle du^{a}-p_{i}^{a}dx^{i},\dots ,dp_{I}^{a}-p_{Ij}^{p}dx^{j}{}_{1\leq |I|\leq k-1}}
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system as an exterior differential system with independence condition. Suppose that we have a
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229:
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36:
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40:
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729:
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of any solution of this partial differential equation system is a submanifold
1116:
H. W. Raudenbush, Jr. "Ideal Theory and
Algebraic Differential Equations",
1148:
1121:
80:
is an ideal which is mapped to itself by each differential operator.
950:
is perfect if it has the property that if it contains an element
84:
Exterior differential systems and partial differential equations
320:{\displaystyle u:\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}
1120:, Vol. 36, No. 2. (Apr., 1934), pp. 361–368. Stable URL:
1164:
277:
th order partial differential equation system for maps
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1118:Transactions of the American Mathematical Society
31:in the ring of smooth differential forms on a
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8:
1081:Toward a geometry of differential equations
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226:having the property that the pullback to
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97:
246:of all differential forms contained in
150:{\displaystyle I\subset \Omega ^{*}(M)}
1106:, Springer--Verlag, Heidelberg, 1991.
728:, and is an integral manifold of the
7:
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1143:
165:of an exterior differential system
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622:
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392:
368:
129:
14:
1128:10.1090/S0002-9947-1934-1501748-1
1109:Thomas A. Ivey, J. M. Landsberg,
50:, meaning that for any form α in
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43:, that is further closed under
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92:consists of a smooth manifold
1:
1104:Exterior Differential Systems
976:then it contains any element
271:partial differential equation
1163:. You can help Knowledge by
90:exterior differential system
1225:Differential geometry stubs
925:Perfect differential ideals
919:Cartan's equivalence method
1241:
1142:
219:{\displaystyle N\subset M}
54:, the exterior derivative
112:and a differential ideal
1139:, Dover, New York, 1950.
1055:{\displaystyle n>0\,}
45:exterior differentiation
1028:{\displaystyle b^{n}=a}
76:in a differential ring
1159:-related article is a
1102:, Hubert Goldschmidt,
1056:
1029:
996:
995:{\displaystyle b\in I}
970:
969:{\displaystyle a\in I}
944:
904:
884:
718:
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321:
266:vanishes identically.
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240:
220:
191:
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1157:differential geometry
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997:
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929:A differential ideal
915:Cartan–Kähler_theorem
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322:
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190:{\displaystyle (M,I)}
152:
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1220:Differential systems
1215:Differential algebra
1137:Differential Algebra
1111:Cartan for beginners
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894:
735:
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486:
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334:
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269:One can express any
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67:differential algebra
943:{\displaystyle I\,}
829:
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543:
519:
35:, in other words a
1210:Differential forms
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71:differential ideal
22:differential ideal
18:differential forms
1172:
1171:
1100:Phillip Griffiths
1092:Shiing-Shen Chern
1076:Phillip Griffiths
903:{\displaystyle k}
717:{\displaystyle N}
660:
607:
475:{\displaystyle k}
462:The graph of the
406:
259:{\displaystyle I}
239:{\displaystyle N}
163:integral manifold
105:{\displaystyle M}
65:In the theory of
16:In the theory of
1232:
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1144:
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39:in the sense of
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1234:
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1199:
1198:
1197:
1078:and Lucas Hsu,
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978:
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931:
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290:
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202:
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167:
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128:
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33:smooth manifold
29:algebraic ideal
12:
11:
5:
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1188:
1181:
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1152:
1141:
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1114:
1107:
1096:Robert Gardner
1085:
1067:
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1024:
1021:
1016:
1012:
991:
988:
985:
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959:
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730:contact system
713:
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314:
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255:
235:
215:
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197:consists of a
186:
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159:
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143:
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131:
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101:
85:
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9:
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3:
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1213:
1211:
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1194:
1189:
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1174:
1168:
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1162:
1158:
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1129:
1125:
1122:
1119:
1115:
1112:
1108:
1105:
1101:
1097:
1093:
1089:
1088:Robert Bryant
1086:
1083:
1082:
1077:
1073:
1072:Robert Bryant
1070:
1069:
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1063:
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1045:
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1022:
1019:
1014:
1010:
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983:
963:
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924:
922:
920:
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911:
910:-jet bundle.
897:
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869:
866:
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850:
847:
838:
834:
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820:
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785:
782:
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631:
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609:
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586:
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566:
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523:
520:
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387:
377:
361:
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63:
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58:α is also in
57:
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38:
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30:
26:
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1165:expanding it
1154:
1136:
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1103:
1080:
928:
912:
461:
274:
268:
162:
160:
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37:graded ideal
28:
24:
21:
15:
327:, given by
199:submanifold
41:ring theory
1204:Categories
1133:J. F. Ritt
1066:References
1002:such that
1035:for some
987:∈
961:∈
873:−
867:≤
851:≤
810:−
786:…
752:−
726:jet space
687:≤
671:≤
647:∂
623:∂
613:…
594:∂
579:∂
524:…
443:≤
427:≤
393:∂
369:∂
303:→
211:⊂
134:∗
130:Ω
126:⊂
890:on the
724:of the
27:is an
1155:This
482:-jet
1161:stub
1046:>
69:, a
20:, a
1124:doi
161:An
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