28:
595:
436:
488:
274:
968:
517:
1171:
363:
791:
1054:
276:
where the first vector in the sum is the tangential component and the second one is the normal component. It follows immediately that these two vectors are perpendicular to each other.
1099:
655:
315:
842:
222:
177:
1007:
617:
703:
508:
1283:
1256:
657:
used (there exist two unit normals to any surface at a given point, pointing in opposite directions, so one of the unit normals is the negative of the other one).
441:
227:
356:
335:
198:
155:
135:
882:
716:
1347:
1325:
1377:
1104:
1305:
868:
1320:
590:{\displaystyle \mathbf {v} _{\parallel }=-{\hat {\mathbf {n} }}\times ({\hat {\mathbf {n} }}\times \mathbf {v} ),}
1013:
794:
431:{\displaystyle \mathbf {v} _{\perp }=\left(\mathbf {v} \cdot {\hat {\mathbf {n} }}\right){\hat {\mathbf {n} }}}
1061:
1205:
629:
289:
1201:
1301:
856:
706:
1193:
852:
799:
205:
160:
976:
64:
40:
1343:
602:
1309:
1216:
1197:
682:
493:
1261:
1234:
1315:
971:
341:
320:
280:
183:
140:
120:
27:
1371:
1228:
710:
620:
85:
56:
17:
1220:
1289:
666:
511:
284:
71:
36:
1224:
1200:), then the derivative gives a spanning set for the tangent bundle (it is a
483:{\displaystyle \mathbf {v} _{\parallel }=\mathbf {v} -\mathbf {v} _{\perp }}
269:{\displaystyle \mathbf {v} =\mathbf {v} _{\parallel }+\mathbf {v} _{\perp }}
1326:
Differential geometry of surfaces § Tangent vectors and normal vectors
31:
Illustration of tangential and normal components of a vector to a surface.
673:
78:
48:
47:, that vector can be decomposed uniquely as a sum of two vectors, one
1358:
1219:(as in the above description of a surface, (or more generally as) a
44:
26:
279:
To calculate the tangential and normal components, consider a
963:{\displaystyle T_{p}M=T_{p}N\oplus N_{p}N:=(T_{p}N)^{\perp }}
626:
These formulas do not depend on the particular unit normal
1292:; the cross product is special to 3 dimensions however.
1166:{\displaystyle v_{\perp }\in N_{p}N:=(T_{p}N)^{\perp }}
1264:
1237:
1107:
1064:
1016:
979:
885:
802:
719:
685:
632:
605:
520:
496:
444:
366:
344:
323:
292:
230:
208:
186:
163:
143:
123:
96:, it can be decomposed into the component tangent to
514:. Another formula for the tangential component is
63:of the vector. Similarly, a vector at a point on a
1277:
1250:
1165:
1093:
1048:
1001:
962:
836:
785:
697:
649:
611:
589:
502:
482:
430:
350:
329:
309:
268:
216:
192:
171:
149:
129:
786:{\displaystyle T_{p}N\to T_{p}M\to T_{p}M/T_{p}N}
1288:In both cases, we can again compute using the
8:
1049:{\displaystyle v=v_{\parallel }+v_{\perp }}
1308:are where the tangential component of the
844:is a generalized space of normal vectors.
1269:
1263:
1242:
1236:
1204:if and only if the parametrization is an
1157:
1144:
1125:
1112:
1106:
1082:
1069:
1063:
1040:
1027:
1015:
990:
978:
954:
941:
922:
906:
890:
884:
825:
816:
807:
801:
774:
765:
756:
740:
724:
718:
684:
636:
634:
633:
631:
604:
576:
562:
560:
559:
542:
540:
539:
527:
522:
519:
495:
474:
469:
460:
451:
446:
443:
417:
415:
414:
398:
396:
395:
387:
373:
368:
365:
343:
322:
296:
294:
293:
291:
260:
255:
245:
240:
231:
229:
209:
207:
185:
164:
162:
142:
122:
1094:{\displaystyle v_{\parallel }\in T_{p}N}
1185:is given by non-degenerate equations.
650:{\displaystyle {\hat {\mathbf {n} }}}
310:{\displaystyle {\hat {\mathbf {n} }}}
7:
25:
67:can be broken down the same way.
1342:. New York: Dover Publications.
1340:Electromagnetic fields and waves
637:
577:
563:
543:
523:
470:
461:
447:
418:
399:
388:
369:
297:
256:
241:
232:
210:
165:
157:be a point on the surface. Let
55:of the vector, and another one
1154:
1137:
951:
934:
749:
733:
641:
581:
567:
556:
547:
422:
403:
301:
1:
837:{\displaystyle T_{p}M/T_{p}N}
875:and the component normal to
871:of the component tangent to
217:{\displaystyle \mathbf {v} }
202:Then one can write uniquely
172:{\displaystyle \mathbf {v} }
100:and the component normal to
1338:Rojansky, Vladimir (1979).
1002:{\displaystyle v\in T_{p}M}
859:, and the tangent space of
283:to the surface, that is, a
1394:
1357:Crowell, Benjamin (2003).
1192:is given explicitly, via
59:to the curve, called the
51:to the curve, called the
1258:, then the gradients of
665:More generally, given a
70:More generally, given a
1285:span the normal space.
612:{\displaystyle \times }
1321:Frenet–Serret formulas
1279:
1252:
1167:
1095:
1050:
1003:
964:
838:
787:
699:
698:{\displaystyle p\in N}
651:
613:
591:
504:
503:{\displaystyle \cdot }
484:
432:
352:
331:
311:
270:
218:
194:
173:
151:
131:
84:, and a vector in the
32:
1378:Differential geometry
1280:
1278:{\displaystyle g_{i}}
1253:
1251:{\displaystyle g_{i}}
1168:
1096:
1051:
1004:
965:
855:, the above sequence
839:
788:
700:
652:
614:
592:
505:
485:
433:
353:
332:
312:
271:
219:
195:
174:
152:
132:
30:
1302:Lagrange multipliers
1262:
1235:
1194:parametric equations
1105:
1062:
1014:
977:
883:
800:
717:
707:short exact sequence
683:
630:
603:
518:
494:
442:
364:
342:
321:
290:
228:
206:
184:
161:
141:
121:
53:tangential component
18:Tangential component
1227:or intersection of
853:Riemannian manifold
117:More formally, let
1275:
1248:
1163:
1091:
1046:
999:
960:
834:
783:
695:
647:
609:
587:
500:
480:
428:
348:
327:
307:
266:
214:
190:
169:
147:
137:be a surface, and
127:
33:
644:
570:
550:
425:
406:
351:{\displaystyle x}
330:{\displaystyle S}
317:perpendicular to
304:
193:{\displaystyle x}
150:{\displaystyle x}
130:{\displaystyle S}
108:Formal definition
16:(Redirected from
1385:
1364:
1360:Light and Matter
1353:
1310:total derivative
1284:
1282:
1281:
1276:
1274:
1273:
1257:
1255:
1254:
1249:
1247:
1246:
1198:parametric curve
1172:
1170:
1169:
1164:
1162:
1161:
1149:
1148:
1130:
1129:
1117:
1116:
1100:
1098:
1097:
1092:
1087:
1086:
1074:
1073:
1057:
1055:
1053:
1052:
1047:
1045:
1044:
1032:
1031:
1008:
1006:
1005:
1000:
995:
994:
969:
967:
966:
961:
959:
958:
946:
945:
927:
926:
911:
910:
895:
894:
867:decomposes as a
843:
841:
840:
835:
830:
829:
820:
812:
811:
792:
790:
789:
784:
779:
778:
769:
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760:
745:
744:
729:
728:
704:
702:
701:
696:
656:
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635:
618:
616:
615:
610:
596:
594:
593:
588:
580:
572:
571:
566:
561:
552:
551:
546:
541:
532:
531:
526:
509:
507:
506:
501:
489:
487:
486:
481:
479:
478:
473:
464:
456:
455:
450:
437:
435:
434:
429:
427:
426:
421:
416:
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409:
408:
407:
402:
397:
391:
378:
377:
372:
359:
357:
355:
354:
349:
336:
334:
333:
328:
316:
314:
313:
308:
306:
305:
300:
295:
275:
273:
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267:
265:
264:
259:
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196:
191:
178:
176:
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170:
168:
156:
154:
153:
148:
136:
134:
133:
128:
61:normal component
43:at a point on a
21:
1393:
1392:
1388:
1387:
1386:
1384:
1383:
1382:
1368:
1367:
1356:
1350:
1337:
1334:
1306:critical points
1298:
1265:
1260:
1259:
1238:
1233:
1232:
1179:
1153:
1140:
1121:
1108:
1103:
1102:
1078:
1065:
1060:
1059:
1036:
1023:
1012:
1011:
1010:
986:
975:
974:
950:
937:
918:
902:
886:
881:
880:
821:
803:
798:
797:
770:
752:
736:
720:
715:
714:
681:
680:
663:
628:
627:
601:
600:
521:
516:
515:
492:
491:
468:
445:
440:
439:
386:
382:
367:
362:
361:
340:
339:
338:
319:
318:
288:
287:
254:
239:
226:
225:
204:
203:
182:
181:
180:
179:be a vector at
159:
158:
139:
138:
119:
118:
115:
110:
23:
22:
15:
12:
11:
5:
1391:
1389:
1381:
1380:
1370:
1369:
1366:
1365:
1354:
1348:
1333:
1330:
1329:
1328:
1323:
1318:
1316:Surface normal
1313:
1304:: constrained
1297:
1294:
1272:
1268:
1245:
1241:
1229:level surfaces
1178:
1175:
1160:
1156:
1152:
1147:
1143:
1139:
1136:
1133:
1128:
1124:
1120:
1115:
1111:
1090:
1085:
1081:
1077:
1072:
1068:
1043:
1039:
1035:
1030:
1026:
1022:
1019:
998:
993:
989:
985:
982:
972:tangent vector
957:
953:
949:
944:
940:
936:
933:
930:
925:
921:
917:
914:
909:
905:
901:
898:
893:
889:
833:
828:
824:
819:
815:
810:
806:
795:quotient space
782:
777:
773:
768:
764:
759:
755:
751:
748:
743:
739:
735:
732:
727:
723:
711:tangent spaces
709:involving the
694:
691:
688:
662:
659:
643:
639:
619:" denotes the
608:
586:
583:
579:
575:
569:
565:
558:
555:
549:
545:
538:
535:
530:
525:
510:" denotes the
499:
477:
472:
467:
463:
459:
454:
449:
424:
420:
412:
405:
401:
394:
390:
385:
381:
376:
371:
347:
326:
303:
299:
263:
258:
253:
248:
243:
238:
234:
212:
189:
167:
146:
126:
114:
111:
109:
106:
92:at a point of
24:
14:
13:
10:
9:
6:
4:
3:
2:
1390:
1379:
1376:
1375:
1373:
1362:
1361:
1355:
1351:
1349:0-486-63834-0
1345:
1341:
1336:
1335:
1331:
1327:
1324:
1322:
1319:
1317:
1314:
1311:
1307:
1303:
1300:
1299:
1295:
1293:
1291:
1286:
1270:
1266:
1243:
1239:
1230:
1226:
1222:
1218:
1214:
1209:
1207:
1203:
1199:
1195:
1191:
1186:
1184:
1176:
1174:
1158:
1150:
1145:
1141:
1134:
1131:
1126:
1122:
1118:
1113:
1109:
1088:
1083:
1079:
1075:
1070:
1066:
1041:
1037:
1033:
1028:
1024:
1020:
1017:
996:
991:
987:
983:
980:
973:
955:
947:
942:
938:
931:
928:
923:
919:
915:
912:
907:
903:
899:
896:
891:
887:
878:
874:
870:
866:
862:
858:
854:
850:
845:
831:
826:
822:
817:
813:
808:
804:
796:
780:
775:
771:
766:
762:
757:
753:
746:
741:
737:
730:
725:
721:
712:
708:
692:
689:
686:
678:
675:
671:
668:
660:
658:
624:
622:
621:cross product
606:
597:
584:
573:
553:
536:
533:
528:
513:
497:
475:
465:
457:
452:
410:
392:
383:
379:
374:
345:
324:
286:
282:
277:
261:
251:
246:
236:
187:
144:
124:
112:
107:
105:
103:
99:
95:
91:
87:
86:tangent space
83:
80:
76:
73:
68:
66:
62:
58:
57:perpendicular
54:
50:
46:
42:
38:
29:
19:
1359:
1339:
1296:Applications
1287:
1221:hypersurface
1212:
1210:
1189:
1187:
1182:
1180:
1177:Computations
876:
872:
864:
860:
848:
846:
679:and a point
676:
669:
664:
625:
598:
278:
116:
101:
97:
93:
89:
81:
74:
69:
60:
52:
34:
1290:dot product
1196:(such as a
970:Thus every
705:, we get a
667:submanifold
661:Submanifold
512:dot product
285:unit vector
281:unit normal
72:submanifold
37:mathematics
1332:References
1217:implicitly
1009:splits as
869:direct sum
39:, given a
1225:level set
1215:is given
1206:immersion
1159:⊥
1119:∈
1114:⊥
1076:∈
1071:∥
1042:⊥
1029:∥
984:∈
956:⊥
916:⊕
750:→
734:→
690:∈
642:^
607:×
574:×
568:^
554:×
548:^
537:−
529:∥
498:⋅
476:⊥
466:−
453:∥
438:and thus
423:^
404:^
393:⋅
375:⊥
302:^
262:⊥
247:∥
224:as a sum
1372:Category
1181:Suppose
674:manifold
79:manifold
1312:vanish.
1223:) as a
599:where "
490:where "
113:Surface
65:surface
49:tangent
1346:
1058:where
857:splits
360:Then,
41:vector
1202:basis
851:is a
672:of a
77:of a
45:curve
1344:ISBN
1231:for
1101:and
793:The
1211:If
1208:).
1188:If
863:at
847:If
337:at
88:to
35:In
1374::
1173:.
1135::=
932::=
879::
713::
623:.
104:.
1363:.
1352:.
1271:i
1267:g
1244:i
1240:g
1213:N
1190:N
1183:N
1155:)
1151:N
1146:p
1142:T
1138:(
1132:N
1127:p
1123:N
1110:v
1089:N
1084:p
1080:T
1067:v
1056:,
1038:v
1034:+
1025:v
1021:=
1018:v
997:M
992:p
988:T
981:v
952:)
948:N
943:p
939:T
935:(
929:N
924:p
920:N
913:N
908:p
904:T
900:=
897:M
892:p
888:T
877:N
873:N
865:p
861:M
849:M
832:N
827:p
823:T
818:/
814:M
809:p
805:T
781:N
776:p
772:T
767:/
763:M
758:p
754:T
747:M
742:p
738:T
731:N
726:p
722:T
693:N
687:p
677:M
670:N
638:n
585:,
582:)
578:v
564:n
557:(
544:n
534:=
524:v
471:v
462:v
458:=
448:v
419:n
411:)
400:n
389:v
384:(
380:=
370:v
358:.
346:x
325:S
298:n
257:v
252:+
242:v
237:=
233:v
211:v
200:.
188:x
166:v
145:x
125:S
102:N
98:N
94:N
90:M
82:M
75:N
20:)
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