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´X¦ó©M¸s½× (Geometry and group theory)
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´X¦ó©M¸s½× Geometry and Group theory            

§@ªÌ: ½²Âצ{Âå®v

³Ì·s¤º®e¥i¥H¬Ý§Úªº³¡¸¨®æhttp://www.wretch.cc/blog/plastylife/15207813

¤j¼Æ¾Ç®aÃe¥[µÜ´¿¸g¤@»y¹D¯}: [¾ã­Ó¼Æ¾Ç´N¬O¸sªºªº¾Ç°Ý.] ¦]¦¹, °Q½×´X¦óµL¥iÁ×§K·|©M¸s¾ã¦X°_¨Ó.

­º¥ý¥ý±q°ò¥»ªº´X¦ó¬[ºc»¡°_. ©Ý¾ëªÅ¶¡ (topological space) ©M ¬y§Î (Manifolds) ¬O«Ü­«­nªº´X¦ó·§©À. Manifold ´N¹³¬O¬ì¾Çªº»R¥x¤@¼Ë, ªíºtªº¤¸¯À©M¨¤¦â³£§e²{¦b¤W­±, ¤]¦]¬°§e²{¦b¤W­±, §Ú­Ì»Ý­n¼@¥», ¨¤¦â©MºØºØ¹ïÀ³Ãö«Y....µ¥µ¥. ¨C­Ó¤H, ¥]¬A§Ú¦b¤º, ¥E¬Ý³o¨ÇªF¦è, ¤ß¤¤¤@©w·|²£¥ÍºÃ°Ý, ¬°¤°»ò­n·d³o»ò½ÆÂø?! ¦Y¹¡¨S¨Æ°µ. ¨ä¹ê§Ú­Ó¤H»{¬°, ¼Æ¾Çµo®i¦Ü²{¥N, ¬O¦³¨ä¾ú¥v¯ßµ¸ªº, ¤]´N¬O¹L¥h³Ì²³æªº¼Ú¤ó´X¦ó, ¦Û±q¦è¤¸«e¤T¦Ê¦~, §ÆÃ¾¼Æ¾Ç®a¼Ú´X¨½±oµoªí<´X¦ó­ì¥»>¤§«á, §Ú­Ì·Q·íµM¦Óªº¹L¤F¨â¤d¦~¥H¤W, ´X¦ó¬O¥­­±´X¦ó, ª½¨ì¹J¤W¦±­±ªº«D¼Ú´X¦ó¤§«á, ¤HÁ`·|¶}©l­«·s«ä¦Ò, ¥­±`¥­­±¤Wªº¨â­ÓÂIªº¶ZÂ÷¥i¥H¹º¤@±øª½½u¨Ó´ú¶q, ¦ý¬O, ¦b¦a²yªº¦±­±¤W, ÅsÅsªº, «ç»ò´ú¶q? Á`¤£¯àÆp¶i¦a²y§a? ¹L¥h»{¬°¹ïªº©w¸q, ¨ì¤F¦±­±³£·d¤£©w, ¨º¦pªG§ó½ÆÂø«ç»ò¿ì? ¨ì©³¤°»ò¬O¶ZÂ÷? ¤°»ò¬OªÅ¶¡? ¦³¨S¦³¤@¨Ç©w¸q¥i¥H©ñ½Ñ¥|®ü¬Ò·Ç, ¼Æ¾Ç®a¤£­n¦Ñ¬O³Q¤H¦R¹Ë: [§A¬Ý, ³oùؤ£¾A¥Î¤F³á]... µ¥µ¥, ¹L¥h²z©Ò·íµMªº¨Æ, ­ì¨Ó¤£¬O¨º»ò¦³¹D²z. ¬ã¨s¬ì¾Ç, ¦pªG¨C­Ó³£¥Î®t¤£¦h, ©ÎªÌ¦³¨Ç¥i¥H, ¦³¨Ç¤£¦æªº¸Ü, ³s¼Æ¾Ç³£¤£Ã­©T, ¨ä¥L³þ°ò¦b³Ì°ò¥»¼Æ¾Çªº¨ä¥L¬ì¾Ç, ´N·|¤j¶Ã. ¤HÃþ¥Í¬¡ªº¶i¨B, ´N¬O¾a³o¨Ç, ¤£¯à¥ú¾a­A¼L¥Ö¤l, ÁÙ¬O½Ö®±ÀY¤j, ©w³W«h, ©wªk«ß? ©Ò¥H, ¦^Âk¨ì¤@­Ó§Ú­Ì¤£¤Ó»{ÃѪº¦hºûªÅ¶¡, ¶W¹Lª½Ä±ªºªÅ¶¡®É, ÂI©MÂI¤§¶¡­n¬ã¨s¥¦ªº©Ê½è, ´N±o±q³Ì°ò¥»ªº¶°¦X(set)¨Ó¤U¤â.
topological space´N¬O¤@­Ó³Ì°ò¥», ¤£·|¥X¿ùªºªÅ¶¡©w¸q, ±q¼Æ¾Ç³Ì°ò¥»ªº¶°¦X·§©À¨Ó©w¸q, ³Ì¬°¤ã¹ê, ¤£³´¤JªÅªxªº¦U»¡¦U¸Ü, ¤]´N¬OÄYÂÔªº©w¸q, ·íµM¤]¥i¥H±q¦UºØ¤¬³qªº¨¤«×¨Ó»¡©ú:
¼Æ¾Ç©M¨ä¥L¬ì¾Ç¤@¼Ë, »Ý­n©w¸q²M·¡¤§«á, ¦A§@­l¥Í»P¬ã¨s.

A topological space is a set S together with O, a collection of subsets of S, satisfying the following axioms:
1. The empty set and S are in O.
2. The union of any collection of sets in O is also in O.
3. The intersection of any finite collection of sets in O is also in O.
±q¤W­±ªº°ò¥»©w¸q´N¥i¥Hª¾¹D: ¤@­Ó¶°¦X¤ºªº²Õ¦¨¤£ºÞ¥ÎÁp¶°ÁÙ¬O¥æ¶°³£ÁÙ¬O¨ä¤¤ªº¤@¥÷¤l¤~ºâ©ÝåNªÅ¶¡ªº©Ê½è!! ÃP´²ÀH«Kªº§Î®e: ¼Æ¾Ç¸g¹L"¹Bºâ"¤§«á, «ç»ò°µ³£·|¤@¼Ë´N¬O©Ý¾ë, Ä´¦p»¡: ¤@­Ó¶êÅܤjÅܤp, ©ÔªøÀ£«óÁÙ¬O¤@­Ó¶ê§Îªºª¬ºA. ¥ý¬Ý¤@¤U¡i¸s¡jªº©w¸q¤]¬OÃþ¦ü:
group is a set, G, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:
1. «Ê³¬©Ê Closure. For all a, b in G, the result of the operation a • b is also in G.
2. µ²¦X©Ê Associativity. For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds.
3. °ò¥»¤¸¯À Identity element. There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds.
4. ¤Ï¤¸¯À Inverse element. For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element. ¹Bºâ(operation)¤§«á, ¶°¦X¤ºªº¤¸¯À°ò¥»¤W³£ÁÙ¦b¸Ì­±, ´N¥s°µ²Å¦X"¸s"ªº©w¸q. §Ú­Ì¥ýÀÁ¦b¤@®Ç. ¥ý½Í½×´X¦óªº¤@¨Ç°ò¥»Æ[©À.

¥u­n©w¸q²M·¡, ´£¨Ñ±j©Î®zªº±ø¥ó, §Ú­Ì´N¥i¥H°Q½×¬ã¨s¦U¦¡¦U¼ËªºªÅ¶¡, ½Ñ¦p Hausdorff space --- ²Å¦XHausdorff axiom¤]´N¬Ofor any pair of distinct points p1 and p2 in S, there exist disjoints open sets O1 and O2, each containing just one of two sets. ´«¥y¸Ü»¡, §Ú­Ì¥i¥H°w¹ï"ÂI"§ä¨ì¤@­Ó°÷¤pªº¶°¦X, ¥B¥]§t¤F¦U¦Û¤@­ÓÂIªº¶°¦X¤£¬Û¥æ. ½Ñ¦p¦¹Ãþ, ¼Æ¾Ç®a©¹©¹°w¹ï¦U¦¡ªºªÅ¶¡¨Ó¬ã¨s¥¦ªº©Ê½è.

¤W­ztoplogical spaceÁÙ·|²£¥Í¤@¨ÇÆ[©À:
¦popen covering (¤S¥s°µpatches, {Ui}, «á­±·|¦A´£¨ì): a collection of open sets such that every point in S is contained in at least one of the Ui;
compact
: if every open covering {Ui} has a finite sub-collection {U1,.....,Uin} that also covers S.

­ì¥»©M½ä³Õ¦³Ãöªº¾÷²v, «Ø¥ß¦bÃP´²¾aª½Æ[¸gÅç¬[ºcªº¾÷²v, «á¨ÓÄY®æ¤Æ¤§«áªº¾÷²vªÅ¶¡(Probability space), ¤]¬OÃþ¦ü¤W­z¦p¦¹ªº©w¸q¤èªk!!


Fig. 1. ¬y§ÎManifoldªº©Ê½è

¤W­±¤j·§ª¾¹D¤@¤U´N¦n, ­«ÂI¬O§Ú­ÌÁÙ¬O±o°µ§Ú­Ì¼ô±xªº¥@¬É, ¤]´N¬O§ä¨ì"¶Ã¤C¤KÁV"ªF¦è©M§Ú­Ì¼ô±xªF¦è¤§¶¡ªºÃöÁp, ³o¤]¬O¬y§Îªººë¯«©Ò¦b. ¦]¬°, ¦b¶Ã¤C¤KÁVªF¦è¸Ìªº¤@°ïÂI, ÂI¥»¨­¨ä¹ê¥»¨­¨S¦³¤°»ò·N¸q­C!! ¥¦¥u¬O¦b¨º¸Ì, «ç»ò¿ì? ´N¹³§Ú­Ì¬ÝµÛ¤ÑªÅªº¬P¬P, ¥¦¤@ª½¦b¨º¸Ì, ¨S¦³»¡¸Ü, ·íµM¤]¤£·|§i¶D§A, ¥¦¦b¨º¸Ì¦³¤°»ò·N¸q? ¥¦¬O¤°»ò? ¦WºÙ³£¬O¤HÃþµ¹¤©ªº, ¶ZÂ÷µ¥Æ[©À§ó¥u¬O§Ú­Ì½á¤©ªº. ¸U¤@¦b¦h¦¸¤¸©Î²§§ÎªÅ¶¡§ó¤£ª¾©Ò±¹, ¤£¼ô±xÅý§Ú­Ì¤ßµê, «Ü©â¶H. §Ú­Ì±q¥X¥Í¤§«á, ¨ì¤HÃþ©Ò³BªºÀô¹Ò, ¥u¼ô±x¤Tºûªº®y¼Ð¨t, Ä´¦p»¡ª½ª½, ¥­¥­ªº²Ã¥d¨à®y¼Ð¨t, ¤]¥u·|¥Î¥¦°µ´ú¶q. ¦]¦¹, ¤@©w±o¥Î¹ïÀ³ªºÃö«Y¨Ó½á¤©·N¸q, µw§â©â¶HªºÂI©M§Ú­Ì©Ò¼ô±xªº, ¥i«×¶qªº®y¼Ð¨tÃl¦b¤@°_, ¤~¦³©Ò¨Ì¾Ú, ¤]´N¬Omapping,  ´N¦n¹³§A­n¤ñ¨­°ª, Á`­n¦³­Ó¤H©Î¤Ø·í°Ñ¦Ò, ¤ñ¸û, ¹ïÀ³, ¤~¯à»¡°ª¸G, ¦h°ªµ¥µ¥ . Ä´¦p¬Y­ÓÂI¹ïÀ³¬Y­Ó¼Æ­È, ¬Æ¦Ü¤èµ{¦¡. §Y¨Ï"©Ç©Ç"ªºªÅ¶¡, ©¼¦¹¥Î¼ô±x©Î¯S©wªº®y¼Ð¨t, §ä¨ì¹ïÀ³Ãö«Y, ¥i¥H[Âà´«], ª¾¹D§Ú­Ì©Ò­n¤F¸Ñ»P¬ã¨sªºªÅ¶¡©Ê½è. ³oºØ¼Æ¾Ç¼Ò¦¡´N¥s°µmanifold

¦A´«¤@ºØ»¡ªk, ©ÎªÌ°²·Q§Ú­Ì¬O¥j¤H, ·í§Ú­Ì¤£ª¾¹D¦a²y¬O²yª¬ªº®É­Ô, ¤@ª½"¶Ì¶Ì¤£²M·¡"¥H¬°¬O¥­ªº, §Ú­ÌÁÙ¬O«Ü¶}¤ßªº´ú¶q, ¦ý¬O¤]¤£·|¥X¤j¿ù¤@¼Ë, ¦]¬°¥u­n¶ZÂ÷¤£¤Ó»·, ¨âÂIªº©Ê½è¥i¥H»¡¬Oª½½u, ¦Ó¤£¬O²y­±¤WÅs¦±ªº½u. ¥u¤£¹L¶ZÂ÷¤@©Ôªø, ³oºØªñ¦üªº§@ªk´N¤£·Ç¤F!!  ³o´N¬Omanifoldªº·Qªk, §Ú­Ì¨ú¤@­ÓÂIªþªñªº°Ï°ì, ·í«Ü¾aªñ®É, µL½u¤pªº·¥­­·§©À, ´N¥i¥H°²³]¥¦Ãþ¦ü©Îµ¥¦P¼Ú´X¨½±oªÅ¶¡¨Ó´ú¶q»P°Q½×. ·íµM, ­n³o¼Ë»¡, ±o¸g¹LÃÒ©ú, ¨Æ¹ê¤W¤w¸g¦³¤HÃÒ©ú¥X¨Ó¤F---- A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated----- µL½×ªÅ¶¡¦h½ÆÂø, ³£¥i¥H§ä¨ì«Ü§½³¡ªº°Ï°ìÃþ¦ü¼Ú¦¡ªÅ¶¡, ³o¼Ë´N¦n¿ì¤F.
 
manifold«ç»ò°µ©O? ¥Î¤£¥¿¦¡ªº»¡ªk´N¬O: §Ú­Ì¥ý¦Ò¼{¤@­Ó topological space S, µM«á§â¥¦¤À¦¨¤@¶ô¶ô, ¤@¤ù¤ùªºpatches, ·íµM§Ú­Ì¥i¥H¿ï¾Ü¨¬°÷ªºpatches¨ÓÂл\¾ã­ÓS, ¤]·|³y¦¨­«Å|. µM«á, ¹ï©ó¥ô·N¤@­Ópatch, ºÙ¥¦§@U1, §Ú­Ì¥i¥H§ä¨ì¤@­Ómap (P1), ¨Ó¤@¹ï¤@¥i°f1-1(unique invertible relation)¹ïÀ³¦Ü¹ê¼ÆªÅ¶¡Rn. ³o¸Ì¥ýÄY®æ©w¸q±M¦³¦Wµü: A set (called an atlas) of maps Pi called charts, which define a 1-1 relation between points in Ui and points in an open ball in Rn.

¦n, ²{¦b¦A§ä¥t¤@¶ôpatch, ¥s¥¦°µU2, ¥¦ªº¹ïÀ³mappingºÙ§@P2, ¤]¹ïÀ³¦Ü¥t¤@­Ó¹ê¼Æ®y¼Ð¨t. ¼Æ¾Ç¥©§®ªº¦a¤è, ´N¬O©äÅs¨S¨¤, ¶¸ô¶±o¤j®aÀY©ü¸£µÈ, ¤S¦n¹³¦³¤@ÂI¹D²z. «e­±¤£¬O»¡¨C¤@¶ôpatch¥i¯à¦³­«Å|ªº¦a¤è, ³á, §Ú­Ì´N¥i¥H§Q¥Î¤¬´«¨Ó§ä¨ì¨â­Ó®y¼Ð¨tªº¹ïÀ³Ãö«Y. «ç»ò»¡? ³o´N¹³¨â­Ó¤£¦PªºÄÒ, ±o§ä¨ì¦@¦PªºªB¤Í, ¤~¯à«Ø¥ß¦X§@Ãö«Y©Î½Í§P. ­º¥ý, §Q¥Î­«Å|ªº°Ï°ìÀ³¸Ó¬O¤@¼Ëªº¦@¦P"¸ÜÃD"©Î"ªB¤Í"(¹ï¤£°_, ¥Õ¸ÜµL²á, ¤£±M·~¤@ÂI, ¤]¦]¬°§Ú¬O·~¾l·R¦nªÌ¦Ó¤w), ©Ò¥H, ±q²Ä¤@­Ó¹ê¼Æ®y¼Ð¨tªº"ªF¦è"¤Ï¹L¨Ómapping(P1-1)´N·|¨ì­«Å|ªº¦a¤è, ¨S¿ù(·Q¤@¤U, ¤£Ãø, ¤]´N¬Oinvertible map), µM«á, ¦A¥ÎP2¤£´N¨ì¥t¤@­Ó¹ê¼Æ®y¼Ð¨t. §ä¨ì¤F¹ïÀ³Âà´«(transformation), ®y¼Ð¨t´«¨Ó´«¥h´N¤£¦A§xÂZ (³o¤@¬qªº±Ô­z, ¦p¤W¹ÏFig. 1 shown). manifold¥¿¦¡¤@ÂIªºÂ²µu»¡ªk¬O: 1, locally homeomorphic to Rn¤¤ªº¤@­Óopen set; 2, ¦P®É has differentiable transition functions. map¬JµM¬O¦a¹ÏÆ[©À, ·íµM¬O§½³¡ªº, ©Î¶m©ÎÂí©Î¥«, ©Ò¥H´N¦³¦Wµü: ¤@¹ï¤@Ãö«Y¦a¹Ïªº¹Ïªí(chart) ©M ¹Ïªíªº¶°¦X"¹ÏÃÐ"(atlas)ªºÆ[©À. ¼Æ¾Ç¤W, ¤W­z¬y§Î¸Ìªº¹ïÀ³, ¦]¬°©Ý¹²ªÅ¶¡¦³«Ü¦h¶ôpatch, ©Ò¥H¤~·|¦³¤W­z­^¤å¥¿¦¡¤@ÂIªº¼gªk¬O An atlas of charts . §Ú­Ì½á¤©³Ì­ì©l¤@°ïÂIªº©Ý¹²ªÅ¶¡©M§Ú­Ì¹ê¼Æ°ìªº¹ïÀ³¦a¹ÏÃö«Y(map), ´N¤£¦A°g±¦°g¸ô, ³o¼Ë§Ú­Ì¤]¥i¥HÀ³¥Î¹ê¼Æ°ìªº©Ò¦³¾Ç°Ý, §â¨â­Ó»â°ì±µ­y¦b¤@°_ºÙ§@topological algebra. ÁöµM¼Æ¾Çµo®i¦Ü«á­±, ¤@¥¹¥Îtangent bundle, ´N¤£»Ý­n¤W­zªº§½³¡®y¼Ð¨t, ¤]¤£»Ý­natlasªºÆ[©À, ¦ý¬O³o²¦³ºÁÙ¬O¤£¿ùªº¤Jªù, ¥B¨ã¦³¾ú¥v·N¸qªº¸g¨å·§©À.

manifold, ¬JµMmapping§ä¨ì, ©ÎªÌ§ä¨ìfunction, ³o­Ófunction´N­±Á{¬O§_¥i·Lµ¥°ÝÃD, ¥i·Lªº´N¥s°µ Differentiable manifold,  ¥i·L´X¦¸´N¥Î¤jCªí¥Ü, ¦p¥i·L¤T¦¸, C3, ½Ñ¦p¦¹Ãþ. ÁÙ¦³, ¤@©w©Ò¦³manifold¥i¥H§ä¨ì¹ïÀ³ªº¼Ú¦¡ªÅ¶¡¹ïÀ³¶Ü? µª®×¬OªÖ©wªº, ¦ý¬Oºû«×­n¤j¤@ÂI. ³o¬O´¿¸g·h¤W¹q¼vªº¼Æ¾Ç©_¤~Nash(«á¨Ó´¿¸gµoºÆ¹L) ÃÒ©ú¥X¨Óªº--- Nash embedding theorem: Any manifold can be embedded isometrically into a Euclidean space of large dimension. (©w¸q:  immersion: locally one-to-one; embedding: locally diffeomorphism). ¥t¥~ strong Whitney embedding theorem: any connected smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean 2m-space, if m>0.
¨ä¹ê¤@¶}©l, ¦]¬°ªÅ¶¡ªº¬ã¨s»Ý­n, µL½×¬O¤Ñ¤å, ¦a²z, ª«²zµ¥µ¥, ©Ò¥H¤~µo®i¥X´X¦ó¾Ç. µM«á¹J¨ì¤£¯à¸ÑÄÀªº, ´N§ä¥X®Ú¥»°ÝÃD ­×¥¿©w¸q, ³o´N¬O¼Æ¾Ç©¹©¹³£¬O"±À¼s"¦Ó§@³\¦h¦nª±ªº¬ã¨s.
¹ï§Ú¦Ó¨¥, °µ¾Ç°Ý¦³½ìªº¦a¤è¤£¦b©ó¤°»ò³£À´, ¦Ó¬O°l¨Dª¾ÃѪº¹Lµ{¤¤µo²{IJÃþ®Ç³qªº§Ö¼Ö!! 
¦³¨S¦³µo²{¤W­zªº¶°¦X, ¦³ÂI¹³¥N¼Æ(Algebra), ¥[¥[, ´î´î, ²¾°Ê¦ì¸mµ¥µ¥. Ä´¦p»¡: ²Å¦X¤£¦Pªº¥N¼Æ¹Bºâ¤§«á ¨C¤@¨Ç©w¸q´N·|¦³¤@¨Ç©Ê½è:
Groupoid(¸s­F): a, bÄÝ©óR---a+b¤]¬O
Semigroup(¥b¸s): a, b, cÄÝ©óR---a+(b+c)=(a+b)+c µ²¦X«ß(distributive)
Monoid(³æ¤l): a, eÄÝ©óR---a+e=e+a=a
Group(¸s): a, a-1 ÄÝ©óR---a+a-1=a-1+a=e
¥t¥~¦A¥[¤W¥æ´««ßªº¸Ü, ¨C­Ó¦Wµü«e­±¥[¤Wcommutative.
Ring(Àô)ªº©w¸q«h¬O: A ring (R,+,.) is a nonempty set R with two binary operations + and . on R. ¦³¤U­z©Ê½è: (1) (R,+) is a commutative group; (2) (R,.) is a semigroup; (3) the two operations are related by the distributive laws.
³o¨Ç¥O¤Hı±oµL²áªºªF¦è, ¨ä¹ê¬O¤Ï¹L¨Óµo®iªº. ·í¤HÃþ¥ý¥Ñ¤@¨Çª½Æ[ªºªF¦è, ©w¸q²³æªº¼Æ¾Ç, ¤§«áµo²{¦n¹³¦³¨Ç¦a¤è¤£¾A¥Î, ´N°µ¤F­×¥¿; ¤S©ÎªÌ, ¦n¹³¥E¬Ý¤£¦Pªº»â°ì, ¦³¨Ç¦@¦Pªº³W«h, ¥i¥HÂǥѬۦPªº°ò¦¨Ó¬[ºc. ©ó¬O, ¥N¼Æ, ´X¦ó, ·¥­­µ¥µ¥´NÅܦ¨¤F¼Æ¾Ç±`¥Î¨Ó§â¤£¦P¾Ç°Ý²Î¦Xªº¤u¨ã. ³oµLºÃ¬O¤HÃþ´¼¼zªº·¥­Pµo´§, ¸g¹L¼Æ¦Ê¦~¤HÃþºë­^ªº§ë¤J, ¤~³vº¥µo®i¥X²{¦bŪ¨Ó½ÆÂøªº²z½×.
¥N¼Æ»P·¥­­¥[¤JªÅ¶¡ªº·§©À, ´N¥i¥H¥¿¦¡¬°¤@¨ÇªÅ¶¡©w¸q, ¥]¬A¦V¶qªÅ¶¡(vector space). ¥ý²Ê²¤¬Ý¤@¤U©w¸q:

´£¨ì¦V¶qªÅ¶¡¤§«e, ¥ý°Q½×¦V¶q¥»¨­. ¦V¶q¦b§Ú­Ì¨D¾Çªº®É­Ô, ¤@ª½§â¥¦·í§@ª½½u, A©MB¨âÂI©Ô¤@±ø±a¦³½bÀYªº½u, ¥s°µ¦V¶q, ªí­±¤W¨S¿ù, ¦ý¬O¨º¥u¾A¥Î©ó¤Tºûªº¼Ú¦¡ªÅ¶¡, ÄY®æ¥s°µ position vector. ¦b²y­±2-sphere¤W´N¤£¹ï¤F, ¨âÂI¤@³s´N·|¬ï¹L²y, ¦Ó³Q¤ÁÂ_¤¤Â_, ¨ºªuµÛ²y­±³sµ²¨âÂIªº©·½u¥s°µ¦V¶q¶Ü? ¤]¤£¬O,  ©Ç©Çªº, ¦]¬°³oºØ¦V¶q¨S¿ìªk²Å¦X¥N¼Æ©Ê½è, ¤£¯àª½±µ¥[´î, ¤£¯à³B²z.  ¦n, ¼Æ¾Ç®a«ÜÁo©ú, ¨º¥u¦n¥Î·¥­­, µL­­¤p(infinitesimal)ªº·§©À, ²y­±¤Wµe«Ü¤p«Ü¤p¤@¬q, ¤£´N«Ü¹³"¤pª½½u", ¤£´N¥i¥H¦A«×À³¥Î¦V¶qªº·§©À¶Ü? ©ÎªÌ±q¥t¤@­Ó¨¤«×¬Ý, §Ú¥Î¤@­Ó¥­­±¥u¤Á³o­ÓÂI, ¨º¤£´N¬O¦b¤Á¥­­±¤W¨«¤@¤p¬q¥s°µ¦V¶q, ©ÎªÌºÙ°µ ¤Á¦V¶q(tangent vector). ¥J²Ó¤@¬Ý, ³o¼Ëªº·Qªk¤£´N¬O·L¿n¤Àªº·§©À¶Ü? ¨S¿ù. ©Ò¥H tangent vector ¤S¥i¥HºÙ¬° derivative operator.
²{¦b­É¥Î¤W­zªº¬y§ÎÆ[©À, °²³]manifold M ¤W­±ªº patch U, ¤W­±¦s¦b¤@­Ó§½³¡®y¼Ð local coordinates xi, patch U ¤W­±¤@±øpath. §Ú­Ì°²³]ªuµÛpath¤Wªº½u©Ê»¼¼W¼W¥[ªº°Ñ¼Æs, §Ú­Ì¥i¥H±o¨ìmanifold M¤WªuµÛ³o±øpathªºÂI, ªí¥Ü¬° xi=xi(s). µM«á±µµÛ¦Ò¼{¤@­Ó¦bM¤W­±ªºsmooth function f. ªuµÛ¤W­z¨º±øpath¤WªºÂI, ¥i¥H¥Îf²£¥Í¼Æ­È¬°f(xi(s)).
®Ú¾Úchain rule, §Ú­Ì¥i¥H¥N¥X
, ¥ªÃä³Ì¤U¤èªºªí¥Üªk, ¬O·R¦]´µ©Z¹ï©ó"©M"ªºÂ²¼g¤è¦¡(Einstein summation convention).
§Ú­Ì¥i¥H©w¸q¤U­z ªuµÛ¨º±øpathªº directed derivative operator V
, ³o¼Ë¥i¥H±Nsmooth function f ¾É¦Vmap¦ÜR ¹ê¼Æ°ì, ¤]´N¬O²£¥Í¼Æ­È .
³o¼Ëªº©w¸q´N¥i¥H²Å¦X   linearity property    V(f+g)=Vf+Vg ; Leibnitz property  V(fg)=(Vf)g+f(Vg).
¦bM¤WªºpÂIªºtangent vectors©Òºc¿v¦Ó¦¨ªº¦V¶qªÅ¶¡ ºÙ°µ tangent space, °O°µTp(M) .
±µµÛ, §Ú­Ì¥i¥H¹B¥Î®õ°Ç®i¶}¦¡(Taylor's theorem), ªí¥Ülocal coordinates xi:
, xip denotes the coordinate system to the point p.
§Ú­Ì©w¸q:
---Vi ´N¬O¦V¶q V ªº²Õ¦¨.
µM«á´N±o¨ì, ¦]¦¹´N¥i¥Hµo²{¦V¶qªÅ¶¡ªº°ò©³(coordinate basis)´N¬O, ³o¸Ì¥i¥Hµo²{¤Á¦V¶qªÅ¶¡ªººû«×µ¥©ó®y¼Ð¨txiªº¼Æ¥Ø, ¤]©Mmanifold M ªººû«×¤@¼Ë. Á`µ²¤@¤U, ¦³§O©ó°á®Ñ®É¥N©w¸q©ó¼Ú¤óªÅ¶¡ªºª½½u, ¦V¶q V ²{¦bªº©w¸q¬O¿W¥ß©ó®y¼Ð¨t(coordinate-independent) , ¤]´N¬O¨S¦³Ãö«Y; ¦ý¬O¥¦ªº²Õ¦¨ Vi ¬O©M®y¼Ð¨t¬ÛÃöªº(coordinate-dependent). ¤j¤è¦V¨SÅÜ, ªF¦è¨SÅÜ, ¦ý¬OÆ[¹îªÌ§ïÅܪº¸Ü, ¥u¬O¸Ì­±ªº¨C¤@­Ó®y¼Ð¤è¦V§ïÅܦӤw, ¨º»ò§ïÅܪº¤½¦¡«Ü²³æ, §Q¥Îchain rule, , ³o´N¬Ogeneral coordinate transformations
³o­Ó¤Á¦V¶qªÅ¶¡¤W¨C¤@ÂI¤£¬O·|¦³«Ü¦h±ø¸g¹L¥¦ªº©Ò¦³¥i¯à¤Á¦V¶q, §â¥¦­Ì¶°¦X°_¨Ó´N¬O tangent bundle, , ©w¸q¬°T(M)©ÎTM.

¦^¨ì«e­±´£¨ìªº¦V¶qªÅ¶¡(vector space), ©w¸q¬O»¡¦V¶q¸g¹L¦³­­¦¸¼Æªº¥[ªk(addition)©Î­¼¤W"¯Â¶q"scalar multiplication)¤´ÂÂÁÙ¬OÄÝ©ó¦V¶qªÅ¶¡¤º, ³o¬O«Ê³¬©Ê.
¤@¯ë¦Ó¨¥, Á|¨Ò: ¦bRnùتº¦V¶qªÅ¶¡, ¦V¶qªºªí¥Ü¤èªk¬°: n-tuplet (a1, a2, ....., an)
¥[ªk(vector addition): (a1, a2, ....., an) + (b1, b2, ....., bn)=(a1+b1, a2+b2, ....., an+bn)
scalar multiplication: r ¬O¹ê¼Æ, r(a1, a2, ....., an)=(ra1, ra2, ....., ran)
·íµM, ¦b¹ê¼Æ°ì¸Ì, commutativity, associativityµ¥¤@¤j°ï©Ê½è¨Ì¦¨¥ß.
¤£¦Pªº¦V¶qªÅ¶¡X, Y ¸g¹L©¼¦¹¹Bºâ, ¤]·|¦¨¬°¥t¤@ºØ§Î¦¡ªº¦V¶qªÅ¶¡, ¤]´N¬O±i¶qªÅ¶¡(tensor-product (vector) space) : ±i¶q¿n¤]²Å¦Xdistributive law , °²³]xi, yj¤À§O¬OX,Yªºbasis, ©Ò¥H±i¶qªÅ¶¡ tensor product space Z=zijxiyj, ¥¦ªººû«×´N¬O¨â­ÓªÅ¶¡ºû«×ªº­¼ÁZ.


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