463:
354:
231:
292:
170:
504:
523:
497:
303:
392:
533:
490:
387:, Die Grundlehren der mathematischen Wissenschaften, vol. 153, New York: Springer-Verlag New York Inc., pp. xiv+676,
84:
The theory is "homological" because currents themselves are defined by duality with differential forms. To wit, the space
184:
426:
103:
245:
140:
54:
528:
430:
470:
78:
50:
39:
176:
388:
367:
237:
74:
31:
474:
442:
406:
17:
438:
402:
446:
434:
418:
410:
398:
380:
517:
46:
99:
349:{\displaystyle \langle \partial T,\alpha \rangle =\langle T,d\alpha \rangle }
425:, Princeton Mathematical Series, vol. 21, Princeton, NJ and London:
70:
66:
35:
462:
30:
This article is about an extension of the theory of the
478:
306:
248:
187:
143:
348:
286:
225:
164:
226:{\displaystyle d:\Omega ^{k-1}\to \Omega ^{k}}
498:
8:
343:
328:
322:
307:
156:
144:
65:is a method for extending the notion of the
505:
491:
287:{\displaystyle \partial :D^{k}\to D^{k-1}}
165:{\displaystyle \langle T,\alpha \rangle .}
305:
272:
259:
247:
217:
198:
186:
142:
524:Definitions of mathematical integration
366:. This is a homological rather than
7:
459:
457:
118:. Thus there is a pairing between
477:. You can help Knowledge (XXG) by
310:
249:
214:
195:
25:
461:
175:Under this duality pairing, the
77:, the integral is defined over
265:
210:
1:
423:Geometric Integration Theory
73:. Rather than functions or
38:. For numerical method, see
18:Geometric integration theory
534:Differential geometry stubs
550:
456:
427:Princeton University Press
29:
385:Geometric measure theory
94:-currents on a manifold
55:geometric measure theory
431:Oxford University Press
59:homological integration
473:-related article is a
350:
288:
227:
166:
471:differential geometry
351:
289:
228:
167:
63:geometric integration
51:differential geometry
304:
246:
185:
141:
40:geometric integrator
433:, pp. XV+387,
177:exterior derivative
27:Mathematics concept
346:
284:
223:
162:
134:, denoted here by
106:, of the space of
102:, in the sense of
98:is defined as the
75:differential forms
486:
485:
394:978-3-540-60656-7
238:boundary operator
32:Lebesgue integral
16:(Redirected from
541:
507:
500:
493:
465:
458:
449:
413:
381:Federer, Herbert
365:
355:
353:
352:
347:
293:
291:
290:
285:
283:
282:
264:
263:
232:
230:
229:
224:
222:
221:
209:
208:
171:
169:
168:
163:
133:
129:
125:
121:
117:
113:
109:
97:
93:
89:
21:
549:
548:
544:
543:
542:
540:
539:
538:
514:
513:
512:
511:
454:
417:
395:
379:
376:
360:
302:
301:
268:
255:
244:
243:
236:goes over to a
213:
194:
183:
182:
139:
138:
131:
127:
123:
119:
115:
111:
107:
95:
91:
85:
81:on a manifold.
43:
28:
23:
22:
15:
12:
11:
5:
547:
545:
537:
536:
531:
529:Measure theory
526:
516:
515:
510:
509:
502:
495:
487:
484:
483:
466:
452:
451:
415:
393:
375:
372:
370:construction.
364: ∈ Ω
357:
356:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
295:
294:
281:
278:
275:
271:
267:
262:
258:
254:
251:
234:
233:
220:
216:
212:
207:
204:
201:
197:
193:
190:
173:
172:
161:
158:
155:
152:
149:
146:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
546:
535:
532:
530:
527:
525:
522:
521:
519:
508:
503:
501:
496:
494:
489:
488:
482:
480:
476:
472:
467:
464:
460:
455:
448:
444:
440:
436:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
390:
386:
382:
378:
377:
373:
371:
369:
368:cohomological
363:
340:
337:
334:
331:
325:
319:
316:
313:
300:
299:
298:
279:
276:
273:
269:
260:
256:
252:
242:
241:
240:
239:
218:
205:
202:
199:
191:
188:
181:
180:
179:
178:
159:
153:
150:
147:
137:
136:
135:
105:
104:distributions
101:
88:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
41:
37:
33:
19:
479:expanding it
468:
453:
422:
384:
361:
358:
296:
235:
174:
86:
83:
62:
58:
47:mathematical
44:
419:Whitney, H.
297:defined by
518:Categories
447:0083.28204
411:0176.00801
374:References
122:-currents
100:dual space
49:fields of
344:⟩
341:α
329:⟨
323:⟩
320:α
311:∂
308:⟨
277:−
266:→
250:∂
215:Ω
211:→
203:−
196:Ω
157:⟩
154:α
145:⟨
71:manifolds
36:manifolds
421:(1957),
383:(1969),
359:for all
79:currents
67:integral
439:0087148
403:0257325
130:-forms
110:-forms
45:In the
445:
437:
409:
401:
391:
469:This
475:stub
429:and
389:ISBN
126:and
53:and
443:Zbl
407:Zbl
114:on
90:of
69:to
61:or
34:to
520::
441:,
435:MR
405:,
399:MR
397:,
57:,
506:e
499:t
492:v
481:.
450:.
414:.
362:α
338:d
335:,
332:T
326:=
317:,
314:T
280:1
274:k
270:D
261:k
257:D
253::
219:k
206:1
200:k
192::
189:d
160:.
151:,
148:T
132:α
128:k
124:T
120:k
116:M
112:Ω
108:k
96:M
92:k
87:D
42:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.