DIFFERENTIAL FORMS AND INTEGRATION

Transcription

1 DIFFERENTIAL FORMS AND INTEGRATION TERENCE TAO The concept of integrtion is of course fundmentl in single-vrible clculus. Actully, there re three concepts of integrtion which pper in the subject: the indefinite integrl f (lso known s the nti-derivtive), the unsigned definite integrl f(x) dx (which one would use to find re under curve, or the mss [,b] of one-dimensionl object of vrying density), nd the signed definite integrl b f(x) dx (which one would use for instnce to compute the work required to move prticle from to b). For simplicity we shll restrict ttention here to functions f : R R which re continuous on the entire rel line (nd similrly, when we come to differentil forms, we shll only discuss forms which re continuous on the entire domin). We shll lso informlly use terminology such s infinitesiml in order to void hving to discuss the (routine) epsilon-delt nlyticl issues tht one must resolve in order to mke these integrtion concepts fully rigorous. These three integrtion concepts re of course closely relted to ech other in singlevrible clculus; indeed, the fundmentl theorem of clculus reltes the signed definite integrl b f(x) dx to ny one of the indefinite integrls F = f by the formul b f(x) dx = F (b) F () (1) while the signed nd unsigned integrl re relted by the simple identity b f(x) dx = f(x) dx = f(x) dx (2) which is vlid whenever b. b When one moves from single-vrible clculus to severl-vrible clculus, though, these three concepts begin to diverge significntly from ech other. The indefinite integrl generlises to the notion of solution to differentil eqution, or of n integrl of connection, vector field, or bundle. The unsigned definite integrl generlises to the Lebesgue integrl, or more generlly to integrtion on mesure spce. Finlly, the signed definite integrl generlises to the integrtion of forms, which will be our focus here. While these three concepts still hve some reltion to ech other, they re not s interchngeble s they re in the single-vrible setting. The integrtion on forms concept is of fundmentl importnce in differentil topology, geometry, nd physics, nd lso yields one of the most importnt exmples of cohomology, nmely de Rhm cohomology, which (roughly speking) mesures precisely the extent to which the fundmentl theorem of clculus fils in higher dimensions nd on generl mnifolds. 1 [,b]

2 2 TERENCE TAO To motivte the concept, let us informlly revisit one of the bsic pplictions of the signed definite integrl from physics, nmely to compute the mount of work required to move one-dimensionl prticle from point to point b, in the presence of n externl field (e.g. one my move chrged prticle in n electric field). At the infinitesiml level, the mount of work required to move prticle from point x i R to nerby point x i+1 R is (up to smll errors) linerly proportionl to the displcement x i := x i+1 x i, with the constnt of proportionlity f(x i ) depending on the initil loction x i of the prticle 1, thus the totl work required here is pproximtely f(x i ) x i. Note tht we do not require tht x i+1 be to the right of x i, thus the displcement x i (or the infinitesiml work f(x i ) x i ) my well be negtive. To return to the non-infinitesiml problem of computing the work b f(x) dx required to move from to b, we rbitrrily select discrete pth x 0 =, x 1, x 2,..., x n = b from to b, nd pproximte the work s b n 1 f(x) dx f(x i ) x i. (3) Agin, we do not require x i+1 to be to the right of x i (nor do we require b to be to the right of ); it is quite possible for the pth to bcktrck repetedly, for instnce one might hve x i < x i+1 > x i+2 for some i. However, it turns out in the one-dimensionl setting, with f : R R ssumed to be continuous, tht the effect of such bcktrcking eventully cncels itself out; regrdless of wht pth we choose, the right-hnd side of (3) lwys converges to the left-hnd side s long s we ssume tht the mximum step size sup 0 i n 1 x i of the pth converges to zero, nd the totl length n 1 i=0 x i of the pth (which controls the mount of bcktrcking involved) stys bounded. In prticulr, in the cse when = b, so tht ll pths re closed (i.e. x 0 = x n ), we see tht signed definite integrl is zero: i=0 f(x) dx = 0. (4) In the lnguge of forms, this is sserting tht ny one-dimensionl form f(x)dx on the rel line R is utomticlly closed. (The fundmentl theorem of clculus then sserts tht such forms re lso utomticlly exct.) The concept of closed form corresponds to tht of conservtive force in physics (nd n exct form corresponds to the concept of hving potentil function). From this informl definition of the signed definite integrl it is obvious tht we hve the conctention formul c f(x) dx = b f(x) dx + c b f(x) dx (5) regrdless of the reltive position of the rel numbers, b, c. In prticulr (setting = c nd using (4)) we conclude tht b f(x) dx = b f(x) dx. 1 In nlogy with the Riemnn integrl, we could use f(x i ) here insted of f(x i ), where x i is some point intermedite between x i nd x i+1. But s long s we ssume f to be continuous, this technicl distinction will be irrelevnt.

3 DIFFERENTIAL FORMS AND INTEGRATION 3 Thus if we reverse pth from to b to form pth from b to, the sign of the integrl chnges. This is in contrst to the unsigned definite integrl f(x) dx, [,b] since the set [, b] of numbers between nd b is exctly the sme s the set of numbers between b nd. Thus we see tht pths re not quite the sme s sets; they crry n orienttion which cn be reversed, wheres sets do not. Now we move from one dimensionl integrtion to higher-dimensionl integrtion (i.e. from single-vrible clculus to severl-vrible clculus). It turns out tht there will be two dimensions which will be relevnt: the dimension n of the mbient spce 2 R n, nd the dimension k of the pth, oriented surfce, or oriented mnifold S tht one will be integrting over. Let us begin with the cse n 1 nd k = 1. Here, we will be integrting over continuously differentible pth (or oriented rectifible curve 3 ) in R n strting t some point R n nd ending t point b R n (which my or my not be equl to, depending on whether the pth is closed or open); from physicl point of view, we re still computing the work required to move from to b, but re now moving in severl dimensions insted of one. In the one-dimensionl cse, we did not need to specify exctly which pth we used to get from to b (becuse ll bcktrcking cncelled itself out); however, in higher dimensions, the exct choice of the pth becomes importnt. Formlly, pth from to b cn be described (or more precisely, prmeterised) s continuously differentible function : [0, 1] R n from the stndrd unit intervl [0, 1] to R n such tht (0) = nd (1) = b. For instnce, the line segment from to b cn be prmeterised s (t) := (1 t) + tb. This segment lso hs mny other prmeteristions, e.g. (t) := (1 t 2 ) + t 2 b; it will turn out though (similrly to the one-dimensionl cse) tht the exct choice of prmeteristion does not ultimtely influence the integrl. On the other hnd, the reverse line segment ( )(t) := t + (1 t)b from b to is genuinely different pth; the integrl on will turn out to be the negtive of the integrl on. As in the one-dimensionl cse, we will need to pproximte the continuous pth by discrete pth x 0 = (0) =, x 1 = (t 1 ), x 2 = (t 2 ),..., x n = (1) = b. Agin, we llow some bcktrcking: t i+1 is not necessrily lrger thn t i. The displcement x i := x i+1 x i R n from x i to x i+1 is now vector rther thn sclr. (Indeed, one should think of x i s n infinitesiml tngent vector to the mbient spce R n t the point x i.) In the one-dimensionl cse, we converted the sclr displcement x i into new number f(x i ) x i, which ws linerly relted to the originl displcement by proportionlity constnt f(x i ) depending on the position x i. In higher dimensions, the nlogue of proportionlity constnt of 2 We will strt with integrtion on Eucliden spces R n for simplicity, lthough the true power of the integrtion on forms concept is only pprent when we integrte on more generl spces, such s bstrct n-dimensionl mnifolds. 3 Some uthors distinguish between pth nd n oriented curve by requiring tht pths to hve designted prmeteristion : [0, 1] R n, wheres curves do not. This distinction will be irrelevnt for our discussion nd so we shll use the terms interchngebly. It is possible to integrte on more generl curves (e.g. the (unrectifible) Koch snowflke curve, which hs infinite length), but we do not discuss this in order to void some techniclities.

4 4 TERENCE TAO liner reltionship is liner trnsformtion. Thus, for ech x i we shll need liner trnsformtion ω xi : R n R tht tkes n (infinitesiml) displcement x i R n s input nd returns n (infinitesiml) sclr ω xi ( x i ) R s output, representing the infinitesiml work required to move from x i to x i+1. (In other words, ω xi is liner functionl on the spce of tngent vectors t x i, nd is thus cotngent vector t x i.) In nlogy to (3), the net work ω required to move from to b long the pth is pproximted by n 1 ω ω xi ( x i ). (6) i=0 If ω xi depends continuously on x i, then (s in the one-dimensionl cse) one cn show tht the right-hnd side of (6) is convergent if the mximum step size sup 0 i n 1 x i of the pth converges to zero, nd the totl length n 1 i=0 x i of the pth stys bounded. The object ω, which continuously ssigns 4 cotngent vector to ech point in R n, is clled 1-form, nd (6) leds to recipe to integrte ny 1-form ω on pth (or, to shift the emphsis slightly, to integrte the pth ginst the 1-form ω). Indeed, it is useful to think of this integrtion s binry opertion (similr in some wys to the dot product) which tkes the curve nd the form ω s inputs, nd returns sclr ω s output. There is in fct dulity between curves nd forms; compre for instnce the identity (ω 1 + ω 2 ) = ω 1 + ω 2 (which expresses (prt of) the fundmentl fct tht integrtion on forms is liner opertion) with the identity ω = 1+ 2 ω + 1 ω 2 (which generlises (5)) whenever 5 the initil point of 2 is the finl point of 1, where is the conctention of 1 nd 2. This dulity is best understood using the bstrct (nd much more generl) formlism of homology nd cohomology. Becuse R n is Eucliden vector spce, it comes with dot product (x, y) x y, which cn be used to describe 1-forms in terms of vector fields (or equivlently, to identify cotngent vectors nd tngent vectors): specificlly, for every 1-form ω there is unique vector field F : R n R n such tht ω x (v) := F (x) v for ll x, v R n. With this representtion, the integrl ω is often written s F (x) dx. However, we shll void this nottion becuse it gives the misleding impression tht Eucliden structures such s the dot product re n essentil spect of the integrtion on differentil forms concept, which cn led to confusion when one generlises this concept to more generl mnifolds on which the nturl nlogue of the dot product (nmely, Riemnnin metric) might be unvilble. 4 More precisely, one cn think of ω s section of the cotngent bundle. 5 One cn remove the requirement tht 2 begins where 1 leves off by generlising the notion of n integrl to cover not just integrtion on pths, but lso integrtion on forml sums or differences of pths. This mkes the dulity between curves nd forms more symmetric.

5 DIFFERENTIAL FORMS AND INTEGRATION 5 Note tht to ny continuously differentible function f : R n R one cn ssign 1-form, nmely the derivtive df of f, defined s the unique 1-form such tht one hs the Tylor pproximtion f(x + v) f(x) + df x (v) for ll infinitesiml v, or more rigorously tht f(x + v) f(x) df x (v) / v 0 s v 0. Using the Eucliden structure, one cn express df x = f(x) dx, where f is the grdient of f; but note tht the derivtive df cn be defined without ny ppel to Eucliden structure. The fundmentl theorem of clculus (1) now generlises s df = f(b) f() (7) whenever is ny oriented curve from point to point b. In prticulr, if is closed, then df = 0. A 1-form whose integrl ginst every closed curve vnishes is clled closed, while 1-form which cn be written s df for some continuously differentible function is clled exct. Thus the fundmentl theorem sserts tht every exct form is closed. This turns out to be generl fct, vlid for ll mnifolds. Is the converse true (i.e. is every closed form exct)? If the domin is Eucliden spce (or more ny other simply connected mnifold), then the nswer is yes (this is specil cse of the Poincré lemm), but it is not true for generl domins; in modern terminology, this demonstrtes tht the de Rhm cohomology of such domins cn be non-trivil. Now we turn to integrtion on k-dimensionl sets with k > 1; for simplicity we discuss the two-dimensionl cse k = 2, i.e. integrtion of forms on (oriented) surfces in R n, s this lredy illustrtes mny fetures of the generl cse. Physiclly, such integrls rise when computing flux of some field (e.g. mgnetic field) cross surfce; more intuitive exmple 6 would rise when computing the net mount of force exerted by wind blowing on sil. We prmeterised one-dimensionl oriented curves s continuously differentible functions : [0, 1] R n on the stndrd (oriented) unit intervl [0, 1]; it is thus nturl to prmeterise two-dimensionl oriented surfces s continuously differentible functions φ : [0, 1] 2 R n on the stndrd (oriented) unit squre [0, 1] 2 (we will be vgue here bout wht oriented mens). This will not quite cover ll possible surfces one wishes to integrte over, but it turns out tht one cn cut up more generl surfces into pieces which cn be prmeterised using nice domins such s [0, 1] 2. In the one-dimensionl cse, we cut up the oriented intervl [0, 1] into infinitesiml oriented intervls from t i to t i+1 = t i + t, thus leding to infinitesiml curves from x i = (t i ) to x i+1 = (t i+1 )) = x i + x i. Note from Tylor expnsion tht x i nd t re relted by the pproximtion x i (t i ) t i. In the two-dimensionl cse, we will cut up the oriented unit squre [0, 1] 2 into infinitesiml oriented squres 7, 6 Actully, this exmple is misleding for two resons. Firstly, net force is vector quntity rther thn sclr quntity; secondly, the sil is n unoriented surfce rther thn n oriented surfce. A more ccurte exmple would be the net mount of light flling on one side of sil, where ny light flling on the opposite side counts negtively towrds tht net mount. 7 One could lso use infinitesiml oriented rectngles, prllelogrms, tringles, etc.; this leds to n equivlent concept of the integrl.

6 6 TERENCE TAO typicl one of which my hve corners (t 1, t 2 ), (t 1 + t, t 2 ), (t 1, t 2 + t), (t 1 + t, t 2 + t). The surfce described by φ cn then be prtitioned into (oriented) regions with corners x := φ(t 1, t 2 ), φ(t 1 + t, t 2 ), φ(t 1, t 2 + t), φ(t 1 + t, t 2 + t). Using Tylor expnsion in severl vribles, we see tht this region is pproximtely n (oriented) prllelogrm in R n with corners x, x + 1 x, x + 2 x, x + 1 x + 2 x, where 1 x, 2 x R n re the infinitesiml vectors 1 x := φ t 1 (t 1, t 2 ) t; 2 x := φ t 2 (t 1, t 2 ) t. Let us refer to this object s the infinitesiml prllelogrm with dimensions 1 x 2 x with bse point x; t this point, the symbol is meningless plceholder. In order to integrte in mnner nlogous with integrtion on curves, we now need some sort of functionl ω x t this bse point which should tke the bove infinitesiml prllelogrm nd return n infinitesiml number ω x ( 1 x 2 x), which physiclly should represent the mount of flux pssing through this prllelogrm. In the one-dimensionl cse, the mp x ω x ( x) ws required to be liner; or in other words, we required the xioms ω x (c x) = cω x ( x); ω x ( x + x) = ω x ( x) + ω x ( x) for ny c R nd x, x R n. Note tht these xioms re intuitively consistent with the interprettion of ω x ( x) s the totl mount of work required or flux experienced long the oriented intervl from x to x + x. Similrly, we will require tht the mp ( 1 x, 2 x) ω x ( 1 x 2 x) be biliner, thus we hve the xioms ω x (c x 1 x 2 ) = cω x ( x 1 x 2 ) ω x (( x 1 + x 1 ) x 2 ) = ω x ( x 1 x 2 ) + ω x ( x 1 x 2 ) ω x ( x 1 c x 2 ) = cω x ( x 1 x 2 ) ω x ( x 1 ( x 2 + x 2 )) = ω x ( x 1 x 2 ) + ω x ( x 1 x 2 ) for ll c R nd x 1, x 2, x 1, x 2. These xioms re lso physiclly intuitive, though it my require little more effort to see this thn in the one-dimensionl cse. There is one dditionl importnt xiom we require, nmely tht ω x ( x x) = 0 (8) for ll x R n. This reflects the geometriclly obvious fct tht when 1 x = 2 x = x, the prllelogrm with dimensions x x is degenerte nd should thus experience zero net flux. Any continuous ssignment ω : x ω x tht obeys the bove xioms is clled 8 2-form. 8 There re severl other equivlent definitions of 2-form. For instnce, s hinted t erlier, 1- forms cn be viewed s sections of the cotngent bundle T R n, nd similrly 2-forms re sections of the exterior power V2 T R n of tht bundle. Similrly, expressions such s v w, where v, w T xr n re tngent vectors t point x, cn be given mening by using bstrct lgebr to construct the exterior power V2 T xr n, t which point (v, w) v w cn be viewed s biliner nti-symmetric mp from T xr n T xr n to V2 T xr n (indeed it is the universl mp with this properties). One cn lso construct forms using the mchinery of tensors.

7 DIFFERENTIAL FORMS AND INTEGRATION 7 By pplying (8) with x := 1 x+ 2 x nd then using severl of the bove xioms, we rrive t the fundmentl nti-symmetry property ω x ( x 1 x 2 ) = ω x ( x 2 x 1 ). (9) Thus swpping the first nd second vectors of prllelogrm cuses reversl in the flux cross tht prllelogrm; the ltter prllelogrm should then be considered to hve the reverse orienttion to the former. If ω is 2-form nd φ : [0, 1] 2 R n is continuously differentible function, we cn now define the integrl ω of ω ginst φ (or more precisely, the imge of the φ oriented squre [0, 1] 2 under φ) by the pproximtion ω ω xi ( x 1,i x 2,i ) (10) φ i where the imge of φ is (pproximtely) prtitioned into prllelogrms of dimensions x 1,i x 2,i bsed t points x i. We do not need to decide wht order these prllelogrms should be rrnged in, becuse ddition is both commuttive 9 nd ssocitive. One cn show tht the right-hnd side of (10) converges to unique limit s one mkes the prtition of prllelogrms incresingly fine, though we will not mke this precise here. We hve thus shown how to integrte 2-forms ginst oriented 2-dimensionl surfces. More generlly, one cn define the concept of k-form 10 on n n-dimensionl mnifold (such s R n ) for ny 0 k n nd integrte this ginst n oriented k-dimensionl surfce in tht mnifold. For instnce, 0-form on mnifold X is the sme thing s sclr function f : X R, whose integrl on positively oriented point x (which is 0-dimensionl) is f(x), nd on negtively oriented point x is f(x). By convention, if k k, the integrl of k-dimensionl form on k -dimensionl surfce is understood to be zero. We refer to 0-forms, 1-forms, 2-forms, etc. (nd forml sums nd differences thereof) collectively s differentil forms. Sclr functions enjoy three fundmentl opertions: ddition (f, g) f + g, pointwise product (f, g) fg, nd differentition f df, lthough the ltter is only obviously well-defined when f is continuously differentible. These opertions obey vrious reltionships, for instnce the product distributes over ddition f(g + h) = fg + fh nd differentition is derivtion with respect to the product: d(fg) = (df)g + f(dg). It turns out tht one cn generlise ll three of these opertions to differentil forms: one cn dd or tke the wedge product of two forms ω, η to obtin new forms 9 For some other notions of n integrl, such s tht of n integrl of connection with non-belin structure group, one loses commuttivity, nd so one cn only integrte long onedimensionl curves. 10 One cn lso define k-forms for k > n, but it turns out tht the multilinerity nd ntisymmetry xioms for such forms will force them to vnish, bsiclly becuse ny k vectors in R n re necessrily linerly dependent.

8 8 TERENCE TAO ω + η nd ω η; nd, if k-form ω is continuously differentible, one cn lso form the derivtive dω, which is k +1-form. The exct construction of these opertions requires little bit of lgebr nd is omitted here. However, we remrk tht these opertions obey similr lws to their sclr counterprts, except tht there re some sign chnges which re ultimtely due to the nti-symmetry (9). For instnce, if ω is k-form nd η is n l-form, the commuttive lw for multipliction becomes ω η = ( 1) kl η ω, nd the derivtion rule for differenttion becomes d(ω η) = (dω) η + ( 1) k ω (dη). A fundmentlly importnt, though initilly rther unintuitive 11 rule, is tht the differentition opertor d is nilpotent: d(dω) = 0. (11) The fundmentl theorem of clculus generlises to Stokes theorem dω = ω (12) S for ny oriented mnifold S nd form ω, where S is the oriented boundry of S (which we will not define here). Indeed one cn view this theorem (which generlises (1), (7)) s definition of the derivtive opertion ω dω; thus differentition is the djoint of the boundry opertion. (Thus, for instnce, the identity (11) is dul to the geometric observtion tht the boundry S of n oriented mnifold itself hs no boundry: ( S) =.) As prticulr cse of Stokes theorem, we see tht dω = 0 whenever S is closed mnifold, i.e. one with no boundry. S This observtion lets one extend the notions of closed nd exct forms to generl differentil forms, which (together with (11)) llows one to fully set up de Rhm cohomology. We hve lredy seen tht 0-forms cn be identified with sclr functions, nd in Eucliden spces 1-forms cn be identified with vector fields. In the specil (but very physicl) cse of three-dimensionl Eucliden spce R 3, 2-forms cn lso be identified with vector fields vi the fmous right-hnd rule 12, nd 3-forms cn be identified with sclr functions by vrint of this rule. (This is n exmple of Hodge dulity.) In this cse, the differentition opertion ω dω is identifible to the grdient opertion f f when ω is 0-form, to the curl opertion X X when ω is 1-form, nd the divergence opertion X X when ω is 2-form. Thus, for instnce, the rule (11) implies tht f = 0 nd ( X) = 0 for ny suitbly smooth sclr function f nd vector field X, while Stokes theorem (12), with this interprettion, becomes the Stokes theorems for integrls of curves nd surfces in three dimensions tht my be fmilir to you from severl vrible clculus. 11 It my help to view dω s relly being wedge product d ω of the differentition opertion with ω, in which cse (11) is forml consequence of (8) nd the ssocitivity of the wedge product. 12 This is n entirely rbitrry convention; one could just hve esily used the left-hnd rule to provide this identifiction, nd prt from some hrmless sign chnges here nd there, one gets essentilly the sme theory s consequence. S

9 DIFFERENTIAL FORMS AND INTEGRATION 9 Just s the signed definite integrl is connected to the unsigned definite integrl in one dimension vi (2), there is connection between integrtion of differentil forms nd the Lebesgue (or Riemnn) integrl. On the Eucliden spce R n one hs the n stndrd co-ordinte functions x 1, x 2,..., x n : R n R. Their derivtives dx 1,..., dx n re then 1-forms on R n. Tking their wedge product one obtins n n-form dx 1... dx n. We cn multiply this with ny (continuous) sclr function f : R n R to obtin nother n-form fdx 1... dx n. If Ω is ny open bounded domin in R n, we then hve the identity f(x)dx 1... dx n = f(x) dx Ω where on the left we hve n integrl of differentil form (with Ω viewed s positively oriented n-dimensionl mnifold), nd on the right we hve the Riemnn or Lebesgue integrl of f on Ω. If we give Ω the negtive orienttion, we hve to reverse the sign of the left-hnd side. This correspondence generlises (2). There is one lst opertion on forms which is worth pointing out. Suppose we hve continuously differentible mp Φ : X Y from one mnifold to nother (we llow X nd Y to hve different dimensions). Then of course every point x in X pushes forwrd to point Φ(x) in Y. Similrly, if we let v T x X be n infinitesiml tngent vector to X bsed t x, then this tngent vector lso pushes forwrd to tngent vector Φ v T Φ(x) (Y ) bsed t Φ(x); informlly speking, Φ v cn be defined by requiring the infinitesiml pproximtion Φ(x + v) = Φ(x) + Φ v. One cn write Φ v = DΦ(x)(v), where DΦ : T x X T Φ(x) Y is the derivtive of the severl-vrible mp Φ t x. Finlly, ny k-dimensionl oriented mnifold S in X lso pushes forwrd to k-dimensionl oriented mnifold Φ(S) in Y, lthough in some cses (e.g. if the imge of Φ hs dimension less thn k) this pushed-forwrd mnifold my be degenerte. We hve seen tht integrtion is dulity piring between mnifolds nd forms. Since mnifolds push forwrd under Φ from X to Y, we thus expect forms to pullbck from Y to X. Indeed, given ny k-form ω on Y, we cn define the pull-bck Φ ω s the unique k-form on X such tht we hve the chnge of vribles formul ω = Φ (ω). Φ(S) In the cse of 0-forms (i.e. sclr functions), the pull-bck Φ f : X R of sclr function f : Y R is given explicitly by Φ f(x) = f(φ(x)), while the pull-bck of 1-form ω is given explicitly by the formul S Ω (Φ ω) x (v) = ω Φ(x) (Φ v). Similrly for other differentil forms. The pull-bck opertion enjoys severl nice properties, for instnce it respects the wedge product, Φ (ω η) = (Φ ω) (Φ η), nd the derivtive, d(φ ω) = Φ (dω). By using these properties, one cn recover rther pinlessly the chnge-of-vribles formule in severl-vrible clculus. Moreover, the whole theory crries effortlessly over from Eucliden spces to other mnifolds. It is becuse of this tht the theory

10 10 TERENCE TAO of differentil forms nd integrtion is n indispensble tool in the modern study of mnifolds, especilly in differentil topology. Deprtment of Mthemtics, UCLA, Los Angeles CA E-mil ddress: to@mth.ucl.edu