Kenneth Kuttler

472 CHAPTER 18. DIFFERENTIAL FORMS

all of Rk. Now let I denote an ordered list of k indices taken from {1,2, · · · , p}. ThusI = (i1, · · · , ik). Then

det(

drI

)≡ det

xi1,u1 xi1,u2 · · · xi1,ukxi2,u1 xi2,u2 · · · xi2,uk

......

...xik,u1 xik,u2 · · · xik,uk

≡ ∂(xi1 , · · ·xik

)∂ (u1, · · · ,uk)

It is the same as det(DrI

)where rI has values in Rp and is obtained by keeping the rows

of r in the order determined by I and leaving out the other rows. More generally, supposeI is an ordered list of l indices and that J is an ordered list of l indices. Then

det(

drI

duJ

)≡ det

xi1,u j1

xi1,u j2· · · xi1,u jl

xi2,u j1xi2,u j2

· · · xi2,u jl...

......

xil ,u j1xil ,u j2

· · · xil ,u jl

≡ ∂(xi1 , · · ·xil

)∂(u j1 , · · · ,u jl

) , x= r (u)

Now with this definition, here is the generalization of the differential forms defined incalculus.

Definition 18.0.2 A differential form of order k is ω ≡ ∑I aI (x)dxI where

dxI ≡ dxi1 ∧dxi2 ∧·· ·∧dxik

To save space, let [a,b] ≡ ∏kk=1 [a j,b j] ,(a,b) ≡ ∏

kk=1 (a j,b j) etc. For I = (i1, · · · , ik) ,∫

(·) ω is a function mapping functions r in C1(

∏kj=1 [a j,b j] ,Rp

)or Lipschitz functions to

R , defined by∫r

ω ≡∫[a,b]

∑I

aI (r (u))det(

dxI (u)

)dmk =

∫[a,b]

∑I

aI (r (u))det(

drI (u)

)dmk

The sum is taken over all ordered lists of indices from {1, · · · , p}. Note that if there are any

repeats in an ordered list I, then det(

drI(u)du

)= 0 and so it suffices to consider the sum

only over lists of indices in which there are no repeats. Thus the sum can be consideredto consist of no more than P(p,k) terms where this denotes the permutations of p thingstaken k at a time.

Consider the free Abelian group of mappings having some specified regularity from agiven [a,b] to Rp. This consists of finite sums of the form ∑mrr and one can define∫

∑mrrω ≡∑mr

∫r

Thus the integral defined on such an r can be extended to give meaning to an arbitraryelement of this free Abelian group.

Actually, if I,J are the same set of indices, listed in different order, then det(

dxI(u)du

)will be±det

(dxJ(u)

)so you could always write the differential form in terms of sums over

472 CHAPTER 18. DIFFERENTIAL FORMSall of R*. Now let I denote an ordered list of k indices taken from {1,2,---,p}. ThusT= (i,,-++ ,ig). ThenNijuu, Xipuy Xi ydr! Xinuy Minug °° Xin, 7) (xi, yt Xi)det | —— ] =det : : : = — +du : : : O (uy,+++ ,Ug)Nig — Xiguy 0 NigsgIt is the same as det (Dr! ) where r! has values in R? and is obtained by keeping the rowsof r in the order determined by I and leaving out the other rows. More generally, supposeI is an ordered list of | indices and that J is an ordered list of | indices. ThenXiq uj, Xiy uj, us Xiy uj,dr! _ Xin ,uj, Xin ,uj, —_ Xin,ujy _ ] (xi, an Xi)det | —— ] = det . . . =o. x=r(u)uy : : O (uj,,--+ Wj.)Xij uj, Xij,uj ue Xipu jyNow with this definition, here is the generalization of the differential forms defined incalculus.Definition 18.0.2 4 differential form of order k is © =Y,q (x) da! whereda! = dx;, N\dxj, \+++ \ dXxi,To save space, let [a,b] = Ti, [a;,b;],(a,b) = Tit, (aj,bj) etc. For I = (i,--+ ix),kSo @ is a function mapping functions r in C! (n [aj,bj|,R? ) or Lipschitz functions toj=lR, defined byda! ) (4 )o= ay (r (w)) det dm = | ay (r (w)) det dm,f Low (r(a)) ( du fay et) du) *The sum is taken over all ordered lists of indices from {1,--- , p}. Note that if there are anyIrepeats in an ordered list I, then det (42) = 0 and so it suffices to consider the sumonly over lists of indices in which there are no repeats. Thus the sum can be consideredto consist of no more than P(p,k) terms where this denotes the permutations of p thingstaken k at a time.Consider the free Abelian group of mappings having some specified regularity from agiven [a,b] to R?. This consists of finite sums of the form Ym,7r and one can defineThus the integral defined on such an r can be extended to give meaning to an arbitraryelement of this free Abelian group.Actually, if 7,J are the same set of indices, listed in different order, then det (“e2)will be + det (4) so you could always write the differential form in terms of sums over