By Kichoon Yang

This monograph offers a concise but basic account of external differential method concept in order that it may be fast utilized to difficulties. the 1st a part of the monograph, Chapters 1-5, bargains with the final thought: the Cartan-Kaehler theorem is proved, the notions of involution and prolongation are rigorously laid out, quasi-linear differential platforms are tested intimately, and specific examples of the Spencer cohomology teams and the attribute sort are given. the second one a part of the monograph, Chapters 6 and seven, offers with functions to difficulties in differential geometry: the isometric embedding theorem of Cartan-Janet and its quite a few geometric ramifications are mentioned, an explanation of the Andreotti-Hill theorem at the O-R embedding challenge is given, and embeddings of summary projective constructions are mentioned.

For researchers and graduate scholars who would favor an outstanding creation to external differential platforms. This quantity can also be relatively worthwhile to these whose paintings contains differential geometry and partial differential equations.

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**Additional resources for Exterior Differential Systems and Equivalence Problems**

**Example text**

The ideal I( dO') is then generated by a two-form IP in 2r variables y1, "', y2rj dO' differs from IP by a factor. Write dO' == blP, b == b(wI, "', w2r+1) t o. ELEMENTARY DIFFERENTIAL SYSTEMS 36 Now (dO,)r = br~r t- 0, hence ~r = for some c = c(y\ ... , y2r) t- C dyl A ... A dy2r O. Using the fact that (d O,)r is a closed form, db A dyl A ... A dy2r This means that b is a function of the yi,s. = o. So dO' is a form in the yi,s. dO' is closed there is a nonzero I-form 'Y in the yi,s with d'Y = dO'.

We have So + sl = sl = dim span {iy~ = iyE}, where v E Mx is such that [v] E Gl (M) is a regular integral element. To put it another way, the first character sl is the dimension of the span of i yE C M*, x where v is taken to be a generic element in the tangent bundle TM. We write v as viC aJ ax\ and compute iy E2: . lylPl = vIdx3 - v3dxl + v2dx4 · 2 3 3 2 lyIP2 = v dx - v dx , · 2 5 5 2 ly IP3 = v dx - v dx , · 5 6 6 5 lyIP4 = v dx - v dx . , v 0, 0, 0, 0, -v 6, ~ ]. v5 Clearly sl is the rank of the polar matrix, where v is generic.

P-2)sp-2 + (p-l) up-I' The fibre 'If~l(E:-l) corresponds to all regular integral p-elements containing In particular, the fibre 'If-ll (EP-l) is an open set in the projective space x EP-l. x IP(H(EP-l)/EP-l): recall that v E H(EP-l) if and only if either v E EP-l or v x x x x together with EP-l span an integral EP. Since x x dim H(EP-l) = n - (s + .. ·+s x 0 p-l ) it follows that dim 'If-ll (EP-l) = u , and x P 1 (E) = dim 1 l(E) + dim 'If-ll (EP-l) P- ,P Px = n + sl + ... + (p-2)sp_2 + (p-l)up_ l + up dim 1 = n + sl + + (p-2)sp-2 + (p-l)sp-l + pUp + (p-l).