Description
Book SynopsisThis book gives a treatment of exterior differential systems. It will in clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficien
Table of ContentsI. Preliminaries.- §1. Review of Exterior Algebra.- §2. The Notion of an Exterior Differential System.- §3. Jet Bundles.- II. Basic Theorems.- §1. Probenius Theorem.- §2. Cauchy Characteristics.- §3. Theorems of Pfaff and Darboux.- §4. Pfaffian Systems.- §5. Pfaffian Systems of Codimension Two.- III. Cartan-Kähler Theory.- §1. Integral Elements.- §2. The Cartan-Kähler Theorem.- §3. Examples.- IV. Linear Differential Systems.- §1. Independence Condition and Involution.- §2. Linear Differential Systems.- §3. Tableaux.- §4. Tableaux Associated to an Integral Element.- §5. Linear Pfaffian Systems.- §6. Prolongation.- §7. Examples.- §8. Families of Isometric Surfaces in Euclidean Space.- V. The Characteristic Variety.- §1. Definition of the Characteristic Variety of a Differential System.- §2. The Characteristic Variety for Linearc Pfaffian Systems; Examples.- §3. Properties of the Characteristic Variety.- VI. Prolongation Theory.- §1. The Notion of Prolongation.- §2. Ordinary Prolongation.- §3. The Prolongation Theorem.- §4. The Process of Prolongation.- VII. Examples.- §1. First Order Equations for Two Functions of Two Variables.- §2. Finiteness of the Web Rank.- §3. Orthogonal Coordinates.- §4. Isometric Embedding.- VIII. Applications of Commutative Algebra and Algebraic Geometry to the Study of Exterior Differential Systems.- §1. Involutive Tableaux.- §2. The Cartan-Poincaré Lemma, Spencer Cohomology.- §3. The Graded Module Associated to a Tableau; Koszul Homology.- §4. The Canonical Resolution of an Involutive Module.- §5. Localization; the Proofs of Theorem 3.2 and Proposition 3.8.- §6. Proof of Theorem 3.8 in Chapter V; Guillemin’s Normal Form.- §7. The Graded Module Associated to a Higher Order Tableau.- IX. Partial Differential Equations.- §1. An Integrability Criterion.- §2. Quasi-Linear Equations.- §3. Existence Theorems.- X. Linear Differential Operators.- §1. Formal Theory and Complexes.- §2. Examples.- §3. Existence Theorems for Elliptic Equations.
