This two-volume series arose out of work by the three authors over a number of years both in teaching various courses and also in their research endeavors.
　　Ihe present volume （Book II） is comprised of five chapters that continue the discussion of Book I. In Chapter 4， we examine the geometry of curves which are the solution space of a constant coefficient ordinary differential equation. We give necessary and sufficient conditions that the curves give a proper embedding and we examine when the total extrinsic curvature is finite. We examine similar questions for the total Gaussian curvature of a surface defined by a pair of ODEs and apply the Gauss-Bonnet Theorem to express the total Gaussian curvature in terms of the curves associated to the individual ODEs. We then examine the volume of a small geodesic ball in a Riemannian manifold. We show that if the scalar curvature is positive， then volume grows more slowly than it does in flat space while if the scalar curvature is negative， then volume grows more rapidly than it does in flat space. Chapter 4 concludes with a brief introduction to holomorphic and Kahler geometry.
　　Chapter 5 treats de Rham cohomology. Ihe basic properties are introduced and it is shown that de Rham cohomology satisfies the Eilenberg-Steenrod axioms; these are properties that all homology and cohomology theories have in common. We shall postpone until Chapter 8 a discussion of the Mayer-Vietoris sequence and the homotopy property as these depend upon some results in homological. algebra that willbe treated there. We determine the de Rham cohomology of the sphere and of real projective space. We introduce Clifford algebras and present the Hodge Decomposition Theorem. This is used to establish the Kunneth formula and Poincare duality. We treat the first Chern class in some detail and use it to determine the ring structure of the de Rham cohomology of complex projective space. A brief introduction to the higher Chern classes and the Pontrjagin classes is given.