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Elementary Differential Equations

Elementary Differential Equations - CHAPTER I ELEMENTARY...

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CHAPTER I ELEMENTARY DIFFERENTIAL GEOMETRY §1-§3. When a Euclidean space is stripped of its vector space structure and only its differentiable structure retained, there are many ways of piecing together domains of it in a smooth manner, thereby obtaining a so-called differentiable manifold. Local concepts like a differentiable function and a tangent vector can still be given a meaning whereby the manifold can be viewed "tangentially," that is, through its family of tangent spaces as a curve in the plane is, roughly speaking, determined by its family of tangents. This viewpoint leads to the study of tensor fields, which are important tools in local and global differential geometry. They form an algebra (M), the mixed tensor algebra over the manifold M. The alternate covariant tensor fields (the differential forms) form a submodule 9t(M) of (M) which inherits a 'multiplication from (M), the exterior multiplication. The resulting algebra is called the Grassmann algebra of M. Through the work of E. Cartan the Grassmann algebra with the exterior differentiation d has become an indispensable tool for dealing with submanifolds, these being analytically described by the zeros of differential forms. Moreover, the pair ((M), d) determines the cohomology of Al via de Rham's theorem, which however will not be dealt with here. §4-§8. The concept of an affine connection was first defined by Levi-Civita for Riemannian manifolds, generalizing significantly the notion of parallelism for Euclidean spaces. On a manifold with a countable basis an affine connection always exists (see the exercises following this chapter). Given an affine connection on a manifold M there is to each curve y(t) in M associated an isomorphism between any two tangent spaces M,(,,) and My(t,). Thus, an affine connection makes it possible to relate tangent spaces at distant points of the manifold. If the tangent vectors of the curve y(t) all correspond under these isomorphisms we have the analog of a straight line, the so-called geodesic. The theory of affine connections mainly amounts to a study of the mappings Exp,: M, - M under which straight lines (or segments of them) through the origin in the tangent space M, correspond to geodesics through p in M. Each mapping Exp, is a diffeomorphism of a neigh- borhood of 0 in M, into M, giving the so-called normal coordinates at p. 5
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§1. Manifolds Let R" n and Rn denote two Euclidean spaces of m and n dimensions, respectively. Let O and O' be open subsets, 0 C R m , O' C RI and suppose p is a mapping of 0 into 0'. The mapping q is called differen- tiable if the coordinates y 1 ('(p)) of p(p) are differentiable (that is, inde- finitely differentiable) functions of the coordinates xi(p), p e 0. The mapping is called analytic if for each point p E 0 there exists a neigh- borhood U of p and n power series Pi (I j < n) in m variables such that y(q(q)) = P,(x 1 (q) - x,(p), ..., x,,(q) - xm(p)) (1 < j < n) for q e U. A differentiable mapping p: O -, O' is called a diffeomorphism of O onto O' if (0) = O', p is one-to-one, and the inverse mapping T - is differentiable. In the case when n
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Elementary Differential Equations - CHAPTER I ELEMENTARY...

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