Spain Barry. Vector Analysis

2nd Edition. — D.Van Nostrand Company Ltd., 1965. — 114 p.The treatment in this book emphasizes at all times that a vector is an entity in itself and should not be considered merely as a triple of numbers. Consequently, proofs by means of Cartesian coordinates are avoided.The first chapter contains the basic elements of vector algebra, which are applied in the following chapter to the geometry of the straight line and the plane. The third chapter discusses the fundamentals of vector calculus, and applications to differential geometry are made in the fourth chapter. The contents of this chapter, with the exception of the Frenet formulae, curvature and torsion, are required for later developments of the theory.Line integrals, surface integrals and volume integrals play an important role in vector analysis. Many students will meet these concepts before they have had formal courses in them. Consequently the fifth chapter presents a concise account of these topics.The following three chapters are respectively devoted to the gradient, divergence and curl of a vector. Of these, the first is defined in terms of the directional derivative whilst the latter two are defined by limits of integrals. In this way the theory is developed independently of any coordinate system. The operator V is not introduced, as the author has observed that this is a dangerous tool in the hands of the inexperienced.There follow chapters on Stokes’s theorem, Green’s theorems and orthogonal curvilinear coordinate systems.