Advanced Vector Analysis for Scientists and Engineers

Vector analysis is one of the most useful branches of mathematics. It is a highly scientific field that is used in practical problems arising in engineering and applied sciences. Based on notes gathered from many years of teaching vector calculus, the main purpose of the book is to illustrate the application of vector calculus to physical problems. The theory is explained elegantly and clearly and there is an abundance of solved problems to manifest the application of the theory. The beauty of this book is the richness of practical applications. There are eight chapters, each of which contains ample exercises. The book concludes with three appendices. Appendix A contains solutions to selected exercises, Appendix B contains a quick summary of important vector formulae and Appendix C contains the brief summary of the historical background of vector analysis. This book is suitable for a one-semester course for senior undergraduates and junior graduate students in science and engineering. It is also suitable for the scientists and engineers working on practical problems.

Table of Contents:
Chapter 1: The algebra of vectors Introduction; Addition and subtraction of vectors; Scalar and vector products; Scalar and vector projections; Cartesian frame of reference; Vector algebra using coordinates; Mixed product in coordinate form; Vector representation using coordinates; Lines and planes using vector algebra; The equation of a plane; Summary; Exercises Chapter 2: Vector functions of one variable Introduction; Vector differentiation; Geometric interpretation of R'; Higher-order derivatives; Curves, length and arc length; Motion on a curve, velocity and acceleration vectors; Curvature, components of acceleration; Curvature, tangential and normal components of acceleration; Equation of an osculating plane; Summary; Exercises Chapter 3: Partial derivatives of functions of several variables Introduction; Partial derivatives; Iterated partial derivatives; Solitary wave solution; Implicit functions and the concept of Jacobians; Summary; Exercises Chapter 4: The Del operator Introduction; Gradient of a scalar function; Gradient characterizes maximum increase; Tangent planes and normal lines; Tangent plane; Normal lines; Divergence and curl of a vector field; Physical interpretation of divergence; Physical interpretation of the curl; The Laplacian operator; Vector identities; Summary; Exercises Chapter 5: Line, surface and volume integrals Introduction; Line integrals and vector functions; Work: interpretation of line integrals as ubiquitous; Line integrals independent of path; Conservative vector fields; Surface integrals; Orientation of a surface; Volume integration; Triple integrals in cylindrical coordinates; Triple integrals in spherical coordinates; Summary; Exercises Chapter 6: Integral theorems Introduction; Green's theorem; Region with holes; Integrals over vector fields; Green's theorem in vector form; Stokes' theorem in 3D; The divergence theorem; Summary; Exercises Chapter 7: Orthogonal curvilinear coordinate systems Introduction; Derivation of curvilinear coordinates; The gradient in generalized curvilinear coordinates; The divergence in orthogonal curvilinear coordinates; The curl in orthogonal curvilinear coordinates; Potentials; Summary; Exercises Chapter 8: Applications Introduction; Acceleration vector; Continuity equation of fluid flow; Euler's equation of motion; Bernoulli's equation; Navier-Stokes equation: equation of vorticity; Continuity equation and heat conduction; Poisson's equation; Vectors in electromagnetic theory; The continuity equation again; Maxwell's equations for electromagnetic fields; The Poynting vector; Waves solutions of Maxwell's equations; Ocean wave interactions; Solution techniques for Snl; Graphical simulation of a vector field; A boundary-initial value problem of the heat equation; Irrotational flow of an incompressible and inviscid fluid over a stationary ellipsoid; Summary; Exercises