MULTIPLE INTEGRALS AND VECTOR ANALYSIS: for Engineers and Scientists (VECTORS AND APPLICATIONS)

Multiple integrals and vector analysis is the branch of applied mathematics used for the formulation and the solution of problems in Engineering, Physics and Geometry, when two or three space dimensions are involved. For instance, the fundamental laws in fluid dynamics, heat transfer, electromagnetic fields, strength of materials, and other areas of natural sciences are formulated and analyzed by utilizing methods and techniques of vector analysis.The book contains 12 chapters, as shown analytically in the table of contents. Chapter 1 is devoted to a systematic summary of fundamental definitions, properties and operations of vectors, (dot and cross product of vectors), coordinate systems, (rectangular, cylindrical, and spherical) and scalar and vector fields. Chapter 2 is devoted to the differentiation and the integration of vector functions of a scalar argument. Double integrals, triple integrals, line integrals of the first and second type and surface integrals of the first and second type are studied in considerable depth and details in chapters 3, 4, 5, and 6 respectively. Fundamental definitions, the gradient of scalar fields, directional derivatives and the physical meaning of the gradient, divergence and curl of vector fields, and the definition of the del operator (which combines both differential and vectorial properties), are introduced in chapter 7. Two of the most important theorems in vector analysis, the divergence theorem of Gauss-Ostrogradsky and Stokes’ theorem are studied in chapter 8. Based on these two theorems, the physical meaning of the divergence and the curl of vector fields are obtained. Chapter 9 is devoted to the development of the Laplacian of scalar and vector fields, (Laplace’s equation) and to the derivation of Green’s integral formulas. Expressions for the gradient, the divergence, the curl and the Laplacian in cylindrical and in spherical coordinates are obtained in chapter 10. The same chapter also includes some useful vector identities, used frequently in vector analysis. Two important types of vector fields, solenoidal and irrotational or conservative fields and their related properties are studied in chapter 11. Finally chapter 12 is devoted to the derivation of expressions of the gradient, the divergence, the curl and the Laplacian in general, orthogonal, curvilinear coordinates. A great variety of problems arising in Engineering, Physics and Geometry can be solved with the aid of methods and techniques developed in vector analysis. Centers of gravity, moments of inertia, volumes of revolution, work of variable forces, the three Kepler’s laws of planetary motion, gravitational attraction between two bodies, motion of charged particles in uniform magnetic fields, the fundamental laws of static electric fields, Laplace’s and Poisson’s equation, the derivation of the wave equation for electric and magnetic fields starting from Maxwell’s equations, to name just a few, are some of the applications presented in this book. The book contains 190 illustrative worked out examples and 455 graded problems for solution. These examples and problems have been selected to help students develop a solid background in multiple integrals and vector analysis, to broaden their knowledge and analytical skills and finally to prepare them to pursue successfully more advanced studies in Engineering and Mathematics.