This book examines the theory of vector functions and their applications in the physical sciences. The main part of the content of the book is aimed at first and second-year students in departments of natural sciences. The primary goal of the authors is to introduce students to the in-depth study of fundamental computational methods and their application in fields such as Theoretical Physics, Analytical Mechanics, Applied Mathematics, and others.

In more detail:

In Chapter 1, scalar functions of multiple variables are studied, and the concepts of partial derivatives and gradient are presented. Chapter 2 introduces the basic concepts of vectors and explores their properties. Chapter 3 presents vector functions of a single variable and examines curves in two and three dimensions.

Chapter 4 introduces vector functions of multiple variables, studies the theory of surfaces, and presents the concepts of gradient, divergence, and curl. Chapter 5 analyzes curvilinear coordinate systems, describes the transformations from curvilinear to Cartesian coordinates and vice versa, and provides all types of operators such as divergence, Laplacian, etc., in spherical and cylindrical coordinates.

In Chapter 6, multiple integrals are studied. Chapter 7 presents line and surface integrals, explores the physical concepts of conservative forces, and introduces scalar and vector potentials of vortical and tubular fields. In Chapter 8, the fundamental theorems of vector analysis, divergence, Stokes, Green theorems, etc., are introduced. Finally, in Chapter 9, the concept of tensors is introduced.

The last section of each chapter includes computations with the help of symbolic language programs that require the use of a computer. The examples and exercises have been selected with the aim of reinforcing the theory and presenting applications that pertain to useful physical concepts. Thus, numerous examples and exercises from Mechanics, Hydrodynamics, Electrodynamics, etc., are included.

Georgios Leontaris completed his undergraduate studies in the Physics department of Aristotle University of Thessaloniki and his doctoral dissertation at the University of Ioannina. He has worked as a research associate at the CERN nuclear research center in Geneva and as a visiting professor at the Ecole Polytechique of Palaiseau and at LPTHE Jussieu Sorbonne University in Paris. Nikolaos Bakas studied Physics at the Physics department of the National and Kapodistrian University of Athens and Applied Mathematics at Harvard University, where he obtained his master's and doctoral degrees. He has worked as a postdoctoral researcher in the Physics Department of EKPA. In 2019, he was elected as an Assistant Professor in the Physics department of the University of Ioannina.