The first part of this book reviews some key topics on multi-variable advanced calculus. The approach presented includes detailed and rigorous studies on surfaces in Rn which comprises items such as differential forms and an abstract version of the Stokes Theorem in Rn. The conclusion section introduces readers to Riemannian geometry, which is used in the subsequent chapters. The second part reviews applications, specifically in variational quantum mechanics and relativity theory. Topics such as a variational formulation for the relativistic Klein-Gordon equation, the derivation of a variational formulation for relativistic mechanics firstly through (semi)-Riemannian geometry are covered. The second part has a more general context. It includes fundamentals of differential geometry. The later chapters describe a new interpretation for the Bohr atomic model through a semi-classical approach. The book concludes with a classical description of the radiating cavity model in quantum mechanics.

The book discusses basic concepts of functional analysis, measure and integration theory, calculus of variations and duality and its applications to variational problems of non-convex nature, such as the Ginzburg-Landau system in superconductivity, shape optimization models, dual variational formulations for micro-magnetism and others. Numerical Methods for such and similar problems, such as models in flight mechanics and the Navier-Stokes system in fluid mechanics have been developed through the generalized method of lines, including their matrix finite dimensional approximations. It concludes with a review of recent research on Riemannian geometry applied to Quantum Mechanics and Relativity. The book will be of interest to applied mathematicians and graduate students in applied mathematics. Physicists, engineers and researchers in related fields will also find the book useful in providing a mathematical background applicable to their respective professional areas.

The first part of this book reviews some key topics on multi-variable advanced calculus. The approach presented includes detailed and rigorous studies on surfaces in Rn which comprises items such as differential forms and an abstract version of the Stokes Theorem in Rn. The conclusion section introduces readers to Riemannian geometry, which is used in the subsequent chapters. The second part reviews applications, specifically in variational quantum mechanics and relativity theory. Topics such as a variational formulation for the relativistic Klein-Gordon equation, the derivation of a variational formulation for relativistic mechanics firstly through (semi)-Riemannian geometry are covered. The second part has a more general context. It includes fundamentals of differential geometry. The later chapters describe a new interpretation for the Bohr atomic model through a semi-classical approach. The book concludes with a classical description of the radiating cavity model in quantum mechanics.

The book discusses basic concepts of functional analysis, measure and integration theory, calculus of variations and duality and its applications to variational problems of non-convex nature, such as the Ginzburg-Landau system in superconductivity, shape optimization models, dual variational formulations for micro-magnetism and others. Numerical Methods for such and similar problems, such as models in flight mechanics and the Navier-Stokes system in fluid mechanics have been developed through the generalized method of lines, including their matrix finite dimensional approximations. It concludes with a review of recent research on Riemannian geometry applied to Quantum Mechanics and Relativity. The book will be of interest to applied mathematicians and graduate students in applied mathematics. Physicists, engineers and researchers in related fields will also find the book useful in providing a mathematical background applicable to their respective professional areas.

This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.

This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.