---

This book provides an introduction to the highly developed and multi-faceted subject of mathematical analysis. It rigorously treats topics such as multivariable differential calculus, Lebesgue integral, vector calculus and differential equations.

---

The book begins at the level of an undergraduate student assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, Lebesgue integral, vector calculus and differential equations. After having built on a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis, as understood by a mathematician today._

---

Preface.- Introduction.- Part 1. A Rigorous Approach to Advanced Calculus.- 1. Preliminaries.- 2. Metric and Topological Spaces I.- 3. Multivariable Differential Calculus.- 4. Integration I: Multivariable Riemann Integral and Basic Ideas toward the Lebesgue Integral.- 5. Integration II: Measurable Functions, Measure and the Techniques of Lebesgue Integration.- 6. Systems of Ordinary Differential Equations.- 7. System of Linear Differential Equations.- 8. Line Integrals and Green´s Theorem.- Part 2. Analysis and Geometry.- 9. An Introduction to Complex Analysis.- 10. Metric and Topological Spaces II.- 11. Multilinear Algebra.- 12. Smooth Manifolds, Differential Forms and Stokes´ Theorem.- 13. Calculus of Variations and the Geodesic Equation.- 14. Tensor Calculus and Riemannian Geometry.- 15. Hilbert Spaces I: Definitions and Basic Properties.- 16. Hilbert Spaces II: Examples and Applications.- Appendix A. Linear Algebra I: Vector Spaces.- Appendix B. Linear Algebra II: More about Matrices.- Bibliography.- Index of Symbols.- Index._

---

From the reviews:

"The book is intended as a second-year course of mathematical analysis for advanced undergraduate students. ... The volume is addressed to undergraduate students seriously interested in mathematics and is accessible to students before they start taking graduate classes. Researchers in pure and applied nonlinear analysis will find interesting material in this volume." (Teodora-Liliana Radulescu, zbMATH, Vol. 1279, 2014)

"The authors included in their book some topics from topology, calculus of real functions of one and several real variables ... elements of functional analysis, as well as some applications. ... the present well written book is a valuable addition to the existing ones on similar topics. It can be used by graduate students in mathematics and researchers in mathematics and other areas ... . The instructors can recommend the book as a supplementary material for their courses." (S. Cobzas, Studia Universitatis Babes-Bolyai, Math, Vol. 58 (4), 2013)

---