Advanced calculus in R^n

Resumo

Mathematics plays a central role in modern science. Beyond its intrinsic value as a discipline with its own concepts, theories, and open problems, mathematics permeates essentially all areas of human knowledge. Many mathematical theories have their roots in natural phenomena, and in turn have driven the remarkable development of physics, engineering, computer science, and many other fields.
From this perspective, mathematics should not be viewed as an isolated subject, but rather as a unifying language and framework that connects and clarifies different scientific disciplines.
This book aims to present key topics in mathematical analysis in Rn, together with illustrative examples and figures. Our goal is to help students from a variety of programs—
mathematics, physics, engineering, and related areas— develop a solid and intuitive understanding of the basic tools of real and vector analysis, while maintaining full rigor. The material grew out of lecture notes for courses taught over several years at the State University of Maringá. We have tried to balance conceptual clarity, motivating examples, and detailed proofs, so that the text can be used both for self-study and as a companion to a classroom course.
We hope that this book will contribute to the reader’s mathematical maturity and stimulate further study in analysis, differential equations, geometry, and beyond.