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Based on notes written during the author's many years of teaching, Analysis in Euclidean Space mainly covers Differentiation and Integration theory in several real variables, but also an array of closely related areas including measure theory, differential geometry, classical theory of curves, geometric measure theory, integral geometry, and others. With several original results, new approaches and an emphasis on concepts and rigorous proofs, the book is suitable for undergraduate students, particularly in mathematics and physics, who are interested in acquiring a solid footing in analysis and expanding their background. There are many examples and exercises inserted in the text for the student to work through independently. Analysis in Euclidean Space comprises 21 chapters, each with an introduction summarizing its contents, and an additional chapter containing miscellaneous exercises. Lecturers may use the varied chapters of this book for different undergraduate courses in analysis. The only prerequisites are a basic course in linear algebra and a standard first-year calculus course in differentiation and integration. As the book progresses, the difficulty increases such that some of the later sections may be appropriate for graduate study.
About the Author.
Introduction.
Euclidean Space.
Continuous Functions.
Coordinate Systems, Curves and Surfaces.
Differentiation.
Higher-Order Derivatives,
The Inverse and Implicit Function Theorems.
Regular Sub-Manifolds.
Ordinary Differential Equations.
Linear Partial Differential Equations.
Orthogonal Families of Curves and Surfaces.
Measuring Sets: The Riemann Integral.
The Lebesgue Integral.
Fubini’s Theorem and Change of Variables.
Integration on Sub-Manifolds.
Geometric Measure Theory and Integral Geometry.
Line Integrals and Flux.
The Basic Theorems of Vector Analysis.
Conservative and Solenoidal Fields.
Harmonic Functions.
The Divergence and Rotational Equations, Poisson’s Equation.
The Dirichlet and Neumann Problems.
Additional Exercises.
Bibliography.
Index