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This book covers multivariate calculus with a combination of geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. It features many practical examples involving problems of several variables.

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Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook not only follows this programme, but additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations.

In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions.

Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

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Introduction to Differentiable Functions.- Level Sets and Tangent Spaces.- Lagrange Multipliers.- Maxima and Minima on Open Sets.- Curves in Rn.- Line Integrals.- The Frenet-Serret Equations.- Geometry of Curves in R3.- Double Integration.- Parametrized Surfaces in R3.- Surface Area.- Surface Integrals.- Stokes´ Theorem.- Triple Integrals.- The Divergence Theorem.- Geometry of Surfaces in R3.- Gaussian Curvature.- Geodesic Curvature.

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"The book is very useful for those who wish to learn the theory properly. ... the book is very clearly written-the theory is nicely presented with important topics being well explained and illustrated with examples. ... Each chapter begins with an outline of its content, and ends with suitably constructed exercises, with solutions given at the end of the book. ... it is also an excellent reference text on multivariate calculus and the basics in differential geometry." (Peter Shiu, The Mathematical Gazette, Vol. 100 (547), 2016)

"A textbook aimed at undergraduate mathematics students. ... The text is accompanied with a large number of figures and explanatory text. Each chapter is concluded by a collection of exercises of both routine and more theoretical nature. The textbook is written in a readable way, especially it is one of rare cases of multivariate calculus texts consequently linked to the geometric roots of the subject." (Vladimír Janis, zbMATH 1312.26001, 2015)

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Sean Dineen taught for many years at University College Dublin where he is now an emeritus professor. He is the author of many research articles and monographs and of a number of successful and popular textbooks including: Analysis; A Gateway to Understanding Mathematics (World Scientific, 2012) and Probability Theory in Finance: A Mathematical Guide to the Black-Scholes Formula (AMS, Second Edition, 2013).

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