© 2014

Multivariate Calculus and Geometry


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Seán Dineen
    Pages 13-23
  3. Seán Dineen
    Pages 25-34
  4. Seán Dineen
    Pages 35-45
  5. Seán Dineen
    Pages 47-53
  6. Seán Dineen
    Pages 55-67
  7. Seán Dineen
    Pages 69-81
  8. Seán Dineen
    Pages 83-92
  9. Seán Dineen
    Pages 93-102
  10. Seán Dineen
    Pages 103-120
  11. Seán Dineen
    Pages 121-134
  12. Seán Dineen
    Pages 135-147
  13. Seán Dineen
    Pages 149-159
  14. Seán Dineen
    Pages 161-178
  15. Seán Dineen
    Pages 179-191
  16. Seán Dineen
    Pages 193-205
  17. Seán Dineen
    Pages 207-215
  18. Seán Dineen
    Pages 217-227
  19. Back Matter
    Pages 229-257

About this book


Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It 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.


Curvature and Torsion of Curves Double and Triple Integrals Frenet-Serret Equations Gaussian Curvature Geodesic Curvature Lagrange Multipliers Line Integrals Stokes Theorem Surface Integrals The Hessian The Weingarten Mapping

Authors and affiliations

  1. 1.School of Mathematical SciencesUniversity College DublinDublinIreland

About the authors

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).

Bibliographic information

  • Book Title Multivariate Calculus and Geometry
  • Authors Seán Dineen
  • Series Title Springer Undergraduate Mathematics Series
  • Series Abbreviated Title SUMS
  • DOI
  • Copyright Information Springer-Verlag London 2014
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-1-4471-6418-0
  • eBook ISBN 978-1-4471-6419-7
  • Series ISSN 1615-2085
  • Series E-ISSN 2197-4144
  • Edition Number 3
  • Number of Pages XIV, 257
  • Number of Illustrations 103 b/w illustrations, 0 illustrations in colour
  • Topics Mathematics, general
  • Buy this book on publisher's site


“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 Janiš, zbMATH 1312.26001, 2015)