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Fundamentals of Advanced Mathematics V3 - 1st Edition - ISBN: 9781785482502, 9780081023860

Fundamentals of Advanced Mathematics V3

1st Edition

Author: Henri Bourles
eBook ISBN: 9780081023860
Hardcover ISBN: 9781785482502
Imprint: ISTE Press - Elsevier
Published Date: 18th September 2019
Page Count: 424
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Fundamentals of Advanced Mathematics, Volume Three, begins with the study of differential and analytic infinite-dimensional manifolds, then progresses into fibered bundles, in particular, tangent and cotangent bundles. In addition, subjects covered include the tensor calculus on manifolds, differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds), an analysis on Lie groups, the Haar measure, the convolution of functions and distributions, and the harmonic analysis over a Lie group. Finally, the theory of connections is (linear connections, principal connections, and Cartan connections) covered, as is the calculus of variations in Lagrangian and Hamiltonian formulations.

This volume is the prerequisite to the analytic and geometric study of nonlinear systems.

Key Features

  • Includes sections on differential and analytic manifolds, vector bundles, tensors, Lie derivatives, applications to algebraic topology, and more
  • Presents an ideal prerequisite resource on the analytic and geometric study of nonlinear systems
  • Provides theory as well as practical information


Graduate students in Systems Theory, Robotics, Physics or Mathematics; research engineers in Automatic control and/or robotics; assistant professors and professors in Automatic control and/or robotics

Table of Contents

  1. Differential and analytic manifolds
    2.1. Manifolds
    2.2. Tangent vectors
    2.3. Tangent linear mappings and submanifolds
    2.4. Lie groups
    2. Fibered bundles
    2.1. Tangent bundle and cotangent bundle
    2.2. Fibrations
    2.3. Vector bundles
    2.4. Manifolds of mappings
    3. Tensor calculus on manifolds
    2.1. Tensors
    2.2. Tensor fields
    2.3. Differential forms
    4. Differential and integral calculus on manifolds
    4.1. Distributions and differential operators
    4.2. Lie derivative
    4.3. Exterior differential
    4.4. Stokes formula and its applications
    4.5. Elements of algebraic topology
    4.6. Integral curves and integral manifolds
    5. Connections
    5.1. Linear connections on a vector bundle
    5.2. Principal connexions
    6. Calculus of variations and optimal control
    6.1. Minima
    6.2. Calculus of variations
    6.3. Optimal control


No. of pages:
© ISTE Press - Elsevier 2019
18th September 2019
ISTE Press - Elsevier
eBook ISBN:
Hardcover ISBN:

About the Author

Henri Bourles

Henri Bourlès is Full Professor and Chair at the Conservatoire National des Arts et Métiers, Paris, France.

Affiliations and Expertise

Conservatoire National des Arts et Metiers, France


"The present volume is the third one of a series which presents the fundamental elements of advanced mathematics that is at the basis of a number of contemporary scientific methods. More precisely, it deals with differential and integral calculus in their local and global components. The book is designed not only for mathematicians, but also for everyone who uses mathematics and needs to understand the control of nonlinear systems (in particular physicists and engineers). The ambitious goal is achieved also thanks to an excellent organization of the topics and the use of a very clear and understandable language. Interesting short historical notes introduce the different topics and help to frame the evolution of concept. The exposition is illustrated with some figures that help a lot in understanding the not easy topics. Very useful attachments are provided: a careful list of notation and term indeces, a reach bibliography, a list of cited authors with biographical notes." --ZBMath

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