Fundamentals of Advanced Mathematics 3

Differential Calculus, Tensor Calculus, Differential Geometry, Global Analysis

Fundamentals of Advanced Mathematics 3

Henri Bourlès, Conservatoire National des Arts et Métiers in Paris, France

ISBN : 9781785482502

Publication Date : September 2019

Hardcover 426 pp

150.00 USD



Published in three volumes, this series, Fundamentals of Advanced Mathematics, presents mathematical elements that make up the foundations of a number of contemporary scientific methods: modern systems theory, physics and engineering.

This third volume is dedicated to differential and integral calculus, in their local and global components. “Local” differential calculus is discussed in the framework of Banach and Fréchet spaces. We examine the “global” approach in how one can replace these spaces with differential or analytic manifolds modeled on them.

The generalized Stokes theorem, other than its usual applications, offers us a view on the dualities of Hodge, Poincaré and de Rham. Lie’s group and algebra theories allow us to develop harmonic analysis in a very general context; this analysis encompasses both the Fourier transforms and the Fourier series.

The study of connections and tensor calculus leads to spaces equipped with curvature and torsion, generalizing the Riemann and Lorentz spaces of general relativity.


1. Differential Calculus.
2. Differential and Analytic Manifolds.
3. Fiber Bundles.
4. Tensor Calculus on Manifolds.
5. Differential and Integral Calculus on Manifolds.
6. Analysis on Lie Groups.
7. Connections.

About the authors

Henri Bourlès has taught automatics at engineering schools and at universities for over 30 years. He is Full Professor and Chair at the Conservatoire National des Arts et Métiers in Paris, France, and his research focuses on systems theory.

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