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The book contains a consistent and sufficiently comprehensive
theory of smooth functions and maps insofar as it is connected with
differential calculus. The scope of notions includes, among others,
Lagrange inequality, Taylor's formula, finding absolute and
relative extrema, theorems on smoothness of the inverse map and on
conditions of local invertibility, implicit function theorem,
dependence and independence of functions, classification of smooth
functions up to diffeomorphism. The concluding chapter deals with a
more specific issue of critical values of smooth mappings. In
several chapters, a relatively new technical approach is used that
allows the authors to clarify and simplify some of the technically
difficult proofs while maintaining full integrity. Besides, the
book includes complete proofs of some important results which until
now have only been published in scholarly literature or scientific
journals (remainder estimates of Taylor's formula in a nonconvex
area (Chapter I, 8), Whitney's extension theorem for smooth
function (Chapter I, 11) and some of its corollaries, global
diffeomorphism theorem (Chapter II, 5), results on sets of critical
values of smooth mappings and the related Whitney example (Chapter
IV). The text features multiple examples illustrating the results
obtained and demonstrating their accuracy. Moreover, the book
contains over 150 problems and 19 illustrations. Perusal of the
book equips the reader to further explore any literature basing
upon multivariable calculus.
The book contains a consistent and sufficiently comprehensive
theory of smooth functions and maps insofar as it is connected with
differential calculus. The scope of notions includes, among others,
Lagrange inequality, Taylor's formula, finding absolute and
relative extrema, theorems on smoothness of the inverse map and on
conditions of local invertibility, implicit function theorem,
dependence and independence of functions, classification of smooth
functions up to diffeomorphism. The concluding chapter deals with a
more specific issue of critical values of smooth mappings. In
several chapters, a relatively new technical approach is used that
allows the authors to clarify and simplify some of the technically
difficult proofs while maintaining full integrity. Besides, the
book includes complete proofs of some important results which until
now have only been published in scholarly literature or scientific
journals (remainder estimates of Taylor's formula in a nonconvex
area (Chapter I, 8), Whitney's extension theorem for smooth
function (Chapter I, 11) and some of its corollaries, global
diffeomorphism theorem (Chapter II, 5), results on sets of critical
values of smooth mappings and the related Whitney example (Chapter
IV). The text features multiple examples illustrating the results
obtained and demonstrating their accuracy. Moreover, the book
contains over 150 problems and 19 illustrations. Perusal of the
book equips the reader to further explore any literature basing
upon multivariable calculus.
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