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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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The Geometry of Domains in Space (Paperback, Softcover reprint of the original 1st ed. 1999)
Loot Price: R1,427
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The Geometry of Domains in Space (Paperback, Softcover reprint of the original 1st ed. 1999)
Series: Birkhauser Advanced Texts / Basler Lehrbucher
Expected to ship within 18 - 22 working days
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The analysis of Euclidean space is well-developed. The classical
Lie groups that act naturally on Euclidean space-the rotations,
dilations, and trans lations-have both shaped and guided this
development. In particular, the Fourier transform and the theory of
translation invariant operators (convolution transforms) have
played a central role in this analysis. Much modern work in
analysis takes place on a domain in space. In this context the
tools, perforce, must be different. No longer can we expect there
to be symmetries. Correspondingly, there is no longer any natural
way to apply the Fourier transform. Pseudodifferential operators
and Fourier integral operators can playa role in solving some of
the problems, but other problems require new, more geometric,
ideas. At a more basic level, the analysis of a smoothly bounded
domain in space requires a great deal of preliminary spadework.
Tubular neighbor hoods, the second fundamental form, the notion of
"positive reach", and the implicit function theorem are just some
of the tools that need to be invoked regularly to set up this
analysis. The normal and tangent bundles become part of the
language of classical analysis when that analysis is done on a
domain. Many of the ideas in partial differential equations-such as
Egorov's canonical transformation theorem-become rather natural
when viewed in geometric language. Many of the questions that are
natural to an analyst-such as extension theorems for various
classes of functions-are most naturally formulated using ideas from
geometry.
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