Integral geometry of tensor fields
by Sharafutdinov, V. A., (Vladimir Alʹtafovich)
Integral geometry can be defined as determining some function or a more general quantity, which is defined on a manifold, given its integrals over submanifolds or a prescribed class. In this book, only integral geometry problems are considered for which the submanifolds are one-dimensional. The book deals with integral geometry of symmetric tensor fields. This section of integral geometry can be considered as the mathematical basis for tomography or anisotropic media whose interaction with soun…
Read More ↓Title Statements
- Full title :Integral geometry of tensor fields / V.A. Sharafutdinov.
Summary
Integral geometry can be defined as determining some function or a more general quantity, which is defined on a manifold, given its integrals over submanifolds or a prescribed class. In this book, only integral geometry problems are considered for which the submanifolds are one-dimensional. The book deals with integral geometry of symmetric tensor fields. This section of integral geometry can be considered as the mathematical basis for tomography or anisotropic media whose interaction with sounding radiation depends essentially on the direction in which the latter propagates. The main mathemat.
Read More ↓Edition Statements
- Publication Date:1994.
Publication Statements
- Publisher(s):VSP
- Place of Publication:Utrecht, the Netherlands
- Language:eng
- Publication Date:1994.
Notes
Description based upon print version of record.
8.1 Formulation of the main definitions and results
The Physical Object
- Description:1 online resource (276 p.).
Edition Identifiers
- ISBN:3110900092, 9783110900095
- oclc:on1162271695
- LC Classification:QA672
- Dewey Decimal:516.3
Table of Contents
1 Introduction; 1.1 The problem of determining a metric by its hodograph and a linearization of the problem; 1.2 The kinetic equation on a Riemannian manifold; 1.3 Some remarks; 2 The ray transform of symmetric tensor fields on Euclidean space; 2.1 The ray transform and its relationship to the Fourier transform; 8.2 The modified horizontal derivative; 2.2 Description of the kernel of the ray transform in the smooth case; 2.3 Equivalence of the first two statements of Theorem 2.2.1 in the case n = 2; 2.4 Proof of Theorem 2.2.2; 2.5 The ray transform of a field-distribution
2.17 Further results; 3 Some questions of tensor analysis; 3.1 Tensor fields; 3.2 Covariant differentiation; 3.3 Symmetric tensor fields; 3.4 Semibasic tensor fields; 3.5 The horizontal covariant derivative; 3.6 Formulas of Gauss-Ostrogradskiĭ type for vertical and horizontal derivatives; 4 The ray transform on a Riemannian manifold; 4.1 Compact dissipative Riemannian manifolds; 4.2 The ray transform on a CDRM; 4.3 The problem of inverting the ray transform; 4.4 Pestov's differential identity; 4.5 Poincaré's inequality for semibasic tensor fields
4.6 Reduction of Theorem 4.3.3 to an inverse problem for the kine…
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