Integral Geometry of Tensor Fields.

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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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Sharafutdinov, V. A.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Berlin : De Gruyter, 1994.
Colección:Inverse and ill-posed problems series.
Temas:
Integral geometry.
Calculus of tensors.
Geometry, Differential.
Géométrie intégrale.
Calcul tensoriel.
Géométrie différentielle.
Calculus of tensors
Geometry, Differential
Integral geometry
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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
  • 2.2 Description of the kernel of the 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.
  • 2.6 Decomposition of a tensor field into potential and solenoidal parts
  • 2.7 A theorem on the tangent component
  • 2.8 A theorem on conjugate tensor fields on the sphere
  • 2.9 Primality of the ideal).
  • 2.16 Application of the ray transform to an inverse problem of photoelasticity
  • 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 kinetic equation
  • 4.7 Proof of Theorem 4.3.3
  • 4.8 Consequences for the nonlinear problem of determining a metric from its hodograph
  • 4.9 Bibliographical remarks
  • 5 The transverse ray transform
  • 5.1 Electromagnetic waves in quasi-isotropic media
  • 5.2 The transverse ray transform on a CDRM
  • 5.3 Reduction of Theorem 5.2.2 to an inverse problem for the kinetic equation
  • 5.4 Estimation of the summand related to the right-hand side of the kinetic equation.
  • 5.5 Estimation of the boundary integral and summands depending on curvature
  • 5.6 Proof of Theorem 5.2.2
  • 5.7 Decomposition of the operators A0 and A1
  • 5.8 Proof of Lemma 5.6.1
  • 5.9 Final remarks
  • 6 The truncated transverse ray transform
  • 6.1 The polarization ellipse
  • 6.2 The truncated transverse ray transform
  • 6.3 Proof of Theorem 6.2.2
  • 6.4 Decomposition of the operator Qq
  • 6.5 Proof of Lemma 6.3.1
  • 6.6 Inversion of the truncated transverse ray transform on Euclidean space
  • 7 The mixed ray transform
  • 7.1 Elastic waves in quasi-isotropic media.

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