Chapter
2.5 The ray transform of a field-distribution
pp.：
31 – 37
2.6 Decomposition of a tensor field into potential and solenoidal parts
pp.：
37 – 40
2.7 A theorem on the tangent component
pp.：
40 – 43
2.8 A theorem on conjugate tensor fields on the sphere
pp.：
43 – 48
2.9 Primality of the ideal (|x|2, (〈x,y〉)
pp.：
48 – 52
2.11 Integral moments of the function If
pp.：
52 – 56
2.10 Description of the image of the ray transform
pp.：
52 – 52
2.12 Inversion formulas for the ray transform
pp.：
56 – 58
2.13 Proof of Theorem 2.12.1
pp.：
58 – 61
2.14 Inversion of the ray transform on the space of field-distributions
pp.：
61 – 66
2.15 The Plancherel formula for the ray transform
pp.：
66 – 69
2.16 Application of the ray transform to an inverse problem of photoelasticity
pp.：
69 – 72
2.17 Further results
pp.：
72 – 79
3 Some questions of tensor analysis
pp.：
79 – 83
3.1 Tensor fields
pp.：
83 – 83
3.2 Covariant differentiation
pp.：
83 – 86
3.3 Symmetric tensor fields
pp.：
86 – 90
3.4 Semibasic tensor fields
pp.：
90 – 95
3.5 The horizontal covariant derivative
pp.：
95 – 99
3.6 Formulas of Gauss-Ostrogradskiĭ type for vertical and horizontal derivatives
pp.：
99 – 104
4 The ray transform on a Riemannian manifold
pp.：
104 – 115
4.1 Compact dissipative Riemannian manifolds
pp.：
115 – 116
4.2 The ray transform on a CDRM
pp.：
116 – 119
4.3 The problem of inverting the ray transform
pp.：
119 – 121
4.4 Pestov's differential identity
pp.：
121 – 124
4.5 Poincaré's inequality for semibasic tensor fields
pp.：
124 – 126
4.6 Reduction of Theorem 4.3.3 to an inverse problem for the kinetic equation
pp.：
126 – 130
4.7 Proof of Theorem 4.3.3
pp.：
130 – 134
4.8 Consequences for the nonlinear problem of determining a metric from its hodograph
pp.：
134 – 136
4.9 Bibliographical remarks
pp.：
136 – 140
5 The transverse ray transform
pp.：
140 – 143
5.1 Electromagnetic waves in quasi-isotropic media
pp.：
143 – 144
5.2 The transverse ray transform on a CDRM
pp.：
144 – 157
5.3 Reduction of Theorem 5.2.2 to an inverse problem for the kinetic equation
pp.：
157 – 159
5.4 Estimation of the summand related to the right-hand side of the kinetic equation
pp.：
159 – 161
5.5 Estimation of the boundary integral and summands depending on curvature
pp.：
161 – 166
5.6 Proof of Theorem 5.2.2
pp.：
166 – 168
5.7 Decomposition of the operators A0 and A1
pp.：
168 – 170
5.8 Proof of Lemma 5.6.1
pp.：
170 – 174
5.9 Final remarks
pp.：
174 – 176
6 The truncated transverse ray transform
pp.：
176 – 179
6.2 The truncated transverse ray transform
pp.：
179 – 184
6.1 The polarization ellipse
pp.：
179 – 179
6.3 Proof of Theorem 6.2.2
pp.：
184 – 185
6.4 Decomposition of the operator Qξ
pp.：
185 – 191
6.5 Proof of Lemma 6.3.1
pp.：
191 – 197
6.6 Inversion of the truncated transverse ray transform on Euclidean space
pp.：
197 – 203
7 The mixed ray transform
pp.：
203 – 207
7.1 Elastic waves in quasi-isotropic media
pp.：
207 – 207
7.2 The mixed ray transform
pp.：
207 – 217
7.3 Proof of Theorem 7.2.2
pp.：
217 – 220
7.4 The algebraic part of the proof
pp.：
220 – 222
8 The exponential ray transform
pp.：
222 – 229
8.1 Formulation of the main definitions and results
pp.：
229 – 230
8.2 The modified horizontal derivative
pp.：
230 – 24
8.4 The volume of a simple compact Riemannian manifold
pp.：
238 – 244
8.5 Determining a metric in a prescribed conformal class
pp.：
244 – 248
8.6 Bibliographical remarks
pp.：
248 – 258
Bibliography
pp.：
258 – 261
Integral Geometry of Tensor Fields
( Inverse and Ill-Posed Problems Series )
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