Foundations of Differential Geodesy

I. Role of Coordinates in Geodesy and Geometry.- 1. Coordinates in Geodesy and Geometry.- 2. Marussi-Hotine Approach to Geodesy.- 3. Classical Formulations of the Tensor Calculus.- 4. Abstract Notion of a Tensor.- 5. Preview of the Leg Calculus 39 Problems for Chapter I.- II. Ricci Calculus.- 1. Introduction.- 2. 3-Leg Algebra.- 3. Congruences.- 4. Geometry of Congruences.- 5. Ricci Coefficients.- 6. Calculation of Ricci Coefficients.- 7. Commutativity of Leg Derivatives.- 8. Lie Bracket.- 9. Jacobi and Schouten Identities.- 10. Interpretation of Ricci Coefficients.- 11. Miscellaneous Leg Representations.- 12. Lamé Equations 67 Problems for Chapter II.- III. Cartan Calculus.- 1. Introduction.- 2. Exterior Product.- 3. Exterior Differentiation.- 4. Basic Properties of the Exterior Product and Differentiation.- 5. Cartan’s Viewpoint.- 6. Structural Equations of Cart an.- 7. Commutators and Schouten Identities.- 8. Cartan Calculus and the Classical Tensor Calculus.- 9. Coordinates in the Cartan and Ricci Calculus.- 10. Conclusions and Summary 99 Problems for Chapter III.- IV. General Leg Calculus.- 1. Introduction.- 2. Fundamental Idea of the Leg Calculus.- 3. Identification of the Ricci Coefficients.- 4. Permutation Algorithm.- 5. Interpretation of the Leg Coefficients.- 6. Basic Equations in the Leg Calculus.- 7. Commutators and Lie Brackets.- 8. Laplacian of a Scalar Function.- 9. Lamé Equations and Identities.- 10. Coordinates in the Leg Calculus.- 11. Reconstruction Theorem and General Viewpoint.- Problems for Chapter IV.- V. Gaussian Differential Geometry.- 1. Introduction.- 2. The Framework of Gaussian Differential Geometry.- 3. Gauss Operator and First Basic Form.- 4. Second Basic Form.- 5. Third and Fourth Basic Forms.- 6. Congruence and Surface Eigenstructure Theorems.- 7. Two Digressions.- 8. Basic Equations of the Surface Leg Calculus.- 9. Frenet Equations.- 10. Meusnier and Bonnet Formulas.- 11. Geodesic Curvature and Torsion in an Arbitrary Direction, and Euler’s Formula.- 12. Euler’s Osculating Paraboloid and Dupin’s Indicatrix of Curvature.- 13. Triply-Orthogonal Systems and Dupin’s Theorem.- Problems for Chapter V.- VI. Basic Equations for Differential Geodesy.- 1. Introduction.- 2. Newtonian Gravitation and the Geopotential Field.- 3. Marussi Ansatz and the Hotine 3-Leg.- 4. Hotine Leg Calculus and N-Integrability Conditions.- 5. Marussi Tensor.- 6. Bruns Curvature Equation.- 7. Bruns Equation and Hotine’s Theorem.- 8. n-Integrability Conditions.- 9. The Hotine-Marussi Equations.- 10. 3-Dimensional Laplacian of Local Gravity.- 11. 2-Dimensional Laplacian of Local Gravity.- Problems for Chapter VI.- VII. The Fundamental Theorem of Differential Geodesy.- 1. Introduction.- 2. Basic Assumptions.- 3. The Fundamental Theorem of Differential Geodesy.- 4. Discussion.- 5. Extension Problem.- 6. Two Subsidiary Problems.- 7. The Congruence-Forming Property.- 8. The Surface-Forming Property.- 9. Conclusions.- Problems for Chapter VII.- VIII. Algebraic Theory of the Marussi Tensor.- 1. Introduction.- 2. Eigenstructure of Symmetric Tensors.- 3. Eigenstructure of the Marussi Tensor.- 4. The Singularity Condition.- 5. Conclusions 268 Problems for Chapter VIII.- IX. Conformal Differential Geodesy.- 1. Introduction.- 2. Fundamental Notions of Conformal Geometry.- 3. Conformal Structural Equations.- 4. Conformal Ricci Coefficients.- 5. Conformal Surface Theory.- 6. Conformal Curve Theory.- 7. Generalized Conformal Transformations.- 8. Conclusions 300 Problems for Chapter IX.- X. Coordinates in Differential Geodesy.- 1. Introduction.- 2. Marussi Hypothesis and Hotine Problem.- 3. Hotine’s Hierarchy of Local Coordinate Systems.- 4. The Marussi-Hotine Approach: A Critique.- 5. Generalized Marussi-Hotine Approach: A Proposal.- 6. Conclusion 338 Problems for Chapter X.