* Introduction Geophysics and its division. Geodesy and gravimetry. *

1. Historical review of investigations of the figure of the Earth Ancient mythical notions. The spherical Earth and its size. Triangulation. The ellipsoidal Earth. Geocentric and geodetic latitudes. Dispute about the type of Earth ellipsoid. Arc measurements in Peru, Lapland and others. Importance of the arc measurements for geodesy, physics and metrology. The geoid as the figure of the Earth. Geometric and physical geodesy. Satellite geodesy; satellite determination of the Earth's flattening. Present values of fundamental constants. *

2. Motions of the Earth - Part I Motions with respect to background radiation. Other motions of the Galaxy, its rotation. Motions of the Solar System. Revolution of the Earth round the Sun. Earth's rotation. *

3. Mechanics in non-inertial reference frames Time variation of an arbitrary vector in different coordinate systems. Equation of motion of a particle in a non-inertial system. Equations of motion of a system of particles in an inertial system; impulse momentum theorem; angular momentum theorem. Equations of motion of a rigid body in an inertial system. Equations of motion of a rigid body in a non-inertial system; Euler's equations. More general equations of motion in a non-inertial system; Liouville's equations. *

4. Motions of the Earth - Part II Time variations of the vector of the angular velocity of the Earth's rotation. Precession and nutation. Polar motion. Other long-period motions. Changes of the length of day. Dynamics of the Earth-Moon system. *

5. Earth tides Tidal effects on a rigid Earth; origin of tidal forces. Tidal potential and its properties. Vertical and horizontal components of the tidal acceleration, tidal deformations of equipotential surfaces. Angular distance of two point on a sphere. Three types of tides. Tidal effects on an elastic Earth; Love numbers. *

6. Legendre polynomials and associated Legendre functions Generating function. Some values and special properties. Recurrence relations. Legendre's differential equation. Bounds for Legendre polynomials. Other properties of Legendre polynomials. Expansion of the reciprocal distance of two arbitrary points. Associated Legendre functions. The addition theorem for Legendre polynomials. *

7. Foundations of the theory of the Earth's gravity field Basic notions; gravity field. Expansion of the external gravity potential into a series of spherical harmonics. Equipotential surfaces; geoid and spheroid. Equation of the spheroid. Normal gravity. Clairaut's theorem. Methods of determining the flattening of the Earth. *

8. The geoid Distance between the geoid and spheroid: Bruns' theorem, fundamental equation of physical geodesy, Stokes' theorem. Vening Meinesz formulae for the deflections of the vertical. Maps of the geoid. *

9. Isostasy Pratt-Hayford and Airy-Heiskanen isostatic systems. Vening Meinesz regional isostatic system. *

10. Gravity measurements and their reductions Methods of gravity measurements: absolute pendulum measurements, free-fall methods, relative measurements by means of gravimeters. Reductions of gravity measurements and gravity anomalies. Free-air reduction; Bouguer reduction. Topographic correction, its practical computing. Isostatic reductions. A physical analysis of the gravity reductions. Applications of various gravity anomalies. *

11. Interpretation of gravity anomalies General features of gravimetric interpretations. Resolution of gravity anomalies: regional and residual anomalies; the "mean value methods" for a circle, square and the surface of a circle; polynomial and other approximations; upward analytic continuation. Identification of perturbing bodies: downward analytic continuation; derivatives of the gravity acceleration; Linsser's method of determining density discontinuities. Determination of the total differential mass. Interpretation: forward problems for general three-dimensional bodies and "two-dimensional" ones; forward and inverse problems for simple homogeneous bodies. A "two-dimensional" prism. A rectangular parallelepiped. *

12. Satellite methods of studying the gravitational field - an elementary theory Methods of studying the Earth's gravitational field. Principles of satellite methods, Kepler's laws. Orbital elements. Circular orbits in a central field. The effect of the Earth's flattening on a circular orbit. Relation between precession and perturbations of a circular orbit. Eliptical orbit in a central field. *

13. Satellite methods of studying the gravitational field - applications of analytical mechanics Lagrange's equations of the second kind. Kepler's problem. Solution of the satellite motion in a general potential field by means of the Hamilton-Jacobi equation. Perturbations of elliptical orbits due to the first term of the perturbing potential. Perturbations of orbits due to other terms of the perturbing potential. Models of the Earth's gravitational potential. *

14. The figure of the real Earth's surface Heights above sea level. Problems of the classical method of determining the Earth's surface. Principles of Molodenskii's method for determining the figure of the Earth's surface. Satellite altimetry. Global positioning system (GPS). * Basic references: - G.D. Garland: The Earth's Shape and Gravity. Pergamon Press, Oxford

1965. - M. Pick, J. Pícha, V. Vyskočil: Theory of the Earth's Gravity Field. Academia, Prague, and Elsevier Scientific Publishing Company, Amsterdam

1973. - M. Burša, K. Pěč: Gravity Field and Dynamics of the Earth. Springer-Verlag, Berlin

1993. - O. Novotný: Motions, Gravity Field and Figure of the Earth. Lecture notes. UFBA, Salvador, Bahia

1998. - F.D. Stacey: Physics of the Earth. J. Wiley, New York

1969. * Further references: - G. Arfken: Mathematical Methods for Physicists. Academic Press, New York

1970. - W.A. Heiskanen, F.A. Vening Meinesz: The Earth and Its Gravity Field. McGraw Hill, New York

1958. - P. Melchior: The Tides of the Planet Earth. Pergamon Press, Oxford

1983. - I. Fischer: The figure of the Earth - changes in concepts. Geophysical Surveys 2 (1975), 3-54.

Historical review of the study of the figure of the Earth. Mechanics in non-inertial reference frames.

Motions of the Earth. Earth tides.

Legendre polynomials and associated Legendre functions. Foundations of the theory of the Earth's gravity field.

The geoid. Isostasy.

Gravity measurements and their reductions. Interpretation of gravity anomalies.

Satellite methods of studying the gravitational field; an elementary theory, applications of analytical mechanics. The figure of the real Earth's surface.