The early thermal history of the Earth is a matter of some speculation. Current scientific consensus is that planet Earth formed by accretion of material with the same composition as chondritic meteorites. Accretion, a process that generated heat as colliding material gave up kinetic energy, led to differentiation of the planetary constituents into concentric layers. When the temperature of the early Earth reached the melting point of iron, the dense iron, accompanied by other siderophile elements such as nickel and sulfur, sank towards the center of the planet to form a liquid core. Meanwhile lighter elements rose to form an outer layer, the primitive mantle. Further differentiation took place later, creating a chemically different thin crust atop the mantle. Only the outer core is now molten, surrounding a solid inner core of iron that solidified out of the core fluid. Lighter elements left behind in the core rise through the core fluid and result in a composition-driven convection in the outer core, which is in addition to thermal convection. Although the short-term behavior of the mantle is like that of a solid, allowing the passage of seismic shear waves, its long-term behavior is characterized by plastic flow, so heat transport by convection or advection is possible. In the solid lithosphere and inner core heat is transported dominantly by thermal conduction.

The physical states of the Earth's mantle and core are well understood, but the variation of temperature with depth is not well known.

The existence of a magnetic force was known for centuries before William Gilbert pointed out in 1600 that the Earth itself behaved like a huge magnet. Gradually maps were made of the geomagnetic elements. Systematic investigation of magnetic behavior was undertaken in the late eighteenth and early nineteenth centuries. The French scientist Charles Augustin de Coulomb showed experimentally that forces of attraction and repulsion exist between the ends of long thin magnetized rods, and that they obey rules similar to those determining the interaction of electrical charges. A freely suspended magnet was observed to align approximately north–south; the north-seeking end became known as its north pole, the opposite end as its south pole. The origin of magnetic force was attributed to magnetic charges, which, through association, became known as magnetic poles. Subsequently, it was shown that individual magnetic poles, or monopoles, do not exist. All magnetic fields originate in electric currents. This is true even at atomic dimensions; circulating (and spinning) electrical charges impart magnetic properties to atoms. However, the concept of multiple pole combinations (e.g., the dipole, quadrupole, and octupole) proved to be very useful for describing the geometries of magnetic fields.

The dipole magnetic field and potential

The most important field geometry is that of a magnetic dipole. This was originally imagined to consist of two equal and opposite magnetic poles that lie infinitesimally close to each other (Appendix A2).

This work was written as a supplementary text to help students understand the mathematical steps in deriving important equations in classical geophysics. It is not intended to be a primary textbook, nor is it intended to be an introduction to modern research in any of the topics it covers. It originated in a set of handouts, a kind of “do-it-yourself” manual, that accompanied a course I taught on theoretical geophysics. The lecture aids were necessary for two reasons. First, my lectures were given in German and there were no comprehensive up-to-date texts in the language; the recommended texts were in English, so the students frequently needed clarification. Secondly, it was often necessary to explain classical theory in more detail than one finds in a multi-topic advanced textbook. To keep such a book as succinct as possible, the intermediate steps in the mathematical derivation of a formula must often be omitted. Sometimes the unassisted student cannot fill in the missing steps without individual tutorial assistance, which is usually in short supply at most universities, especially at large institutions. To help my students in these situations, the “do-it-yourself” text that accompanied my lectures explained missing details in the derivations. This is the background against which I prepared the present guide to geophysical equations, in the hope that it might be helpful to other students at this level of study.

The Earth is not rigid and its rotation causes it to deform, flattening at the poles and bulging at the equator. The gravitational attractions of Sun and Moon on the equatorial bulge result in torques on the Earth, which cause additional motions of the rotation axis, known as precession and nutation. These motions occur relative to a coordinate system fixed in space, for example in the solar system. The rotation axis is inclined to the pole to the ecliptic plane at a mean angle of 23.425°; this angle is the obliquity of the axis. Precession is a very slow motion of the tilted rotation axis around the pole to the ecliptic, with a period of 25,720 yr. The nutation is superposed on this motion and consists of slight fluctuations in the rate of precession as well as in the obliquity.

The other planets also affect the Earth's rotation, causing small but significant cyclical changes on a very long timescale. These are observable directly by precise measurement of the position of the rotation axis using very-long-baseline interferometry (VLBI). The fluctuations influence the intensity of solar radiation incident on the Earth and produce cyclical climatic effects that are evident in sedimentary processes, where they are known as the Milankovitch (or Milanković) cycles. They correspond to retrograde precession of the rotation axis (period ~ 26 kyr), changes in the angle of obliquity (period ~ 41 kyr), prograde precession of Earth's elliptical orbit (period ~ 100 kyr), and variation of the ellipticity of the orbit (period ~ 100 kyr).