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In words, this equation says that as a vertical column of water is squashed, it moves toward the Equator; as it is stretched, it moves toward the pole. Assuming, as did Sverdrup, that there is a level below which motion ceases, the vorticity equation can be integrated from this level to the base of the Ekman surface layer to obtain:
The curve can also be shown in non-dimensional or standardized form by scaling elevation and area by the maximum values. The non-dimensional hypsometric curve provides a hydrologist or a geomorphologist with a way to assess the similarity of watersheds — and is one of several characteristics used for doing so. The hypsometric integral is a ...
Also shown (in green) are curves of constant s 12, which are the geodesic circles centered A. Gauss (1828) showed that, on any surface, geodesics and geodesic circle intersect at right angles. The red line is the cut locus , the locus of points which have multiple (two in this case) shortest geodesics from A .
The inverse problem for earth sections is: given two points, and on the surface of the reference ellipsoid, find the length, , of the short arc of a spheroid section from to and also find the departure and arrival azimuths (angle from true north) of that curve, and . The figure to the right illustrates the notation used here.
The original and most common aspect of the Mercator projection for maps of the Earth is the normal aspect, for which the axis of the cylinder is the Earth's axis of rotation which passes through the North and South poles, and the contact circle is the Earth's equator.
Earth's shape is also slightly lumpy because it is composed of different materials of different densities that exert slightly different amounts of gravitational force per volume. The liquidity of a hot, newly formed planet allows heavier elements to sink down to the middle and forces lighter elements closer to the surface, a process known as ...
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These friction factors determine the rotational diffusion of molecules, which affects the feasibility of many techniques such as protein NMR and from which the hydrodynamic volume and shape of molecules can be inferred. Oblate spheroidal coordinates are also useful in problems of electromagnetism (e.g., dielectric constant of charged oblate ...