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The numerical solution of the Richards equation is one of the most challenging problems in earth science. [16] Richards' equation has been criticized for being computationally expensive and unpredictable [ 17 ] [ 18 ] because there is no guarantee that a solver will converge for a particular set of soil constitutive relations.
As is the case with spherical coordinates and spherical harmonics, Laplace's equation may be solved by the method of separation of variables to yield solutions in the form of oblate spheroidal harmonics, which are convenient to use when boundary conditions are defined on a surface with a constant oblate spheroidal coordinate.
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation.
The X/Y plane coincides with Earth's equatorial plane, with the +X axis pointing toward the vernal equinox and the Y axis completing a right-handed set. The ECI reference frame is not truly inertial because of the slow, 26,000 year precession of Earth's axis , so the reference frames defined by Earth's orientation at a standard astronomical ...
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 curve always has x-intercepts at ± c where c 2 = a 2 + b 2. When e < 1 there are two additional real x-intercepts and when e > 1 there are two real y-intercepts, all other x- and y-intercepts being imaginary. [6] The curve has double points at the circular points at infinity, in other words the curve is bicircular. These points are ...
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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 ...