Knowledge Bank

The geodetic boundary value problem consists in determining an unknown closed surface from the boundary values of an external potential and its gradient. A rigorous mathematical formulation of this problem is given leading to a system of non-linear integro-differential equations. The formalism of differentiation in function spaces is applied yielding a linearized version which involves no further neglections and approximations. Tensor calculus is used in linearizing the various differential geometric quantities. The results are specialized to a linearization with respect to the equipotential sphere in which case the formulas of Stokes and Vening Meinesz are simultaneously obtained.