On Some Class of Solutions to the Two-Dimensional Laplace Equation on a Three-Dimensional Manifold.

We find the solution to the two-dimensional Laplace equation on a given set of three independent variables in the three-dimensional Euclidean space. The problem is solved by transforming the two-dimensional Laplace equation into some equation with the sought function of three independent variables. This turns out possible by introducing a spherical coordinate system. The proposed method allows us to find a solution to the two-dimensional Laplace equation in the form of a function of three independent variables. By way of example, we consider the problem of an incompressible fluid flow around a three-dimensional body shaped as an "iron." For this problem, we give the detailed arguments that reduce the three-dimensional Laplace equation describing the distribution of the scalar potential of the flow velocities near the surface of the body which depends on three independent coordinates to some two-dimensional Laplace equation whose solution is strictly justified by analysis. We also observe that similar problems arise not only in hydrodynamics but also in elasticity and electromagnetism theories. The described technique, namely, the possibility of passing from two to three independent variables by a given transformation enables us to find purely physical solutions for a wide range of problems from the various areas of natural sciences. [ABSTRACT FROM AUTHOR]