Application of quadrature rules to the determination of plane equipotential lines and other curves defined by Harmonic functions

摘要

Many important kinds of curves (like equipotentials) in plane problems of physics and engineering are determined by an equation of the form u(x,y) = C, where u(x,y) is a harmonic function and C a constant. Here a solution of that equation is suggested (on the basis of previous analogous results for analytic functions). This solution contains a parameter varying along the curve under consideration and requires the use of convergent quadrature rules for the numerical evaluation of the integral appearing in it. An application to a simple problem of potential theory (e.g., heat transfer) is also made, and points of lines of heat flow are determined by the present method. Finally, a generalization of the present results to the complicated equation of the theory of caustics in dynamic plane fracture mechanics [where u(x,y) is no longer a harmonic function] is made.