# Transformation of the Line-Element

by Anamitra
Tags: lineelement, transformation
 P: 621 An Analytical Endeavor: Equation: $${Cot}{(}{\theta}{)}{=}{[}\frac{{\partial}^{2}{\theta}}{{\partial x}^{2}}{+}\frac{{\partial}^{2}{\theta}}{{\partial y}^{2}}{]}$$ ----- (1) We look for a solution of the form: $$\theta{=}{A}{(}{-}{ln}{Sin}{x}{-}\frac{1}{2}{x}^{2}{)}{+}{B}{(}{-}{ln}{Sin}{y}{-}\frac{1}{2}{y}^{2}{)}{+}{\lambda}{x}^{2}$$------------- (2) A and B are constants Lambda is a parameter independent of x and y. Using the above trial in relation (1) we have $${A}{Cot}^{2}{x}{+}{B}{cot}^{2}{y}{+}{2}{\lambda} {=}{Cot}\theta$$--- (3) We may eliminate lambda between (2) and (3) to find Particular Integrals[surfaces theta=f(x,y)] [Even if one has to use numerical methods, the equation will be an ordinary one instead of a“differential Equation”] Instead of (2) one may use a trial of the form below, to obtain a greater variety of particular integrals: $$\theta{=}{A}{f1}{(}{x}{)}{+}{B}{f2}{(}{y}{)}{+}{\lambda}{x}^{2}$$------------- (4) f1 and f1 are known ,arbitrary functions--well behaved ones of course,in terms of continuity ,differentiability etc. The homogeneous part of equation (1) is simply Laplace’s equation in two dimensions. We have the familiar solutions. [This posting will be revised]