For the purpose of image analysis, we usually take the application method relying on the various mathematical theories. On the respect of image as two variable function one may uses the gradient vector or several type of energy functions induced by the conventional (partial) derivative. We also have used the tangent plane or curvature vector from the concept of differential geometry {**]. However, these mathematical tools my assume that the given function should be sufficiently smoothing enough to depict every local variation continuously. But the real application of these mathematical methods to the natural images or phenomena may occur the ill-posed problem. In this paper, we have defined the weak derivative as a loose form of the derivative so that it my applied to the irregular case with less ill-posed problem.