Relationship formula between nonlinear polynomial equations and the corresponding Jacobian matrix

This paper provides a general proof of a relationship theorem between nonlinear analogue polynomial equations and the corresponding Jacobian matrix, presented recently by the present author. This theorem is also verified generally effective for all nonlinear polynomial algebraic system of equations. As two particular applications of this theorem, we gave a Newton formula without requiring the evaluation of nonlinear function vector as well as a simple formula to estimate the relative error of the approximate Jacobian matrix. Finally, some possible applications of this theorem in nonlinear system analysis are discussed.