Abstract: This paper is concerned with operator variation from the transforming point of view, and presents some new concepts and new relations. Related problems and concepts include: convex operator, reciprocity set, reciprocity principles, unilateral variation principles, (H, H₁, H₂) -generalized solution and operator-differential equation, etc..Variation principle and variation method is a sort of basic concept and method in the analysis of many problems concerning mathematics, mechanics, physics and control theory. The change and transformation of variation or stationary value relations is the most fundamental form and process of these princeples and methods. This paper analyzes some properties of operator variation and stationary value from the transforming point of view and presents some new concepts, unifiedly treats and extends the reciprocity theorem of classical variation, variation theorem of quadratic functional and unilateral variation principles. Furthermore, we investigate the calculus of normed ring and the solution of a sort of operator differential equation.

Key words: Riemannian manifold, fiber bundle, exterior differential, absolute differential, Lie derivative, invariant form, integral invariant, comoving time derivative