Smooth Mappings

This paper considers smooth mappings between Euclidean spaces and their basic differential properties. A mapping defined on an open set is studied through its coordinate functions, differentiability, and continuity. Special attention is given to the Jacobian matrix, the differential of a mapping, and the concept of rank. Definitions of smooth, infinitely differentiable, and continuous mappings are presented, and illustrative examples are discussed to demonstrate the theoretical results.