A smooth map is a function between two differentiable manifolds that preserves the structure of the manifolds. Specifically, it allows for the smooth transition of points from one manifold to another without any abrupt changes. This means that the map can be differentiated any number of times, making it a key concept in differential geometry. In mathematical terms, a smooth map is often defined as a function that has continuous derivatives of all orders. This property is essential in various fields, including calculus, physics, and engineering, as it ensures that the behavior of the system can be analyzed using tools from differential calculus.