<p>This paper delves into the realm of differentiable manifolds and morphisms, emphasizing their differentiability of class C∞. We introduce fundamental concepts, such as the tangle bundle T(M) and the algebra of differentiable functions F(M) defined on the real n-dimensional connected differentiable manifold M. Furthermore, we explore the module of differentiable sections of a vector bundle H, denoted as Γ(H). This investigation sheds light on the intricate interplay between differentiable structures within the realm of manifold theory.</p>