As shown in manifold structure of mapping spaces, the space of smooth maps from a sequentially compact Frölicher space (or diffeological or Chen space) in to a smooth manifold is again a smooth manifold. Its tangent space is straightforward to identify. A tangent vector is an infinitesimal deviation of a smooth map; that is, it defines a direction in which to deform that map. As a smooth map is determined by its values at points, when deforming a smooth map it is enough to explain how to deform each point. Thus a tangent vector at defines, for each , a tangent vector at . Thus we obtain a map . It is not unbelievable that this map is again smooth, whence we have:
T C^\infty(S,M) \to C^\infty(S, T M).