ABSTRACT
Parametrized curves into Rn are continuous functions from an interval of R to Rn. Therefore, the study of local and global properties of curves requires single-variable calculus. As one might expect, the study of surfaces in Euclidean three-space involves continuous functions from R2 to R3. Therefore, in order to properly phrase the theory of surfaces and, more generally, the theory of manifolds, one needs to understand the analysis of multivariable functions f : Rn → Rm.
