Multivariable Differential Calculus

In the very definition of smooth manifold, there is a statement like “…for every point p in the topological space M there exist an open neighborhood U of p, an open subset V of Euclidean space $$ {\mathbb{R}}^{n} , $$ R n , and a homeomorphism $$ \varphi $$ φ from U onto V …”. This phrase clearly indicates that homeomorphism $$ \varphi $$ φ can act as a messenger in carrying out the well-developed multivariable differential calculus of $$ {\mathbb{R}}^{n} $$ R n into the realm of unornamented topological space M. So the role played by multivariable differential calculus in the development of the theory of smooth manifolds is paramount. Many theorems of multivariable differential calculus are migrated into manifolds like their generalizations. Although most students are familiar with multivariable differential calculus from their early courses, but the exact form of theorems we need here are generally missing from their collection of theorems. So this is somewhat long chapter on multivariable differential calculus that is pertinent here for a good understanding of smooth manifolds.