Differential Calculus on ℝ m $$\mathbb{R}^{m}$$

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This chapter provides a complete and systematic introduction to the differential calculus on finite dimensional normed vector spaces. Among the main theorems proved are the mean value theorem, the inverse and implicit function theorems, the rank theorem, symmetry of higher order derivatives, Taylor’s theorem and the strong form of local existence and uniqueness theorem for ODEs using methods based on smooth uniform approximation. Statements and proofs are given of both the Leibniz rule and Faa di Bruno’s formula for the derivatives of products and composites of maps. In an appendix, there is a proof of the theorem of Frigyes Riesz on the equivalence of norms on a finite dimensional vector space.