10 - Calculus in ℝn

Abstract

This chapter discusses Taylor's theorem in n variables. Taylor's theorem is not difficult to apply, but it does become tedious when computing higher-order terms of a function of n variables. It is true that if f(x) is a polynomial of degree m, then f(x) = pm(x). If Ω is a subset of ℝ2, then a function or mapping f from ℝn to ℝm is a rule that assigns to each x = (x1, x2, . , xn) in Ω a unique vector f(x) in ℝm. The set Ω is called the domain of f. The set {f(x): x ∈ Ω} is called the range of f.