# Chapter 10Analysis in ℝ m

## 10.1 Derivatives

In this section, we review some of the key results in the differential calculus of one or more real variables. For the most part, proofs are not provided.

Let U be an open set in ℝ m , let F : U → ℝ n be a map, and let p be a point in U. We say that F is differentiable at p if there is a linear map L p : ℝ m → ℝ n such that

It can be shown that if such a map exists, it is unique. We call this map the differential of F at p and henceforth denote it by

In the literature, the differential of F at p is also called the derivative of F at p or the total derivative of F at p, and is denoted variously by dF p , dF p , DF(p), D p (F), or F'(p). We have chosen to include parentheses in the notation d p (F) to set the stage for viewing d p as a type of map.

It is usual to characterize d p (F) as being a “linear approximation” to F in the vicinity of p. For a more geometric interpretation, let us define the graph of F by

We can think of d p (F)(ℝ m ), which is a vector space, as being the “tangent space” to graph(F) at F(p). This is a generalization of the tangent line and tangent plane familiar from the ...

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