Semi-Riemannian Geometry

Chapter 10 Analysis 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

(10.1.1)

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 ...