The chain rule.

Fundamental to many of the constructions we want to consider in the following sections is the chain rule:

Theorem 1.1 (Chain Rule.)   Let $U \subset \mathbb{R}^n$ be open , $f\colon U \to \mathbb{R}^m$, $V \subset \mathbb{R}^m$ open and $g \colon V \to \mathbb{R}^k$ with $f(U) \subset V$. Let $x \in U$. Then:

$$d(f \circ g) (x) = dg(f(x)) \circ df(x).$$

The composition on the right hand side is the composition of linear operators. In particular if we expand both sides in terms of the standard basis of $\mathbb{R}^n$ then we have

$${\partial _j( g^i \circ f)} (x) = \sum_{l=1}^m {\partial _l g^i } \circ {\partial _j f^l }$$

An important part of the chain rule is the fact that the composition of smooth functions is also smooth. A partial converse of this result will be important in the sequel.

Lemma 1.1   Let $U$ be an open subset of $\mathbb{R}^n$ and $V$ an open subset of $\mathbb{R}^m$. A function $\phi \colon U \to V$ is smooth if and only if for every smooth function $f \colon V \to \mathbb{R}$ the composite $f \circ \phi \colon U \to \mathbb{R}$ is smooth.

Proof. If $\phi$ is smooth then the result follows via the chain rule. If the result is true then take $f$ to be the restriction to $V$ of each of the co-ordinate functions $x^i$. Then $x^i$ is smooth so $x^i \circ \phi = \phi^i$ is smooth. $\qedsymbol$