I am reading "Calculus on Manifolds" by Michael Spivak.
The following Theorem 2-2 and Theorem 2-5 are from this book.
2-2 Theorem (Chain Rule). If $f:\mathbb{R}^n\to\mathbb{R}^m$ is differentiable at $a$, and $g:\mathbb{R}^m\to\mathbb{R}^p$ is differentiable at $f(a)$, then the composition $g\circ f:\mathbb{R}^n\to\mathbb{R}^p$ is differentiable at $a$, and $$D(g\circ f)(a)=Dg(f(a))\circ Df(a).$$
2-5 Theorem. If $D_{i,j} f$ and $D_{j,i} f$ are continuous in an open set containing $a$, then $$D_{i,j} f(a)=D_{j,i} f(a).$$
The following Theorem 7.1 is from "Analysis on Manifolds" by James R. Munkres.
Theorem 7.1. Let $A\subset\mathbb{R}^m$; let $B\subset\mathbb{R}^n$. Let $$f:A\to\mathbb{R}^n\,\,\,\,\,\text{and}\,\,\,\,\,g:B\to\mathbb{R}^p,$$ with $f(A)\subset B$. Suppose $f(a)=b$. If $f$ is differentiable at $a$, and if $g$ is differentiable at $b$, then the composite function $g\circ f$ is differentiable at $a$. Furthermore, $$D(g\circ f)(a)=Dg(b)\cdot Df(a),$$ wherethe indicated product is matrix multiplication.
Spivak assumed $f$ is defined on $\mathbb{R}^n$ and $g$ is defined on $\mathbb{R}^m$ in Theorem 2-2.
Spivak didn't write Theorem 2-2 like Theorem 7.1 in Munkres' book.
Why?
Spivak didn't write Theorem 2-5 as follows.
Why?
2-5' Theorem. If $D_{i,j} f$ and $D_{j,i} f$ are continuous at $a\in\mathbb{R}^n$, then $$D_{i,j} f(a)=D_{j,i} f(a).$$