Did I find an error on page 42 of my edition of Spivak's Calculus on Manifolds regarding computing the partial derivatives of the implicit function defined by the Implicit Function Theorem? I don't find this error listed in any errata online, but can't make sense of what is written.
For convenience, Spivak states the Implicit Function Theorem as:
Suppose $f: R^n \times R^m \to R^m$ is continuously differentiable in an open set containing $(a,b)$ and $f(a,b) = 0$. Let $M$ be the $m \times m$ matrix
$$ (D_{n+j}f^i(a,b)) \qquad 1\leq i,j \leq m.$$
If $\det{M} \neq 0$, there is an open set $A \subset R^n$ containing $a$ and an open set $B \subset R^m$ containing $b$, with the following property: for each $x \in A$ there is a unique $g(x) \in B$ such that $f(x,g(x)) = 0$. The function $g$ is differentiable.
Spivak's goes on to explain how to find the partial derivatives of $g$ by taking $D_j$ of both sides of $f(x,g(x)) = 0$ and he writes:
$$0 = D_jf^i(x,g(x)) + \sum_{\alpha=1}^{m}D_{n+\alpha}f^i(x,g(x)) \cdot D_jg^\alpha(x) \\
i,j = 1, \ldots, m.$$
But shouldn't $j$ go from $1$ to $n$, not from $1$ to $m$? If $g: R^n \to R^m$, then each $g^\alpha$ has $n$ partial derivatives.
Best Answer
In the theorem, the indices are correctly stated as $1\leq i,j\leq m$ (we're looking at a certain $m\times m$ sub-matrix of $f'(a,b)$). But in the subsequent calculation, you're right, we should have $1\leq i\leq m$ and $1\leq j\leq n$ precisely because the functions are $\Bbb{R}^n\to\Bbb{R}^m$.