The Partial Jacobian Matrix and the Chain Rule for Functions on Products of Real n-Spaces

The usual chain rule is for compositions of functions from $R^n$ to $R^m$ , for some $m$ and $n$ . We consider functions whose domains and codomains are products of various $R^n$ ’s, and introduce the notion of a partial Jacobian matrix, using which we formulate a chain rule in this setting that should be useful for calculations as well as for some theoretical purposes.