Remarks

*1) We often say that $D_i u$ is the partial derivative of u with respect to X, ; we also use the notation $D_{x_i} u, \frac{\partial u}{\partial x_i}$ and $u'_{x_i}$. For a single indeterminate X the unique partial derivative $Du$ (also written $\frac{du}{dX}$ or $u'$) is called the derivative of $u$.

2) The formula (9) shows that the endomorphisms $A^p$ of the A-module $A[[X]]$ commute painvise. Hence the same holds of the endomorphisms $D_i$.

3) If $u \in A[(X_i)_{i \in I}]$ is a polynomial, the polynomials $A^p u$ and $D_i u$ defined in IV, p. 6 and 7 coincide with the formal power series denoted by the same symbols.