Let $X_1, \cdots, X_n$ and $Y$ be sets, and let $f$ be a function of $n$ variables: $f:X_1\times X_2\times\cdots\times X_n\to Y$ . Fix $x_i\in X_i$ for $2\leq i\leq n$ . The induced mapping $a\mapsto f(a,x_2,\ldots,x_n)$ is called the partial mapping determined by $f$ corresponding to the first variable.

In the case where $n=2$ , the map defined by $a\mapsto f(a,x)$ is often denoted $f(\cdot,x)$ . Further, any function $f:X_1\times X_2\to Y$ determines a mapping from $X_1$ into the set of mappings of $X_2$ into $Y$ , namely $\overline{f}:x\mapsto(y\mapsto f(x,y))$ . The converse holds too, and it is customary to identify $f$ with $\overline{f}$ . Many of the ``canonical isomorphisms'' that we come across (e.g. in multilinear algebra) are illustrations of this kind of identification.