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Multilinearity is linearity spread, independently, among several variables.
Suppose $X$ is a vector space and suppose we have a function: $$\varphi \colon X^k \rightarrow {\bf R},$$ $$\varphi (x,y,z,...)= r \in {\bf R}.$$
It is called multilinear if
- $\varphi(\cdot, b, c, ...)$ is linear for each $b, c,...$,
- $\varphi(a, \cdot, c, ...)$ is linear for each $a, c,...$,
- $\varphi(a, b, \cdot, ...)$ is linear for each $a, b,...$,
- etc.
As an example, see differential forms.