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\newpage
\section{Tensors}
\label{appx:tensors}

Keep it brief and practical.  State (with don't-be-scared caveat) the
abstract-algebra definition.  Then immediately give examples showing that these
are just pairs (or $n$-tuples) with the manipulation rules (which is the
practical meaning of the equivalence relations) of scalar-through and
multilinearity.  These give different ways to write the same thing, and give us
some flexibility for computations.

Tensor product of functions in general:  given
$$
	f_1, \ldots, f_n : X \to k
$$
where $k$ is a field,
$$
	(f_1 \tct f_n)(\vecx_1, \ldots, \vecx_n) :=
	f_1(\vecx_1) \cdot \cdots \cdot f_n(\vecx_n).
$$

Maybe talk about vectors and covectors (and how they consume one another
by duality) but \emph{only} if necessary for the discussion in this course.
