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http://www.mathpages.com/rr/s5-02/5-02.htm>:
"Figure 1 shows an arbitrary coordinate system, with
the axes X1 and X2, as well as the contravariant and
covariant components of the position vector P with
respect to these coordinates. As can be seen, the jth
component of the "contravariant path" from O to P
consists of a segment parallel to jth coordinate
axis, whereas the jth component of the "covariant
path" consists of a segment perpendicular to all the
axes other than the jth. This is the essential
distinction between the contravariant and covariant
ways of expressing a vector (or, more generally, a
tensor)."
However, Weinreich (p. 7) emphasizes that there is
also a contravariant/covariant distinction in a change of scale of the
units along the axes.