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From: "John S. Denker" <jsd@MONMOUTH.COM>This is one thing that confuses me. In a non-curved non-orthogonal Euclidean coordinate system (I have in mind a two dimensional coordinate system in which the x and y axes are not perpendicular), the term *covariant* components refers to the components that are perpendicular to the axes (the covariant x component is perpendicular to the y axis and the covariant y component is perpendicular to the x axis). The term *contravariant* components refers to the components that are parallel to the axes (the contravariant x component is parallel to the x axis and the contravariant y component is parallel to the y axis). I quote from the web site <http://www.mathpages.com/rr/s5-02/5-02.htm>:
For vectors in ordinary non-curved Euclidean space, there
is no distinction between covariant and contravariant
vectors, so the issue isn't usually mentioned when
vectors are first introduced.
(Note the reference cites the E-field as an example ofAha! See above.
a 1-form, but I'm not 100% happy with that; I suppose
it's OK for electrostatics but it's not relativistically
correct.)