In 2 dimensions, for an oblique rectilinear coordinate system, contravariant coordinates of a directed line segment (in two dimensions this is termed a vector) can be established by placing the origin of the coordinate axis at the tail of the vector. Parallel lines are placed through the head of the vector. The intersection of the line parallel to the x^{1} axis with the x^{2} axis provides the x^{2} coordinate. Similarly, the intersection of the line parallel to the x^{2} axis with the x^{1} axis provides the x^{1} coordinate.
By definition; the oblique, rectilinear, contravariant coordinates of the point P above are summarized as: x^{i} = (x^{1},x^{2})
Notice the superscript, this is a standard nomenclature convention for contravariant tensor components and should not be confused with the subscript; which is used to designate covariant tensor components.
Using the definition above, the contravariant components of a position vector v^{i}, where i=2, can be defined as the differences between coordinates (or position vectors) of the head and tail, on the same coordinate axis. Stated in another way, the vector components are the projection onto an axis from the direction parallel to the other axis.
so, since we have placed our origin at the tail of the vector,
v^{2}=( (x^{1}  0), (x^{2}  0 ) )
v^{2}=(x^{1}, x^{2})
This result is generalized into ndimensions. Contravariant is a fundamental concept or property within tensor theory and applies to tensors of all ranks over all manifolds. Since whether tensor components are contravariant or covariant, how they are mixed, and the order of operations all impact the results it is imperative to track for correct application of methods.
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