contravariant

Contravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projection) used to derive the components by projecting the magnitude of the tensor quantity onto the coordinate system being used as the basis of the tensor. Another method is used to derive covariant tensor components. When performing tensor transformations it is critical that the method used to map to the coordinate systems in use be tracked so that operations may be applied correctly for accurate, meaningful results. 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 x¹ axis with the x² axis provides the x² coordinate. Similarly, the intersection of the line parallel to the x² axis with the x¹ axis provides the x¹ coordinate. By definition; the oblique, rectilinear, contravariant coordinates of the point P above are summarized as: xⁱ = (x¹, x²) 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 vⁱ, where i = {1, 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,

vⁱ = ( (x¹ − 0), (x² − 0 ) )

vⁱ = (x¹, x²)

This result is generalized into n-dimensions. Contravariance 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. In more modern terms, the transformation properties of the covariant indecies of a tensor are given by a pullback; by contrast, the transformation of the contravariant indecies is given by a pushforward. In category theory a functor may be covariant or contravariant, with the dual space being a standard example of a contravariant construction and tensor. Some constructions of multilinear algebra are of 'mixed' variance, which prevents them from being functors as such.

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