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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. Small icon for merging articles File links The following pages link to this file: Friction Jacobin Private branch exchange Pro-feminist Rotary piston engine Tagalog language Saint Veronica Spoiler effect Parser Password length equation Sudovian language Wikipedia:Why arent these pages copy-edited Static scoping Maximum power theorem General... In physics, a covariant transformation is a rule (specified below), that describes how certain physical entities change under a change of coordinate system. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... For more technical Wiki articles on tensors, see the section later in this article. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

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 category theory, see covariant functor. ...

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) Download high resolution version (2042x1280, 56 KB)My very first Metapost creation. ...

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. In category theory, see covariant functor. ...

Novices may be puzzled as to whether there is a fundamental difference in the way contravariant and covariant components can be used, or whether one could simply interchange them everywhere. The answer is that in curved spaces, or in curved coordinate systems in flat space (e.g. cylindrical coordinates in Euclidean space), the quantity dxⁱ is a perfect differential that can be immediately integrated to yield xⁱ, whilst the covariant components of the same differential, dx_i are not in general perfect differentials; the integrated change depends on the path. In the example of cylindrical coordinates, for example, the radial and z components are the same in covariant and contravariant form, but the covariant differential of angle round the z axis is r² times the contravariant one, dθ and its integral depends on the path. In category theory, see covariant functor. ...

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. This article discusses the pullback in differential geometry. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

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. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, a functor is a special type of mapping between categories. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, multilinear algebra extends the methods of linear algebra. ...