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In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In physics, a basis is sometimes thought of as a set of reference axes. A change of scale on the reference axes corresponds to a change of units in the problem. For instance, in changing scale from meters to centimeters (that is, ''dividing'' the scale of the reference axes by 100), the components of a measured velocity vector will ''multiply'' by 100. Vectors exhibit this behavior of changing scale ''inversely'' to changes in scale to the reference axes: they are ''contravariant''. As a result, vectors often have units of distance or distance times some other unit (like the velocity). In contrast, dual vectors (also called ''covectors'') typically have units the inverse of distance or the inverse of distance times some other unit. An example of a dual vector is the gradient, which has units of a spatial derivative, or distance−1. The components of dual vectors change in the ''same way'' as changes to scale of the reference axes: they are ''covariant''. The components of vectors and covectors also transform in the same manner under more general changes in basis: * For a vector (such as a direction vector or velocity vector) to be basis-independent, the components of the vector must ''contra-vary'' with a change of basis to compensate. That is, the matrix that transforms the vector of components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of dual vectors) are said to be contravariant. Examples of vectors with ''contravariant components'' include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity, acceleration, and jerk. In Einstein notation, contravariant components are denoted with ''upper indices'' as in *: * For a dual vector (also called a ''covector'') to be basis-independent, the components of the dual vector must ''co-vary'' with a change of basis to remain representing the same covector. That is, the components must be transformed by the same matrix as the change of basis matrix. The components of dual vectors (as opposed to those of vectors) are said to be covariant. Examples of ''covariant'' vectors generally appear when taking a gradient of a function. In Einstein notation, covariant components are denoted with ''lower indices'' as in *: Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems. Associated to any coordinate system is a natural choice of coordinate basis for vectors based at each point of the space, and covariance and contravariance are particularly important for understanding how the coordinate description of a vector changes in passing from one coordinate system to another. The terms covariant and contravariant were introduced by James Joseph Sylvester in 1853 in order to study algebraic invariant theory. In this context, for instance, a system of simultaneous equations is contravariant in the variables. Tensors are objects in multilinear algebra that can have aspects of both covariance ''and'' contravariance. The use of both terms in the modern context of multilinear algebra is a specific example of corresponding notions in category theory. ==Introduction== In physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list (or tuple) of numbers such as : The numbers in the list depend on the choice of coordinate system. For instance, if the vector represents position with respect to an observer (position vector), then the coordinate system may be obtained from a system of rigid rods, or reference axes, along which the components ''v''1, ''v''2, and ''v''3 are measured. For a vector to represent a geometric object, it must be possible to describe how it looks in any other coordinate system. That is to say, the components of the vectors will ''transform'' in a certain way in passing from one coordinate system to another. A ''contravariant vector'' has components that "transform as the coordinates do" under changes of coordinates (and so inversely to the transformation of the reference axes), including rotation and dilation. The vector itself does not change under these operations; instead, the components of the vector make a change that cancels the change in the spatial axes, in the same way that co-ordinates change. In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, like the co-ordinates, would reduce in an exactly compensating way. Mathematically, if the coordinate system undergoes a transformation described by an invertible matrix ''M'', so that a coordinate vector x is transformed to x′ = ''M''x, then a contravariant vector v must be similarly transformed via v′ = ''M''v. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities. For example, if ''v'' consists of the ''x'', ''y'', and ''z''-components of velocity, then ''v'' is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way. Examples of contravariant vectors include displacement, velocity and acceleration. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract vector, but this vector would not be contravariant, since a change in coordinates on the space does not change the box's length, width, and height: instead these are scalars. By contrast, a ''covariant vector'' has components that change oppositely to the coordinates or, equivalently, transform like the reference axes. For instance, the components of the gradient vector of a function : transform like the reference axes themselves. When only rotations of the axes are considered, the components of contravariant and covariant vectors behave in the same way. It is only when other transformations are allowed that the difference becomes apparent. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「covariance and contravariance of vectors」の詳細全文を読む スポンサード リンク
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