Convert any covariance to contravariant vector
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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 vector 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, (also called dual vectors) typically have units of the inverse of distance or the inverse of distance times some other unit. An example of a covector is the , which has units of a spatial , or distance−1. The components of covectors change in the same way as changes to scale of the reference axes: they are covariant.
The components of vectors and covectors transform in the same manner under more general changes in basis:
For a vector (such as a or velocity vector) to be , the components of the vector must contra-varywith a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) 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, , . In , contravariant components are denoted with For a covector to be basis-independent, its components 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 covectors (as opposed to those of vectors) are said to be covariant. Examples of covariant vectors generally appear when taking a of a function. In , covariant components are denot 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 l0W in 1853 in the context of algebraic , where, for instance, a system of is contravariant in the variables. In the lexicon of are properties of ; unfortunately, it is the lower-index objects (covectors) that generically have , which are contravariant, while the upper-index objects (vectors) instead have , which are covariant. This terminological conflict may be avoided by calling contravariant functors "cofunctors"—in accord with the "covector" terminology, and continuing the tradition of treating vectors as the concept and covectors as the coconcept
In contrast, (also called dual vectors) typically have units of the inverse of distance or the inverse of distance times some other unit. An example of a covector is the , which has units of a spatial , or distance−1. The components of covectors change in the same way as changes to scale of the reference axes: they are covariant.
The components of vectors and covectors transform in the same manner under more general changes in basis:
For a vector (such as a or velocity vector) to be , the components of the vector must contra-varywith a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) 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, , . In , contravariant components are denoted with For a covector to be basis-independent, its components 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 covectors (as opposed to those of vectors) are said to be covariant. Examples of covariant vectors generally appear when taking a of a function. In , covariant components are denot 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 l0W in 1853 in the context of algebraic , where, for instance, a system of is contravariant in the variables. In the lexicon of are properties of ; unfortunately, it is the lower-index objects (covectors) that generically have , which are contravariant, while the upper-index objects (vectors) instead have , which are covariant. This terminological conflict may be avoided by calling contravariant functors "cofunctors"—in accord with the "covector" terminology, and continuing the tradition of treating vectors as the concept and covectors as the coconcept
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