Covariance and contravariance of vectors

In <a href="/facts/Physics/a7aH2y08">physics</a>, especially in <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a> and <a href="/facts/Tensor_analysis/cAC6F6rF">tensor analysis</a>, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a>. Briefly, a contravariant vector is a list of numbers that transforms oppositely to a change of basis, and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called vectors and covariant vectors are called covectors or dual vectors. The terms covariant and contravariant were introduced by <a href="/facts/James_Joseph_Sylvester/6NNurKJH">James Joseph Sylvester</a> in 1851.
<a href="/facts/Curvilinear_coordinate_system/RQD7Cajd">Curvilinear coordinate systems</a>, such as <a href="/facts/Cylindrical_coordinates/1Vc6lhxD">cylindrical</a> or <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a>, are often used in physical and geometric problems. Associated with 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 by passing from one coordinate system to another. <a href="/facts/Tensor/pNjFo5tV">Tensors</a> are objects in <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a> that can have aspects of both covariance and contravariance.

Covariance and contravariance of vectors open-in-new

Covariance and contravariance of vectors