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The first is that vectors whose components are covariant (called covectors or 1-forms) actually pull back under smooth functions, meaning that the operation assigning the space of covectors to a smooth manifold is actually a contravariant functor. Likewise, vectors whose components are contravariant push forward under smooth mappings, so the ...
Consequently, a general curvilinear coordinate system has two sets of basis vectors for every point: {b 1, b 2, b 3} is the contravariant basis, and {b 1, b 2, b 3} is the covariant (a.k.a. reciprocal) basis. The covariant and contravariant basis vectors types have identical direction for orthogonal curvilinear coordinate systems, but as usual ...
In particular, if is a diffeomorphism between open subsets of and , viewed as a change of coordinates (perhaps between different charts on a manifold ), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject.
As before, , are covariant basis vectors and b i, b j are contravariant basis vectors. Also, let (e 1, e 2, e 3) be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.
English: To accompany the images File:Transformation2polar.svg and File:Transformation2polar basis vectors.svg. This image shows how under the change of coordinate system presented in those images, the coordinate representation of the vector V transforms contravariantly -- specifically, it is rotated clockwise.
At each point of a manifold, the tangent and cotangent spaces to the manifold at that point may be constructed. Vectors (sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly dual vectors or one-forms) are elements of the cotangent space.
Covariance and contravariance of vectors, in mathematics and theoretical physics; Covariance and contravariance of functors, in category theory; Covariance and contravariance (computer science), whether a type system preserves the ordering ≤ of types; An informal synonym for invariance (physics)
The basis vectors shown above are covariant basis vectors (because they "co-vary" with vectors). In the case of orthogonal coordinates, the contravariant basis vectors are easy to find since they will be in the same direction as the covariant vectors but reciprocal length (for this reason, the two sets of basis vectors are said to be reciprocal ...