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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 ...
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.
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 ...
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.
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.
The radial basis vectors e r and e φ appear rotated anticlockwise with respect to the rectangular basis vectors e x and e y. The covariant transformation, performed to the basis vectors, is thus an anticlockwise rotation, rotating from the first basis vectors to the second basis vectors.
A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, [1] in contrast to orthogonal coordinates.. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since the metric tensor will have nonzero off-diagonal components, preventing many simplifications in formulas for tensor algebra and tensor ...