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Likewise, vectors whose components are contravariant push forward under smooth mappings, so the operation assigning the space of (contravariant) vectors to a smooth manifold is a covariant functor. Secondly, in the classical approach to differential geometry, it is not bases of the tangent bundle that are the most primitive object, but rather ...
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 ...
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 ...
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.
Given a basis for T, we can define a basis, called the dual basis for the dual space in a natural way by taking the set of linear functions mentioned above: the projection functions. Each projection function (indexed by ω) produces the number 1 when applied to one of the basis vectors e i {\displaystyle \mathbf {e} _{i}} .
If ,, are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (,,), then the gradient of the tensor field is given by (see [3] for a proof.) = From this definition we have the following relations for the gradients of a scalar field ϕ {\displaystyle \phi } , a vector field v , and a second ...
A two-vector or bivector [1] is a tensor of type () and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of ...
A prototypical example that gives linear maps their name is a function ::, of which the graph is a line through the origin. [ 7 ] More generally, any homothety v ↦ c v {\textstyle \mathbf {v} \mapsto c\mathbf {v} } centered in the origin of a vector space is a linear map (here c is a scalar).