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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 ...
This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis, or contravariant curvilinear basis, b i. All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each ...
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 ...
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 ...
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.
The numbers of, respectively, vectors: n (contravariant indices) and dual vectors: m (covariant indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law. The order of a tensor is the sum of these two numbers.
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 ...
For any curve and two points = and = on this curve, an affine connection gives rise to a map of vectors in the tangent space at into vectors in the tangent space at : =,, and () can be computed component-wise by solving the differential equation = () = () where () is the vector tangent to the curve at the point ().