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For example, [5] suppose that we are given a basis e 1, e 2 consisting of a pair of vectors making a 45° angle with one another, such that e 1 has length 2 and e 2 has length 1. Then the dual basis vectors are given as follows: e 2 is the result of rotating e 1 through an angle of 90° (where the sense is measured by assuming the pair e 1, e 2 ...
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 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 ...
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}} .
The contravariant basis isn't a very convenient one to use, however it shows up in definitions so must be considered. We'll favor writing quantities with respect to the covariant basis. Since the basis vectors are all constant, vector addition and subtraction will simply be familiar component-wise adding and subtraction.
When working with coordinates in geometric algebra it is usual to write the basis vectors as (e 1, e 2, ...), a convention that will be used here. A vector in real two-dimensional space R 2 can be written a = a 1 e 1 + a 2 e 2, where a 1 and a 2 are real numbers, e 1 and e 2 are orthonormal basis vectors. The geometric product of two such ...
Given two inertial or rotated frames of reference, a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix Λ: ′ =. In index notation, the contravariant and covariant components transform according to, respectively: ′ =, ′ = in which the matrix Λ has components Λ μ ν in row μ and column ν, and the matrix (Λ −1) T has components Λ ...