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Let us now call e_{i} the basic vectors of a vectorial space and x^{i} the coordinates of a given vector x. We may write:
(3.1) |
(3.2) | |
Quantities with a subscript transform in a change of coordinate like the basic vectors and are called covariant ; those with a superscript transform like the coordinates and are called countervariant . Let us now consider the scalar products:
(3.3) |
(3.4) |
x^{i} = x_{j}g^{ij} | (3.5) |
(3.6) |
Let us now introduce the following set of vectors
(3.7) |
This set of vectors constitutes a set of basic vectors. To show this we may simply transform equation (3.1):
(3.8) |
The vectors e^{j} constitute therefore a set of basic vectors and the x_{j} are the coordinates of x with respect to this base. They are called countervariant basic vectors. They are also identical to the basic vectors of the reciprocal space. This can easily be demonstrated by showing that they satisfy the basic relations (2.3) of the reciprocal space vectors. Let us consider the scalar products .Using (3.7), (3.4) and (3.6), we may write:
which is indeed identical to 2.3.
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