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Tensor[DGGramSchmidt] - calculate an orthonormal basis with respect to a metric

Calling Sequences

     DGGramSchmidt(S, g, method, signature)

Parameters

   S         - a list of differential forms, vectors or tensors

   g         - a metric tensor

   method    - (optional) keyword argument method = "normalized" or method = "un-normalized" 

   signature - (optional) keyword argument signature = pm, where pm is a list of +1 or -1

 

 

Description

Examples

See Also

Description

• 

The command DGGramSchmidt applies the usual Gram-Schmidt procedure to a list , of  independent vectors, forms or tensors (the forms or tensors must all be of the same type). The procedure returns a list orthogonal vectors, forms or tensors such that

 for .

• 

The inner products of differential forms and tensors are calculated using TensorInnerProduct.

• 

The default assumption is that the metric  is positive-definite, that is, the signature is . In this case the length of each  is normalized to 1.  With method = "un-normalized" the  are left un-normalized.

• 

If, for example, the signature of  is , use the keyword signature = [+1, -1, -1, -1]. The length of  is normalized to 1 and the lengths of  are normalized to .

• 

If, for example, the signature of  is  and the vectors  are both null, then the usual Gram-Schmidt procedure will fail. In this case set signature = [[1,-1], 1, 1] and DGGramSchmidt will take  to be a vector of non-zero length in the span of

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form DGGramSchmidt(...) only after executing the command with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-DGGramSchmidt.

Examples

 

Example 1.

First define a 4 dimensional manifold  with coordinates .

 

(2.1)

 

Define a metric  on .

M > 

(2.2)

 

Define a list of vectors S1.

M > 

(2.3)
M > 

(2.4)

 

We use the command TensorInnerProduct. to check this result.

M > 

 

We repeat the same computation with method = "un-normalized". The result is free of square roots but now the vectors are not unit vectors.

M > 

(2.5)
M > 

 

Example 2.

We continue with the metric from Example 1 but now apply the Gram-Schmidt procedure to a list of 2-forms.

 

M > 

(2.6)
M > 

(2.7)
M > 

 

Example 3.

Consider now an indefinite metric.

M > 

(2.8)
M > 

(2.9)

 

A direct application of the GramSchmidt process yields complex-valued forms.

M > 

(2.10)

 

We can adjust the normalization of the last two 1-forms to have length -1 to obtain a real basis:

M > 

(2.11)

 

Example 4.

Consider another indefinite metric.

M > 

(2.12)
M > 

(2.13)

 

A direct application of the GramSchmidt process fails since the first two vectors are null vectors. We can work around this problem by changing the initial basis.

M > 

(2.14)
M > 

(2.15)

 

Alternatively, we can adjust the signature option.

M > 

(2.16)

See Also

DifferentialGeometry, Tensor, TensorInnerProduct


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