Indexing facilities in the Atlas package
Any object in the Atlas package can be indexed. The following rules are used to provide the indexing facilities.
Any declaration of an object with symbolic indexes means that the indexes can be of any TypeQ. For instance, the declaration
Consctants[ci] means that
ci is Constant for any i.
Any declaration of an object with numeric indexes means that the indexes can be only the same as has been declared. For instance, the declaration
Forms[
3→ 1, 0→n] means that
3 is 1-form,
xi0 is n-form and
i is 0-form if i is not equal to 3 or 0.
Examples:
| Constants[c1,c2,...,ci,...,cn] | c1,c2,...,ci,...,cn-Constants identifiers. |
| d[expr] | expr - any expression |
Functions[f1→f1[x1,x2,...,xn],f2→f2[y1,y2,...,ym],...,
fk→fk[z1,z2,...,zj]] | fk=fk[z1, z2, ..., zj] - equations where fk-function identifier and zj - variables. |
Necessary functions.
The following declaration means that
h1 are Constants for any
i.
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The following declaration means that
,
0,
are Constants:
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The following declaration means that
fi=fi(y1, y2, .., yn) for any
i where n is the dimension.
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The following declaration means that
hi, j=hi, j(x, y, z) for any
i, j .
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The following declaration means that
f=f(z1, z2, .., zn) where n is the dimension.
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The following declaration means that
F=F(z0, z3).
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The following definition means that
G=G(z0, x1, x2, .., xn) where n is the dimension.
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| Vectors[v1,v2,...,vi,...,vn] | vi - vector identivier. |
| Kind[t] | t -any expression containing tensors, vectors, p-forms etc. |
| iotav1,v2,...,vn[expr] | expr - any expression (on which interior product operator is defined).
v1, v2, ..., vn - vector fields. |
Necessary functions.
The following definition means that
k are vectors for any k and
U0 is a vector:
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The following definition means that
ej is 1-form for any j;
1 and
2 are 1-form and p-form respectively and
i, j are 2-forms for any i, j.
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- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
