Indexing facilities in the atlas package  

Description: 

• 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 type. For instance, the declaration Constants(c[i]) means that c[i] 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(xi[3]=1, xi[0]=n) means that xi[3] is 1-form, xi[0] is n-form and xi[i] is 0-form if i is not equal to 3 or 0.  

 

Examples: 

>	restart: with(atlas):

 

The following declaration means that are constants for any .  

>	Constants(h[i]);

 

(2.1)

 

>	type(h[k],const);

 

(2.2)

 

>	type(h[3],const);

 

(2.3)

 

>	type(h[-1/2],const);

 

(2.4)

 

>	'd(h[j])'=d(h[j]);

 

(2.5)

 

The following declaration means that are constants: 

>	Constants(alpha[0],alpha[1/2],alpha[-1/2]);

 

(2.6)

 

>	type(alpha[0],const);

 

(2.7)

 

>	type(alpha[1/2],const);

 

(2.8)

 

>	type(alpha[-1/2],const);

 

(2.9)

 

>	type(alpha[2],const);

 

(2.10)

 

>	type(alpha[k],const);

 

(2.11)

 

>	'd(alpha[j])'=d(alpha[j]);

 

(2.12)

 

>	'd(alpha[-1/2])'=d(alpha[-1/2]);

 

(2.13)

 

The following declaration means that for any where n is the dimension. 

>	Functions(f[i]=f[i](y[k]));

 

(2.14)

 

>	type(f[k],func);

 

(2.15)

 

>	type(f[0],func);

 

(2.16)

 

>	type(f[-1],func);

 

(2.17)

 

>	'd(f[j])'=d(f[j]);

 

(2.18)

 

The following declaration means that for any . 

>	Functions(h[i,j]=h[i,j](x,y,z));

 

(2.19)

 

>	type(h[m,k],func);

 

(2.20)

 

>	type(h[0,-1],func);

 

(2.21)

 

>	type(h[-1,i],func);

 

(2.22)

 

>	'd(h[k,0])'=d(h[k,0]);

 

(2.23)

 

The following declaration means that where n is the dimension. 

>	Functions(f=f(z[k]));

 

(2.24)

 

>	type(f,func);

 

(2.25)

 

>	'd(f)'=d(f);

 

(2.26)

 

The following declaration means that . 

>	Functions(F=F(z[0],z[3]));

 

(2.27)

 

>	type(F,func);

 

(2.28)

 

>	'd(F)'=d(F);

 

(2.29)

 

The following definition means that where n is the dimension. 

>	Functions(G=G(z[0],x[j]));

 

(2.30)

 

>	'd(G)'=d(G);

 

(2.31)

 

The following definition means that are vectors for any k and is a vector:  

>	Vectors(E[k],U[0]);

 

(2.32)

 

>	type(E[i],vect);

 

(2.33)

 

>	type(E[-3],vect);

 

(2.34)

 

>	type(U[0],vect);

 

(2.35)

 

>	type(U[1],vect);

 

(2.36)

 

>	type(U[i],vect);

 

(2.37)

 

>	iota[U[0]](d(x));

 

(2.38)

 

>	iota[U[3]](d(x));

 

Error, (in atlas/iota2) U[3] is not a vector

 

The following definition means that is 1-form for any j; and are 1-form and p-form respectively and are 2-forms for any i, j. 

>	Forms(e[j]=1,omega[1]=1,omega[2]=p,omega[i,j]=2);

 

(2.39)

 

>	kind(e[b]);

 

(2.40)

 

>	kind(e[6]);

 

(2.41)

 

>	kind(omega[1]);

 

(2.42)

 

>	kind(omega[3]);

 

(2.43)

 

>	kind(omega[1,k]);

 

(2.44)

 

More complex example: 

>	Forms(xi[i,0,k]=3);

 

(2.45)

 

>	kind(xi[a,0,b]);

 

(2.46)

 

>	kind(xi[1,0,-1/2]);

 

(2.47)

 

>	kind(xi[i,0,k]);

 

(2.48)

 

>	kind(xi[i,1,k]);

 

(2.49)

 

>	kind(xi[i,j,k]);

 

(2.50)

 

>

 

See Also:  

atlas.