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[f1f1[x1,x2,...,xn],f2f2[y1,y2,...,ym],...,
fkfk[z1,z2,...,zj]]
fk=fk[z1, z2, ..., zj] - equations where fk-function identifier and zj - variables.

Necessary functions.

In[7]:=
Click for copyable input
The following declaration means that h1 are Constants for any i.
In[8]:=
Click for copyable input
Out[8]=
In[9]:=
Click for copyable input
Out[9]=
In[10]:=
Click for copyable input
Out[10]=
In[11]:=
Click for copyable input
Out[11]=
In[12]:=
Click for copyable input
Out[12]=
The following declaration means that , 0, are Constants:
In[13]:=
Click for copyable input
Out[13]=
In[14]:=
Click for copyable input
Out[14]=
In[15]:=
Click for copyable input
Out[15]=
In[16]:=
Click for copyable input
Out[16]=
In[17]:=
Click for copyable input
Out[17]=
In[18]:=
Click for copyable input
Out[18]=
In[19]:=
Click for copyable input
Out[19]=
In[20]:=
Click for copyable input
Out[20]=
The following declaration means that fi=fi(y1, y2, .., yn) for any i where n is the dimension.
In[21]:=
Click for copyable input
Out[21]=
In[22]:=
Click for copyable input
Out[22]=
In[23]:=
Click for copyable input
Out[23]=
In[24]:=
Click for copyable input
Out[24]=
In[25]:=
Click for copyable input
Out[25]=
The following declaration means that hi, j=hi, j(x, y, z) for any i, j .
In[26]:=
Click for copyable input
Out[26]=
In[27]:=
Click for copyable input
Out[27]=
In[28]:=
Click for copyable input
Out[28]=
In[29]:=
Click for copyable input
Out[29]=
In[30]:=
Click for copyable input
Out[30]=
The following declaration means that f=f(z1, z2, .., zn) where n is the dimension.
In[31]:=
Click for copyable input
Out[31]=
In[32]:=
Click for copyable input
Out[32]=
In[33]:=
Click for copyable input
Out[33]=
The following declaration means that F=F(z0, z3).
In[34]:=
Click for copyable input
Out[34]=
In[35]:=
Click for copyable input
Out[35]=
In[36]:=
Click for copyable input
Out[36]=
The following definition means that G=G(z0, x1, x2, .., xn) where n is the dimension.
In[37]:=
Click for copyable input
Out[37]=
In[38]:=
Click for copyable input
Out[38]=
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:
In[39]:=
Click for copyable input
Out[39]=
In[40]:=
Click for copyable input
Out[40]=
In[41]:=
Click for copyable input
Out[41]=
In[42]:=
Click for copyable input
Out[42]=
In[43]:=
Click for copyable input
Out[43]=
In[44]:=
Click for copyable input
Out[44]=
In[45]:=
Click for copyable input
Out[45]=
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.
In[47]:=
Click for copyable input
Out[47]=
In[48]:=
Click for copyable input
Out[48]=
In[49]:=
Click for copyable input
Out[49]=
In[50]:=
Click for copyable input
Out[50]=
In[51]:=
Click for copyable input
Out[51]=
In[52]:=
Click for copyable input
Out[52]=
More complex example:
In[53]:=
Click for copyable input
Out[53]=
In[54]:=
Click for copyable input
Out[54]=
In[55]:=
Click for copyable input
Out[55]=
In[56]:=
Click for copyable input
Out[56]=
In[57]:=
Click for copyable input
Out[57]=
In[58]:=
Click for copyable input
Out[58]=