Tensors - declaration of tensors

Tensors[t1→ {n1, p1}, t2→ {n2, p2}, ..., ti→ {ni, pi}]
allows one to declare tensors of any kind but it is normally used to declare tensors that are not vectors or forms.
  • ti→ {ni, pi} - equations where ti - tensor identifier and {ni, pi} is a list of variables or integers - the kind of the tensor.
  • In the Atlas package any identifier is treated as 0-form i.e. as non-constant scalar (if it not declared as constant, p-form, tensor etc. (see Types).
  • For vectors and forms it is more convenient to use Vectors and Forms procedures.
  • The Atlas package deals with tensors themselves not their components. Thus indexed quantities such as ej or Ek are treated as sets of tensors e1, e2, ..ej ..or E1, E2, ..Ek . To get components of a tensor use iota procedures.
In[1]:=
Click for copyable input
Declare tensors T, Omega vectors X, Zk and 1-forms ej:
In[2]:=
Click for copyable input
Out[2]=
Verify that ei is 1-form using kind (see Kind) and type procedures:
In[3]:=
Click for copyable input
Out[3]=
In[4]:=
Click for copyable input
Out[4]=
Verify that X is a vector using kind and type procedures:
In[5]:=
Click for copyable input
Out[5]=
In[6]:=
Click for copyable input
Out[6]=
Calculation of interior product (see iota):
In[7]:=
Click for copyable input
Out[7]=
What kind is the result?
In[8]:=
Click for copyable input
Out[8]=
Construct [1,1] tensor using tensor product operator (see CircleTimes):
In[9]:=
Click for copyable input
Out[9]=
Out[9]=
And another one
In[10]:=
Click for copyable input
Out[10]=
Out[10]=
And for tensor T:
In[11]:=
Click for copyable input
Out[11]=
What kind?
In[12]:=
Click for copyable input
Out[12]=
Calculate Lie derivative of e[j] along vector field X (see L):
In[13]:=
Click for copyable input
Out[13]=
Construct new tensor from Omega and T (see CircleTimes):
In[14]:=
Click for copyable input
Out[14]=
What kind?
In[15]:=
Click for copyable input
Out[15]=
Let's see "who is who":
In[16]:=
Click for copyable input
Out[16]//TableForm=
In[17]:=
Click for copyable input
Out[17]//TableForm=