atlas[ToBasis] - To basis decomposition 

Calling Sequence: 

    ToBasis(T) 

Parameters: 

    T - variable or expression of tensor type 

Description: 

The ToBasis procedure allows one to decompose a tensor into its basis. If the coframe and frame have not been previously defined, then no decomposition is available.  

Examples: 

>	restart: with(atlas):

 

Declare forms: 

>	Forms(e[j]=1,xi=1,phi=1);

 

(2.1)

 

Declare vectors: 

>	Vectors(X,Y,Z,E[j]);

 

(2.2)

 

One can not make the decomposition before coframe declaration:
ToBasis(X); 

(2.3)

 

Declare coframe:
Coframe(e[j],j=1..n); 

(2.4)

 

Declare frame:
Frame(E[k]); 

(2.5)

 

"To basis" decomposition 

>	X=ToBasis(X);

 

(2.6)

 

>	xi=ToBasis(xi);

 

(2.7)

 

>	X&.xi=ToBasis(X&.xi);

 

(2.8)

 

Functions declaration : 

>	Functions(xi[i]=xi[i](x[k]));

 

(2.9)

 

New cofame declaration: 

>	Coframe(e[j]=xi[j]*d(x[j]),j=1..3);

 

Warning, You have to restart the Coframe procedure and then Frame procedure!

 

>	Coframe(e[j]=xi[j]*d(x[j]),j=1..3);

 

(2.10)

 

>	'd(xi[i])'=d(xi[i]);

 

(2.11)

 

>	'd(x[k])'=d(x[k]);

 

(2.12)

 

>	'd(e[3])'=d(e[3]);

 

(2.13)

 

Declare frame:
Frame(E[k]); 

(2.14)

 

"To basis" decomposition 

>	X=ToBasis(X);

 

(2.15)

 

Note that   are functions but is 1-form: 

>	xi=ToBasis(xi);

 

(2.16)

 

>	X&.xi=ToBasis(X&.xi);

 

(2.17)

 

>	xi&.xi=ToBasis(xi&.xi);

 

(2.18)

 

>	&@(x)=ToBasis(&@(x));

 

(2.19)

 

>	&@(x[1])=ToBasis(&@(x[1]));

 

(2.20)

 

>

 

See Also:  

atlas, atlas[Coframe], atlas[Frame].