Dimension in the atlas package  

Description: 

• In the atlas package the working dimension can be integer or symbolic. The corresponding variable is automatically assigned by atlas[Coframe] procedure.

 

• If the dimension is not an integer then all calculations are performed in terms of Sum operator (see examples below).

 

Examples: 

>	restart: with(atlas):

 

Declare some functions:  

>	Functions(f=f(y[k]),h=h(x[j]));

 

(2.1)

 

Declare some vectors:  

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

 

(2.2)

 

Declare some forms: 

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

 

(2.3)

 

Declare coframe 1-forms: 

>	Coframe(e[k],k=1..n);

 

(2.4)

 

Declare frame vectors: 

>	Frame(E[j]);

 

(2.5)

 

Declare connection: 

>	Connection(omega);

 

(2.6)

 

Using d-procedure: 

For functions:
'd(f*h)'=d(f*h); 

(2.7)

 

Obviously that:
'd(h*xi)'=d(h*xi); 

(2.8)

 

Using ToBasis procedure: 

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

 

(2.9)

 

Using div-procedure: 

>	div(X)=div(ToBasis(X));

 

(2.10)

 

Using Lie derivative: 

>	'L[E[i]](e[k])'=L[E[i]](e[k]);

 

(2.11)

 

>	'L[E[i]](E[j])'=L[E[i]](E[j]);

 

(2.12)

 

>

 

Declare constant :
Constants(lambda); 

(2.13)

 

Let us declare another coframe 1-forms:
Coframe(e[j]=d(x[j])/(1+lambda*Sum(x[i]^2,i=1..N)),j=1..N); 

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

 

>	Coframe(e[j]=d(x[j])/(1+lambda*Sum(x[i]^2,i=1..N)),j=1..N);

 

(2.14)

 

And another frame vectors:
Frame(E[k]); 

(2.15)

 

>	'd(e[k])'=normal(d(e[k]));

 

(2.16)

 

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

 

(2.17)

 

>	'd(h)'=d(h);

 

(2.18)

 

>	Functions(F=F(f,h));

 

(2.19)

 

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

 

(2.20)

 

>

 

See Also:  

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