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atlas[`&@`] - Natural vector operator 

Calling Sequence: 

    &@(x) 

Parameters: 

       x - variable 

Description: 

  • The `&@` operator allows one to represent natural vectors. The definition is as follows.
 

  • For any scalar x we have .
 

Examples: 

> restart:
with(atlas):
 

Declare Functions  

> Functions(f=f(x,y,z));
 

{f}(2.1)
 

Declare Vectors  

> Vectors(X,Y,Z,E[j]);
 

{X, Y, Z, E[j]}(2.2)
 

Declare forms:  

> Forms(e[i]=1);
 

{e[i]}(2.3)
 

Declare coframe:  

> Coframe(e[1]=d(x),e[2]=d(y),e[3]=d(z));
 

[e[1] = d(x), e[2] = d(y), e[3] = d(z)](2.4)
 

Declare frame:  

> Frame(E[k]);
 

[E[1] = Diff(``, x), E[2] = Diff(``, y), E[3] = Diff(``, z)](2.5)
 

Using &@ procedure: 

> &@(x),&@(y),&@(z);
 

Diff(``, x), Diff(``, y), Diff(``, z)(2.6)
 

> iota[&@(x)](e[k]);
 

iota[Diff(``, x)](e[k])(2.7)
 

To basis decomposition: 

> &@(z)=ToBasis(&@(z));
 

Diff(``, z) = E[3](2.8)
 

> &@(phi)=ToBasis(&@(phi));
 

Diff(``, phi) = `+`(`*`(iota[Diff(``, phi)](e[1]), `*`(E[1])), `*`(iota[Diff(``, phi)](e[2]), `*`(E[2])), `*`(iota[Diff(``, phi)](e[3]), `*`(E[3])))(2.9)
 

>
 

See Also:  

atlas, atlas[iota], atlas[Frame], atlas[Coframe], atlas[ToBasis].