atlas[delta] - Kronecker's delta symbol 

Calling Sequence: 

    delta[i,j] 

Parameters: 

       i, j - variables or integers. 

Description: 

  • The delta table allows one to use Kronecker's delta symbol. The main syntax is as follows: delta[i,j] i.e. delta[i, j] .
 

  • The definition for Kronecker's delta symbol is as follows:
 

  • delta is just the table: `assign`(delta, table(symmetric, identity))
 

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,omega=2,sigma=p);
 

{omega, sigma, 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 delta procedure: 

Interior product of frame vector and coframe form:
'iota[E[j]](e[k])'=iota[E[j]](e[k]); 

iota[E[j]](e[k]) = delta[j, k](2.6)
 

To basis decomposition:
X=ToBasis(X);
 

X = `+`(`*`(iota[X](e[1]), `*`(E[1])), `*`(iota[X](e[2]), `*`(E[2])), `*`(iota[X](e[3]), `*`(E[3])))(2.7)
 

And then:
iota[X](e[k])=iota[ToBasis(X)](e[k]); 

iota[X](e[k]) = `+`(`*`(iota[X](e[1]), `*`(delta[1, k])), `*`(iota[X](e[2]), `*`(delta[2, k])), `*`(iota[X](e[3]), `*`(delta[3, k])))(2.8)
 

Some more examples:
'iota[E[n],E[i]](e[j]&^e[k])'=iota[E[n],E[i]](e[j]&^e[k]); 

iota[E[n], E[i]](`&^`(e[j], e[k])) = `+`(`*`(delta[i, j], `*`(delta[n, k])), `-`(`*`(delta[i, k], `*`(delta[n, j]))))(2.9)
 

For function f:
'iota[E[k]](d(f))'=iota[E[k]](d(f)); 

iota[E[k]](d(f)) = `+`(`*`(Diff(f, x), `*`(delta[1, k])), `*`(Diff(f, y), `*`(delta[2, k])), `*`(Diff(f, z), `*`(delta[3, k])))(2.10)
 

For exterior product:
'iota[E[j]](e[k]&^d(f))'=iota[E[j]](e[k]&^d(f)); 

iota[E[j]](`&^`(e[k], d(f))) = `+`(`-`(`*`(Diff(f, x), `*`(`+`(`*`(delta[1, j], `*`(e[k])), `-`(`*`(delta[j, k], `*`(e[1]))))))), `-`(`*`(Diff(f, y), `*`(`+`(`*`(delta[2, j], `*`(e[k])), `-`(`*`(delta...
iota[E[j]](`&^`(e[k], d(f))) = `+`(`-`(`*`(Diff(f, x), `*`(`+`(`*`(delta[1, j], `*`(e[k])), `-`(`*`(delta[j, k], `*`(e[1]))))))), `-`(`*`(Diff(f, y), `*`(`+`(`*`(delta[2, j], `*`(e[k])), `-`(`*`(delta...
(2.11)
 

>
 

See Also:  

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