atlas[cov] - covariant derivative 

Calling Sequence: 

    cov(V,expr)
    cov[V](expr) 

Parameters: 

       expr - any expression.
       V - vector field. 

Description: 

• The cov - procedure allows one to calculate the covariant derivative on an expression along given vector field. The derivative has the following properties.

 

• For any vector field X and 0-form we have:   

 

• For vector fields X, Y and Z we have: and for any functions and .

 

• For any vector field X and tensor fields and the Leibniz rule for the derivative takes place:

 

Examples: 

>	restart: with(atlas):

 

Declare constants:  

>	Constants(Lambda);

 

(2.1)

 

Declare functions:  

>	Functions(f=f(x,y),h=h(z));

 

(2.2)

 

Declare p-forms:  

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

 

(2.3)

 

Declare vectors:  

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

 

(2.4)

 

Declare tensors:  

>	Tensors(T=[n,k],Omega=[l,m]);

 

(2.5)

 

Using cov- procedure: 

As h is 0-form by defaults then:
'cov[X](h)'=cov(X,h); 

(2.6)

 

f- declared as function f=f(x,y) thus:
'cov[X](f)'=cov(X,f);
 

(2.7)

 

Obviously that:
'cov[X](x^2*sin(x))'=cov(X,x^2*sin(x)); 

(2.8)

 

As declared as constant thus:
'cov[X](Lambda)'=cov(X,Lambda); 

(2.9)

 

Lie derivative is linear with respect to any argument:
'cov[X](Y+Z)'=cov(X,Y+Z); 

(2.10)

 

And:
'cov[X+Y](Z)'=cov(X+Y,Z); 

(2.11)

 

More complex examples:
'cov[f*X+h*Y](Z)'=cov(f*X+h*Y,Z); 

(2.12)

 

And:
'cov[Z](f*X+h*Y)'=cov(Z,f*X+h*Y); 

(2.13)

 

Verify that Leibniz rule takes place for tensor product:
'cov[X](T&.Omega)'=cov(X,T&.Omega); 

(2.14)

 

Examples of calculations with definite metric 

Declare coframe:
Coframe(e[1]=d(x),e[2]=d(y)); 

(2.1.1)

 

Declare frame:
Frame(E[k]); 

(2.1.2)

 

Declare metric:
Metric(g=4*(d(x)&.d(x)+d(y)&.d(y))/(1+Lambda*(x^2+y^2))^2); 

(2.1.3)

 

Calculate connection:
Connection(theta); 

(2.1.4)

 

For frame vectors E[j]:
'cov[E[j]](sin(x)*cos(y))'=cov(E[j],sin(x)*cos(y)); 

(2.1.5)

 

As g is a metric then for any vector field X:
'cov[X](g)'=cov(X,g); 

(2.1.6)

 

By definition:
'cov[E[i]](E[1])'=cov(E[i],E[1]); 

(2.1.7)

 

For coframe 1-forms:
'cov[E[k]](e[2])'=cov(E[k],ToBasis(e[2])); 

(2.1.8)

 

For exterior product:
'cov[E[k]](e[1]&^e[2])'=cov(E[k],e[1]&^e[2]); 

(2.1.9)

 

For tensor product:
'cov[E[k]](e[2]&.E[2])'=cov(E[k],e[2]&.E[2]); 

(2.1.10)

 

>

 

See Also:  

atlas, atlas[d], atlas[cov], atlas[`&.`], atlas[`&^`], atlas[iota].