atlas[cov] - covariant derivative

Calling Sequence:

     cov(V,expr)

Parameters:

         expr - any expression.
        V - vector field.

Description:

• The cov  procedure calculates 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);

Declare functions:
Functions(f=f(x,y),h=h(z));

Declare p-forms:
Forms(e[k]=1,xi=1);

Declare vectors:
Vectors(E[j],X,Y,Z);

Declare tensors:
Tensors(T=[n,k],Omega=[l,m]);

Using cov - procedure:

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

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

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

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

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

And:
'cov(X+Y,Z)'=cov(X+Y,Z);

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

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

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

Examples of calculations with definite metric

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

Declare frame:
Frame(E[k]);

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

Calculate connection:
Connection(theta);

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

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

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

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

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

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

See Also:

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