Abstract revolutionary surface  

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Problem:

Find metric and second fundamental form of the following revolutionary surface:     

Solution:

Load atlas package:
restart:
with(atlas):

First of all we have to describe the space we are working in. The space is 3-dimensional Euclidean (flat) space. To define the space we declare domain, forms, vectors, coframe, frame, flat metric and calculate connection (it equals to zero of cause).
Domain(R^3);

Forms(e[k]=1);

Vectors(E[j]);

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

Frame(E[k]);

Metric(g=d(x)&.d(x)+d(y)&.d(y)+d(z)&.d(z));

Connection(omega);

Now the working space is defined completely and we can start to solve the problem.

Just for right simplification:
`atlas/simp`:=proc(a) factor(simplify(a)) end:

Surface

Define the surface as a manifold:
Domain(S);

Declare functions:
Functions(rho=rho(u),zeta=zeta(u));

Declare 1-form for surface coframe
Forms(phi[i]=1);

Declare vectors for surface frame:
Vectors(Phi[k]);

Declare coframe on the surface:
Coframe(phi[1]=d(u),phi[2]=d(v));

Declare frame of the surface:
Frame(Phi[j]);

Declare mapping of the surface into :
Mapping(pi,S,R^3,
                x=rho*cos(v),
                y=rho*sin(v),
                z=zeta);

One can also calculate metric induced on the surface by the mapping.
Metric(G = g &/ pi);

Calculate invariants of the mapping:
Inv:=Invariants(pi);

Let us extract the mean curvature vector field:
mu:=Inv[meanCurvature];

Let us extract the second fundamental form:
Sf:=eval(Inv[secondForm]);

Now we can calculate the corresponding tensor:
B:=add(add(`&.`(e[i],e[j],eval(Sf)[i,j]),i=1..2),j=1..2);