Surfaces in  

Abstract revolutionary surface  

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 is equal to zero of course). 

>	Domain(R^3);

 

(2.1)

 

>	Forms(e[k]=1);

 

(2.2)

 

>	Vectors(E[j]);

 

(2.3)

 

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

 

(2.4)

 

>	Frame(E[k]);

 

(2.5)

 

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

 

(2.6)

 

>	Connection(omega);

 

(2.7)

 

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); 

(2.1.1)

 

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

(2.1.2)

 

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

(2.1.3)

 

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

(2.1.4)

 

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

(2.1.5)

 

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

(2.1.6)

 

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

 

	(2.1.7)

 

One can also calculate metric induced on the surface by the mapping. 

>	Metric(G = g &/ pi);

 

(2.1.8)

 

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

(2.1.9)

 

Let us extract the mean curvature vector field: 

>	mu:=Inv[meanCurvature];

 

(2.1.10)

 

Let us extract the second fundamental form: 

>	Sf:=eval(Inv[secondForm]);

 

(2.1.11)

 

Now we can calculate the corresponding tensor: 

>	B:=add(add(`&.`(e[i],e[j],eval(Sf)[i,j]) ,i=1..2),j=1..2);

 

(2.1.12)

 

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