Surfaces in R3

Abstract revolutionary surface

Examples:

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

Examples:

Domain[manifold]manifold - string - a manifold name or a name of a manifold domain.
Metric[id→expr]id - variable - metric identifier, expr - expression - metric declaration.
Connection[id]id - variable - connection identifier.

Necessary functions.

Load the Atlas package:
In[1]:=
Click for copyable input
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).
Declare domain:
In[1]:=
Click for copyable input
Out[1]=
Declare some forms:
In[2]:=
Click for copyable input
Out[2]=
Declare some vectors:
In[3]:=
Click for copyable input
Out[3]=
Declare coframe:
Declare frame:
In[5]:=
Click for copyable input
Out[5]=
Declare a flat metric:
In[6]:=
Click for copyable input
Out[6]=
Calculate the connection of the metric:
In[7]:=
Click for copyable input
Out[7]=
Now the working space is defined completely and we can start to solve the problem.
Just for right simplification:
In[8]:=
Click for copyable input

Surface

Define the surface as a manifold:
In[9]:=
Click for copyable input
Out[9]=
Declare functions:
In[10]:=
Click for copyable input
Out[10]=
Declare 1-form for surface coframe
In[11]:=
Click for copyable input
Out[11]=
Declare vectors for surface frame:
In[12]:=
Click for copyable input
Out[12]=
Declare coframe on the surface:
Declare frame of the surface:
In[14]:=
Click for copyable input
Out[14]=
Declare mapping of the surface into R3:
One can also calculate metric induced on the surface by the mapping.
In[16]:=
Click for copyable input
Out[16]=
Calculate invariants of the mapping:
In[17]:=
Click for copyable input
Out[17]=
Let us extract the mean curvature vector field:
In[18]:=
Click for copyable input
Out[18]=
Let us extract the second fundamental form:
In[19]:=
Click for copyable input
Out[19]=
In[20]:=
Click for copyable input
Out[20]//MatrixForm=
Now we can calculate the corresponding tensor:
In[21]:=
Click for copyable input
Out[21]=
In[22]:=
Click for copyable input
Out[22]=