atlas 2 for Mathematica®

atlas package Documentation

Declaration operators

Calculation operators

Standard DG operators

Utility operators

General Tutorials

Visualization Tutorials

Atlas Palette Tutorials

Atlas Wizard Tutorials

Licenses & Pricing

Pullback - Pullback of a [0, k] tensor field under a mapping

Pullback[expr, mapId]
allows one to calculate pullback of a covariant tensor field under a mapping.
  • mapId - variable - the mapping identifier i.e. mapId : dom1 ---> dom2 expr - expression - a tensor expression which has to be restricted
  • The pullback is linear operation defined on [0,k] tensors only. The definition is as follows.
  • Let M and N be manifolds of dimensions m=dim(M), n=dim(N). Let F be mapping between the manifolds: F:MN defined by functions: where {x1, x2, ..xm} are local coordinates on M and {y1, y2, ..yn} are local coordinates on N (in some domains).
  • For any [0,1] tensor field T on N the pullback of T under F is tensor field =Pullback[T, F] on M with components Pullback[((T1)(T2)), F]=Pullback[T1, F]Pullback[T2, F] in local coordinates.
  • For tensor product of any [0, k] tensor fields on N the following formula takes place:
  • The formulas considered above completely define the linear pullback operator Pullback.
  • According to the definition it is necessary to calculate the pullbacks on the domain M. Use Domain procedure to jump on M manifold if needed.
EXAMPLESCLOSE ALL
Basic Examples   (1)
The following example shows how the pullback operator can be used.
Let M be 2-dimentional sphere S2 and N be 3-dimensional Euclidean space R3. Let F:MN be standard embedding of sphere S2 into R3.
In[1]:=
Click for copyable input
Declare 1-forms ej and uk for corresponding coframes:
In[2]:=
Click for copyable input
Out[2]=
Declare vectors for corresponding frames:
In[3]:=
Click for copyable input
Out[3]=
Declare Euclidean space - R3:
In[4]:=
Click for copyable input
Out[4]=
Declare coframe on R3:
In[5]:=
Click for copyable input
Declare frame on R3:
In[6]:=
Click for copyable input
Out[6]=
Declare metric on R3 (standard flat metric):
In[7]:=
Click for copyable input
Out[7]=
Declare sphere - S2:
In[8]:=
Click for copyable input
Out[8]=
Declare coframe on S2:
In[9]:=
Click for copyable input
Declare frame on S2:
In[10]:=
Click for copyable input
Out[10]=
Declare definite mapping F:S2R3:
In[11]:=
Click for copyable input
Visualize the mapping:
In[12]:=
Click for copyable input
Out[12]=
In[13]:=
Click for copyable input
Out[13]//TableForm=
Verify that we are on the sphere:
In[14]:=
Click for copyable input
Out[14]=
Calculate metric induced on the sphere using pullback operator:
In[15]:=
Click for copyable input
Out[15]=
One can calculate pullback of any [0,k] tensor field on R3 under the mapping: pullback of coframe 1-forms:
In[16]:=
Click for copyable input
Out[16]=
Out[16]=
pullback of 0-forms (scalars):
In[17]:=
Click for copyable input
Out[17]=
pullback of "rotation" 1-form:
In[18]:=
Click for copyable input
Out[18]=
pullback of tensor product d(x)d(z):
In[19]:=
Click for copyable input
Out[19]=
pullback of exterior product d(x)^d(y):
In[20]:=
Click for copyable input
Out[20]=
Some more examples:
Declare abstract mapping between S2 and R3:
In[21]:=
Click for copyable input
In[22]:=
Click for copyable input
Out[22]//TableForm=
pullback of exterior product d(x)^d(y) under abstract mapping :
In[23]:=
Click for copyable input
Out[23]=
pullback of coframe 1-forms
In[24]:=
Click for copyable input
Out[24]=
Out[24]=
Who is who?
In[25]:=
Click for copyable input
Out[25]//TableForm=