Wedge - exterior product operator

Wedge[f1, f2, ..., fn]
allows one to calculate the exterior product of given forms.
  • f1, f2, ..., fn - forms.
  • The main syntax is as follows: Wedge[1, 2] i.e. (1)(2) where 1 and 2 are forms. To calculate the exterior product for forms 1, 2, ..k use the following Wedge[1, 2, ...k) i.e. 1 2..k
  • - The exterior product is a linear operation with respect to its arguments. Thus if , are 0-forms then: (1)(2+3)=(1)(2)+(1)(2)
  • - Let 1 be p-form and 2 be q-form then the following formula defines the exterior product:(1)(2)=(-1)pq(2)(1)
  • - Particularly for 1-forms and we have: = -
  • ⋀ can be entered as \[Wedge] or Esc ^ Esc.
In[1]:=
Click for copyable input
Declare p-forms:
In[2]:=
Click for copyable input
Out[2]=
Declare vectors:
In[3]:=
Click for copyable input
Out[3]=
Using Wedge- procedure: Exterior product is linear operation with respect to its arguments.
In[4]:=
Click for copyable input
Out[4]=
As 1 is p-form and 2 is q-form then under main rule for exterior product we have:
In[5]:=
Click for copyable input
Out[5]=
Particularly for 1-forms and we have:
In[6]:=
Click for copyable input
Out[6]=
Some more examples:
In[7]:=
Click for copyable input
Out[7]=
And with Lie derivative:
In[8]:=
Click for copyable input
Out[8]=
And with exterior derivative:
In[9]:=
Click for copyable input
Out[9]=
And again:
In[10]:=
Click for copyable input
Out[10]=
And finally:
In[11]:=
Click for copyable input
Out[11]=