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Differential form

  • Thread starter Canavar
  • Start date
16
0
Hello,

I try to understand differential forms. For istance i want to prove that
[tex]h=e_1\wedge e_2 + e_3\wedge e_4[/tex]
is a differential form, where e_1,..,e_4 are elements of my basis.



Do you have a idea, why this is a differential form?

Regards
 

Answers and Replies

HallsofIvy
Science Advisor
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First, your 'basis' elements are themselves "differentials". If you think of [itex]e_1= dx[/itex], [itex]e_2= dy[/itex], [itex]e_3= dz[/itex], and [itex]e_4= dt[/itex] then [itex]e_1\wedge e_2+ e_3\wedge e_4= dxdy+ dzdt[/itex]. It would convert the function f(x,y,z, t) into
[tex]\int\int f(x,y,z,t) dxdy+ f(x,y,z,t)dzdt[/tex]

The "wedge", [itex]\wedge[/itex], is there because this product is "anti-symmetric" [itex]dx\wedge dy= -dy\wedge dx[/itex] so the, in particular, such things as "[itex]dx\wedge dx[/itex]" will be 0.
 
16
0
Hello,

thank you, but why it is a differential form? We have defined differential form as a smooth section of the projection map.
Therefore i have to show this. But for instance i do not see why it is smooth.

Regards
 

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