why a vector field w ( x , y , z ) = f 1 e ^ x + f 2 e ^ y + f 3 e ^ z can be associated with the 2-form ω 2 = f 1 d y ∧ d z + f 2 d z ∧ d x + f 3 d x ∧ d y .Is it OK to say this works because the wedge product of the 1-forms ω = u 1 d x + u 2 d y + u 3 d z and ν = v 1 d x + v 2 d y + v 3 d z : ω ∧ ν = ( u 2 v 3 − u 3 v 2 ) d y ∧ d z + ( u 3 v 1 − u 1 v 3 ) d z ∧ d x + ( u 1 v 2 − u 2 v 1 ) d x ∧ d yhas the same components as the cross product of the vectors u = u 1 e ^ x + u 2 e ^ y + u 3 e ^ z and v = v 1 e ^ x + v 2 e ^ y + v 3 e ^ z ?

Question

why a vector field      w        (    x    ,    y    ,    z    )    =      f          1                                    e                ^                    x        +      f          2                                    e                ^                    y        +      f          3                                    e                ^                    z      can be associated with the 2-form      ω          2        =      f          1          d  y  ∧  d  z  +      f          2          d  z  ∧  d  x  +      f          3          d  x  ∧  d  y  .Is it OK to say this works because the wedge product of the 1-forms   ω  =      u          1        d  x  +      u          2        d  y  +      u          3        d  z and   ν  =      v          1        d  x  +      v          2        d  y  +      v          3        d  z  :  ω  ∧  ν  =      (          u              2                    v              3              −          u              3                    v              2              )    d  y  ∧  d  z  +      (          u              3                    v              1              −          u              1                    v              3              )    d  z  ∧  d  x  +      (          u              1                    v              2              −          u              2                    v              1              )    d  x  ∧  d  yhas the same components as the cross product of the vectors       u    =      u          1                                    e                ^                    x        +      u          2                                    e                ^                    y        +      u          3                                    e                ^                    z       and       v    =      v          1                                    e                ^                    x        +      v          2                                    e                ^                    y        +      v          3                                    e                ^                    z      ?

Momellaxi · Accepted Answer

Yes, the wedge product of two vectors in             R        3   corresponds to the cross product. But what&#039;s going on here is that to a vector field       w    =  (      f    1    ,      f    2    ,      f    3    ) you can associate both a 1-form  ω  =      f    1      d  x  +      f    2      d  y  +      f    3      d  zand a 2-form  η  =  ⋆  ω  =      f    1      d  y  ∧  d  z  +      f    2      d  z  ∧  d  x  +      f    3      d  x  ∧  d  y  .If you have a curve   γ, then       ∫    γ    ω gives the work done by w along   γ. If you have an oriented surface   Σ, then       ∫    Σ    η gives the flux of w across   ΣWithout any discussions of Stokes&#039;s (or the Divergence) Theorem, the reason for the flux interpretation is as follows. Let u,v span an oriented parallelogram       P   in             R        3  , and let n the unit vector normal to the parallelogram. Then                    η        (                  u                ,                  v                )                            =                  |                                                                      f                  1                                                                              f                  2                                                                              f                  3                                                                                                      u                  1                                                                              u                  2                                                                              u                  3                                                                                                      v                  1                                                                              v                  2                                                                              v                  3                                                              |                =                  w                ⋅        (                  u                ×                  v                )                                          =        (                  w                ⋅                  n                )        (                  n                ⋅        (                  u                ×                  v                )        )        =        (                  w                ⋅                  n                )        ‖                  u                ×                  v                ‖                                          =        (                  w                ⋅                  n                )        area        (                  P                )        .

Vector field and 2-form correspondence - is this explanation correct? why a vector field w(x,y,z)=f_1 hat(e)_x+f_2 hat(e)_y+f_3 hat(e)_z can be associated with the 2-form omega_2=f_1 dy ^^ dz+f_2 dz ^^ dx+f_3 dx ^^ dy.

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