Transformation of a 3 - form in R3

Theorem
Let  
\[f_1, \: f_2 , \: f_3\]
  be differentiable functions of  
\[x_1, \: x_2, \: x_3\]
  in  
\[\mathbb{R}^3\]
  with the same domain.
Then  
\[df_1 \wedge dx_2 \wedge dx_3 = \frac{\partial (f_1,f_2,f_3)}{\partial (x_1 , x_2, x_3)} dx_1 \wedge dx_2 \wedge dx_3\]

Proof
\[df_1 = \frac{\partial f_1}{\partial x_1}dx_1 + \frac{\partial f_1}{\partial x_1}dx_2 + \frac{\partial f_1}{\partial x_3}dx_3\]

\[df_2 = \frac{\partial f_2}{\partial x_1}dx_1 + \frac{\partial f_2}{\partial x_1}dx_2 + \frac{\partial f_2}{\partial x_3}dx_3\]

\[df_3 = \frac{\partial f_3}{\partial x_1}dx_1 + \frac{\partial f_3}{\partial x_1}dx_2 + \frac{\partial f_3}{\partial x_3}dx_3\]

\[\begin{equation} \begin{aligned} df_1 \wedge dx_2 \wedge dx_3 &= (\frac{\partial f_1}{\partial x_1}dx_1 + \frac{\partial f_1}{\partial x_1}dx_2 + \frac{\partial f_1}{\partial x_3}dx_3) \\ &\wedge ( \frac{\partial f_2}{\partial x_1}dx_1 + \frac{\partial f_2}{\partial x_1}dx_2 + \frac{\partial f_2}{\partial x_3}dx_3) \\ &\wedge (\frac{\partial f_3}{\partial x_1}dx_1 + \frac{\partial f_3}{\partial x_1}dx_2 + \frac{\partial f_3}{\partial x_3}dx_3) \\ &= [(\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2}-\frac{\partial f_1}{\partial x_2}\frac{\partial f_2}{\partial x_1} )dx_1 \wedge dx_2 \\ &+ (\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_3}-\frac{\partial f_1}{\partial x_3}\frac{\partial f_2}{\partial x_1} )dx_1 \wedge dx_3 \\ &+ (\frac{\partial f_1}{\partial x_2} \frac{\partial f_2}{\partial x_3}-\frac{\partial f_1}{\partial x_3}\frac{\partial f_2}{\partial x_2} )dx_2 \wedge dx_3] \wedge (\frac{\partial f_3}{\partial x_1}dx_1 + \frac{\partial f_3}{\partial x_1}dx_2 + \frac{\partial f_3}{\partial x_3}dx_3) \\ &= (\frac{\partial f_1}{\partial x_3} \frac{\partial f_2}{\partial x_1} \frac{\partial f_3}{\partial x_2}- \frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_3} \frac{\partial f_3}{\partial x_2} )dx_1 \wedge dx_2 \wedge dx_3 \\ &+ (\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} \frac{\partial f_3}{\partial x_3}- \frac{\partial f_1}{\partial x_2} \frac{\partial f_2}{\partial x_1} \frac{\partial f_3}{\partial x_3} )dx_1 \wedge dx_2 \wedge dx_3 \\ &+ (\frac{\partial f_1}{\partial x_2} \frac{\partial f_2}{\partial x_3} \frac{\partial f_3}{\partial x_1}- \frac{\partial f_1}{\partial x_3} \frac{\partial f_2}{\partial x_2} \frac{\partial f_3}{\partial x_1} )dx_1 \wedge dx_2 \wedge dx_3 \\ &= \left| \begin{array}{ccc} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\ \frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3} \end{array} \right| \\ &= \frac{\partial (f_1,f_2,f_3)}{\partial (x_1,x_2,x_3)} \end{aligned} \end{equation}\]
 

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