## 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{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}