I am quiet new to differential forms so I am not sure if my solution to the following problem is correct.
The Problem:
given: $U \subset \mathbb{R}^3$ and $U$ is an open set $$ V \in C^1(U, \mathbb{R}^3)$$ $$\omega_V(v_1,v_2):= det(V,v_1,v_2)$$
find $f \in C^1 (U, \mathbb{R})$ such that $$(V\cdot dx) \wedge \omega_V = f dx_1 \wedge dx_2 \wedge dx_3$$
My solution:
Since $(V\cdot dx) \wedge \omega_V $ is a 3-Form, the RHS is a Basis representation. So, in order to obtain $f$ I have to plug in the Basis vectors of $\mathbb{R}^3$ in $(V\cdot dx) \wedge \omega_V $ is this correct?
Now my solution is $$ f=\frac{1}{3}(V_1^2-V_2^2+V_3^2) $$ where $V= (V_1,V_2,V_3)^T$.
is this correct?