I am trying to learn differential forms. I have read some scripts about differential forms and now I am trying to solve some problems.
So the problem is:
given $f: \mathbb{R}^2 \to \mathbb{R}^3, f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T$
Now I have to calculate the basis representation of
$f^*(dx_1 \wedge dx_2 \wedge dx_3) $ where $f^*$ is a pullback of $f$
I guess I have to calculate the pullback?
According to the definition $$(f^* \alpha)(v_1,\ldots, v_k)=\sum_{I_k} f^* \alpha (e_{I_1},\ldots, e_{I_k}) dx_{I_1} \wedge \ldots \wedge dx_{I_3}$$
where $\alpha $ is a $k$-Form and $I_k :=\{ \{I_1,..,I_k \} | 1 \leq I_1 \leq I_2 \leq \ldots \leq I_k \leq N \}$
since $N = 2$ we get
$\sum_{I_k} f^* \alpha (e_{I_1},\ldots, e_{I_k}) dx_{I_1} \wedge \ldots \wedge dx_{I_3} = f^* \alpha (e_{1},e_{2}) dx_{1} \wedge dx_{2} + f^* \alpha (e_{1},e_{3}) dx_{1} \wedge dx_{3} + f^* \alpha (e_{2},e_{2}) dx_{2} \wedge dx_{3}$
My $\alpha$ is $dx_1 \wedge dx_2 \wedge dx_3$
but I don't know how to continue. Can someone help me with this ? Or give me an example how to calculate it.
To carry out the computation even though we know the answer will be zero:
We have $df_1 = dx_1 + dx_2$, $df_2 = -dx_1$ and $df_3 = -dx_2$. Thus \begin{align*} f^*(dx_1 \wedge dx_2 \wedge dx_3) &= df_1 \wedge df_2 \wedge df_3 \\ &= (dx_1 + dx_2) \wedge (-dx_1) \wedge (-dx_2) \\&= dx_1 \wedge dx_1 \wedge dx_2 + dx_2 \wedge dx_1 \wedge dx_2 \\ &= 0.\end{align*}