The question reads:
Consider the torus $T^2 = S^1 \times S^1$, parametrized in $\mathbb{R}^3$ by$$\Gamma(\theta, \varphi) = (\cos(\theta)(R+r\cos(\varphi)), \sin(\theta)(R+r\cos(\varphi)), r\sin(\varphi))$$ where $0<r<R$. Compute the differential $I_{*}$, mapping $T_{p}(T^2)$ to $T_{\Gamma(p)}(\mathbb{R}^3)$, in the coordinates $\theta, \varphi$ on the torus and the standard co-ordinate $x,y,z\in\mathbb{R}^3$. Compute the induced map $I^*$ on $T^*_{p}$ and the map on the second exterior power $\Lambda^2T^*_{p}$.
Now I believe I understand the first part, just find the derivative of the map and then parametrize it into $x,y,z$, but I am totally confused about the second part. Am I supposed to compute the map on the dual space of the tangent vectors at $p$? If anyone could help it would be greatly appreciated. Thank you.
Right, I think by the induced map you do it on the dual space, like this:
$I^*:T^*(T^2)\to T^*(\mathbb{R}^3), \alpha\mapsto\omega$
where $\Gamma^*\omega=\alpha$. That is $\alpha_p(\xi)=\omega_{\Gamma(p)}(\Gamma_*\xi)$ for $\xi\in T_pT^2$.
Now for $\wedge^2 T_p^*(T^2)\to \wedge^2 T_{\Gamma(p)}^*\mathbb{R}^3$ you use $I^*$ on the generators: $\alpha\wedge\beta\mapsto I^*\alpha\wedge I^*\beta$.