Let $F : \Bbb R^2 → \Bbb R^2$ be given by If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$,
compute $F^{∗}(u \, du+v \, dv)$.
$$F(x,y) = (x^2 +y^2,xy).$$
I am confused so much. I know the proporties related to the question. But I cannot do. I am studying the exam. ı need to learn such types of the questions. Thank you
The definition of $F^*$ is $F^*(\omega)_p (v_1\ldots v_n) = \omega_{F(p)}(dF(v_1)\ldots dF(v_n)$. Which, in local coordinates means $F^*(\omega)_p = \omega_I F d(y^I F)$ if $\omega = \omega_I dy^I$.
In this case we have $\omega = udu + vdv$, which means $F^*(\omega) = u(x^2 + y^2,xy)d(u((x^2 + y^2,xy))) + v(x^2+y^2,xy) d(v((x^2+y^2,xy)))$ but $u,v$ are your standard local coordinates on $\mathbb{R}^2$, which means that $F^*(\omega) = x^2 d(x^2+y^2) + xy d(xy) = x^2 2x dx + x^2 2y dy + xy (ydx+xdy)= 2x^3 dx + 2y dy + xy^2dx + x^2y dy = (2x^3 + xy^2)dx + 3x^2y dy$.
(I'm not totally sure, though. Someone had better check this and tell me if anything is completely wrong, since I've never really had to do any exercises on this and just had to work "generally".)