Transition formula for 1-forms on Loring W. Tu

$\bf{17.3.}$ Transition formula for $1$-forms
Suppose that $(U,x^1,\ldots,x^n)$ and $(V,y^1,\ldots,y^n)$ are two charts on $M$ with nonempty overlap $U\cap V$. Then a $C^\infty \;1$-form $\omega$ on $U\cap V$ has two different local expressions: $$\omega=\sum a_jdx^j=\sum b_idy^i.$$ Find a formula for $a_j$ in terms of $b_i$.

I try to solve this question but I could not. I am working for my exam. please help me.

Hint: If $y^i=y^i(x^1,\dots,x^n)$, what is $dy^i$ in terms of $dx^j$?

Well, by chain rule $d y^i=\sum_j \frac{\partial y^i}{\partial x^j}dx^j$, plugging this to $\omega=\sum b_idx^i=\sum_i b_i\sum_j \frac{\partial y^i}{\partial x^j}dx^j=\sum a_jd x^j$, so we obtain $$ a_j=\sum_i b_i \frac{\partial y^i}{\partial x^j}, $$ which is the transition formula for 1-forms.

I suggest you to read the appropriate section of the book again. At local patch if you have two coordinates, then you must assume $$x\rightarrow y, \phi_{y}\phi_{x}^{-1}(M_{x})$$ is a $C^{1}$ transit function which is bijective. So locally since this is a map $$\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$$you can write its Jacobian matrix. Therefore you can go from one way to the other using chain rule. Denote the transition function by $\Phi$. It is now trivial to verify that $$dx_{i}=d[p_{i}\Phi(y_{1}\cdots y_{n})]=\sum \frac{\partial [p_{i}\Phi]}{\partial y_{j}}dy_{j}$$ So if you have the original function to be $$\sum a_{i}(x_{1},\cdots x_{n}) dx_{i}$$ then you just need to expand $$\sum a_{i}(\Phi^{-1}(y_{1}\cdots y_{n}))\sum \frac{\partial [p_{i}\Phi]}{\partial y_{j}}dy_{j}$$ and re-arrange it to be $$\sum b_{j}dy_{j}$$