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Let me elaborate more. Suppose that $(U, x^1, ... , x^n)$ and $(V, y^1, ... , y^n)$ are two charts on $M$ with a nonempty overlap $U \cap V$. Then a $C^{\infty}$ 1-form $\omega$ on $U \cap V$ has two different local expressions:

$\omega = \sum a_j dx^j = \sum b_i dy^i$. Find a formula for $a_j$ in terms of $b_i$.

Any help would be great. A complete solution is ideal. I'm studying for an upcoming exam in my Differential manifolds class. Thanks!

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Use the chain rule for differentials: $d x^j = \sum \frac{\partial x^j}{\partial y^i} d y^i$. –  Zhen Lin Apr 14 at 18:38

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up vote 2 down vote accepted

Your exam must have passed a while ago now, but hopefully this is still helpful.

In general, how do you find the local expression for a $1$-form $\omega$? The coefficient on a basis element $dx^j$ in the local coframe is whatever you get when you apply $\omega$ to the corresponding basis element $\partial/\partial x^j$ in the local frame. So: \begin{align*}a_j&=\omega\left(\frac{\partial}{\partial x^j}\right)\\ &=\left(\sum_{i} b_i dy^i\right)\left(\frac{\partial}{\partial x^j}\right)\\ &=\sum_{i} b_i\left(dy^i\frac{\partial}{\partial x^j}\right)\\ &=\sum_ib_i\frac{\partial y^i}{\partial x^j} \end{align*}

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