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In Ana Cannas' Lectures on Symplectic Geometry (page $9$), the tautological $1$-form on the cotangent bundle $T^*Q$ is defined on local coordinates $(U,x_1,...,x_n,\xi_1,...,\xi_n)$ by:

$$\alpha\big|_U:=\sum_{i=1}^n\xi_i dx_i$$

She goes on to prove that for other coordinates $(U',x_1',...,x_n',\xi_1',...,\xi_n')$ the local form $\sum_{i=1}^n\xi_i' dx_i'$ coincides with $\alpha\big|_{U}$ at $U\cap U'$, concluding that $\alpha$ can be defined globally.

She uses $2$ arguments: first, that $dx'_j=\sum_{i=1}^n\left(\frac{\partial x'_j}{\partial x_i}\right)dx_i$ and second that $\xi_j'=\sum_{i=1}^n\xi_j\left(\frac{\partial x_j}{\partial x_i'}\right)$. I understand the first one perfectly, but I have no idea where she took the second from.

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The statement should be $$\xi_{j}'=\sum_{i=1}^{n}\frac{\partial x_{i}}{\partial x_{j}'}\xi_{i}.$$ Note that $\xi_{j}'$ is a fiberwise linear function on $T^{*}M$, and such functions are generated by $\xi_{1},\ldots,\xi_{n}$. So we may write $$\xi_{j}'=\sum_{i=1}^{n}f_{i}\xi_{i}$$ To find the coefficients $f_{i}$, we note that $$\xi_{i}(dx_{j})=\delta_{i,j}$$ and similarly $$\xi_{i}'(dx_{j}')=\delta_{i,j}.$$ Hence, $$f_{i}=\xi_{j}'(dx_{i})=\xi_{j}'\left(\sum_{k=1}^{n}\frac{\partial x_{i}}{\partial x_{k}'}dx_{k}'\right)=\frac{\partial x_{i}}{\partial x_{j}'}.$$

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