This is an attempt to solve problem 4.1 in Tu's Introduction to manifolds.
Given the 1-form $\omega = z dx - dz$ and the vector field $X = y \partial_x + x \partial_y$. Compute $\omega(X)$ and the exterior derivative $d\omega$.
\begin{align} \omega(X) & = \sum a_I b_I \\ & = zy - x \\ \end{align} where $I = \{1, 2, 3\}$ is the multi-index and $a_I,b_I$ the function coeficients in $\omega, X$ respectively. Also \begin{align} d\omega & = (\sum da_I\wedge dx_I) \\ & = \sum_I(\sum_j \partial_x^j a_I dx^j)\wedge dx_I \\ & = d(z) \wedge dx + d(-1)\wedge dz \\ & = (\partial_x z dx + \partial_y z dy + \partial_z z dz) \wedge dx + 0 \wedge dz\\ & = 1 dz \wedge dx \\ &= dz \wedge dx \end{align}
Is this correct?
Your expression for $\omega(X)$ is incorrect. Note that $a_2 = 0$, $b_2 = x$, $a_3 = -1$ and $b_3 = 0$, so $\omega(X) = a_1b_1 = zy$.
Your expression for $d\omega$ is correct.