There is a differential operator $d^{\prime}:A^{p,q}(U)\rightarrow A^{p+1,q}(U)$ given by

$$d^{\prime}\alpha:=\sum_{|I|=p,|J|=q}\sum_{i=1}^{r}\frac{\partial\alpha_{IJ}}{\partial x_{i}}d^{\prime}x_{i}\wedge d^{\prime}x_{I}\wedge{d^{\prime\prime}x_{J}}.$$

This does not depend on the choice of coordinates as $d^{\prime}=d\otimes{\rm id}$ on $A^{p,q}(U)=A^{p}(U,{\mathbb{R}})\otimes_{\mathbb{Z}}\Lambda^{q}M$ is an intrinsic characterization using the classical differential $d$ on the space $A^{p}(U,{\mathbb{R}})$ of real smooth $p$-forms. Similarly, we define a differential operator $d^{\prime\prime}:A^{p,q}(U)\rightarrow A^{p,q+1}(U)$ by

$$d^{\prime\prime}\alpha:=\sum_{|I|=p,|J|=q}\sum_{j=1}^{r}\frac{\partial\alpha_{IJ}}{\partial x_{j}}{d^{\prime\prime}x_{j}}\wedge d^{\prime}x_{I}\wedge{d^{\prime\prime}x_{J}}.$$

By linearity, we extend these differential operators to $A(U)$. Moreover, we set $d:=d^{\prime}+d^{\prime\prime}$.