A supercurrent on $U$ is a continuous linear functional on $A_{c}^{p,q}(U)$ where the latter is a locally convex vector space in a similar way as in the classical case. We denote the space of such supercurrents by $D_{p,q}(U)$. As usual, we define the linear differential operators $d$, $d^{\prime}$ and $d^{\prime\prime}$ on $D(U):=\bigoplus_{p,q}D_{p,q}(U)$ by using $(-1)^{p+q+1}$ times the dual of the corresponding differential operator on $A_{c}^{p,q}(U)$. The sign is chosen in such a way that the canonical embedding $A^{p,q}(U)\rightarrow D_{r-p,r-q}(U)$ is compatible with the operators $d$, $d^{\prime}$ and $d^{\prime\prime}$. Here, $\alpha\in A^{p,q}(U)$ is mapped to $[\alpha]\in D_{r-p,r-q}(U)$ given by $[\alpha](\beta)=\int_{N_{\mathbb{R}}}\alpha\wedge\beta$ for any $\beta\in A_{c}^{r-p,r-q}(U)$.