I am reading the Geometric Measure Theory book by H. Federer and I have some questions about currents:

Assuming $T \in \mathscr{D}_{m}(U),$ we call $T$ locally normal if and only if $T$ is representable by integration and either $\partial T$ is representable by integration or $m=0 .$ Furthermore call $T$ normal if and only if $T$ is locally normal and $\operatorname{spt}T$ is compact.
Now, I want to construct a locally normal current that is not normal! Does the following current work:
$\mathbf{E}^{n} \llcorner \psi$ corresponding to all weakly differentiable realvalued functions $\psi,$ with $$ \begin{aligned} \left[\partial\left(\mathbf{E}^{n} \llcorner \psi\right)\right](\phi) &=\int\left\langle e_{1} \wedge \cdots \wedge e_{n}, \psi(x) d \phi(x)\right\rangle d\left[\begin{array}{l} n \\ z \end{array}\right) \\ &=(1)^{n1} \int \psi(x) \operatorname{div} \xi(x) d \mathfrak{L}^{n} x \end{aligned} $$

$M \subset U$ oriented submanifold, then there is a corresponding $n$current $[M]$ defined by $$ [M](\omega)=\int_{M}\langle\omega(x), \xi(x)\rangle d H^{n}(x), \omega \in D^{n}(U) $$
Now, I want to calculate $\partial [M]$! Is it $\omega$ on $M$??