geometric measure theory – Normal and locally normal currents


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

  1. 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 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 real-valued functions $psi,$ with
$$
begin{aligned}
left(partialleft(mathbf{E}^{n} llcorner psiright)right)(phi) &=intleftlangle e_{1} wedge cdots wedge e_{n}, psi(x) d phi(x)rightrangle dleft(begin{array}{l}
n \
z
end{array}right) \
&=(-1)^{n-1} int psi(x) operatorname{div} xi(x) d mathfrak{L}^{n} x
end{aligned}
$$

  1. $M subset U$ oriented submanifold, then there is a corresponding $n$-current $(M)$ defined by
    $$
    (M)(omega)=int_{M}<omega(x), xi(x)>d H^{n}(x), omega in D^{n}(U)
    $$

Now, I want to calculate $partial (M)$! Is it $omega$ on $M$??